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  • Springer  (109)
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  • 1
    Electronic Resource
    Electronic Resource
    On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: a Very Sensitive Tool (2000)
    Springer
    Celestial mechanics and dynamical astronomy 78 (2000), S. 167-195 
    Details
    ISSN: 1572-9478
    Keywords: chaos ; numerical tools ; Nekhoroshev ; Chirikov
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract It is already known (Froeschlé, Lega and Gonczi, 1997) that the Fast Lyapunov Indicator (FLI), that is the computation on a relatively short time of the largest Lyapunov indicator, allows to discriminate between ordered and weak chaotic motion. We have found that, under certain conditions, the FLI also discriminates between resonant and non-resonant orbits, not only for two-dimensional symplectic mappings but also for higher dimensional ones. Using this indicator, we present an example of the Arnold web detection for four and six-dimensional symplectic maps. We show that this method allows to detect the global transition of the system from an exponentially stable Nekhoroshev’s like regime to the diffusive Chirikov’s one.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011141018230
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  • 2
    Electronic Resource
    Electronic Resource
    Crossing of Various Cantori (2000)
    Springer
    Celestial mechanics and dynamical astronomy 78 (2000), S. 197-210 
    Details
    ISSN: 1572-9478
    Keywords: chaos ; KAM tori ; cantori ; asymptotic curves
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We find the form of cantori surrounding an island of stable motion in the standard map for various values of the nonlinearity parameter K near the value K = 5 (much larger than the critical value K cr = 0.971635...). The asymptotic curves of unstable periodic orbits inside the cantorus cross it after a certain time and then escape to the large chaotic sea. For K = 5 the crossing time (in appropriate units) is t = 1 and the escape time is t = 2. For K = 4.998 the crossing time is t = 7 and the escape time t = 23000. This delay of escape is due to the existence of higher order cantori, with very small gaps. We found that, as K increases the noble torus [2,4,1,1,..] is destroyed before the destruction of the higher order tori [2,4,1,1,1,1,2,1,...] and [2,4,1,1,1,1,3,1,...]. Thus the torus with the simplest noble number is not the last KAM curve to be destroyed. Then we find that nearby orbits deviate considerably, but the average times spent near various resonance before escape are very similar.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011197104118
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  • 3
    Electronic Resource
    Electronic Resource
    Symplectic Tangent map for Planetary Motions (1999)
    Springer
    Celestial mechanics and dynamical astronomy 74 (1999), S. 59-67 
    Details
    ISSN: 1572-9478
    Keywords: planetary systems ; Liapunov exponents ; chaos ; symplectic integration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008312912468
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  • 4
    Electronic Resource
    Electronic Resource
    Chaos and secondary resonances in the mimas–tethys system (1999)
    Springer
    Celestial mechanics and dynamical astronomy 74 (1999), S. 111-146 
    Details
    ISSN: 1572-9478
    Keywords: Mimas–Tethys system ; chaos ; secondary resonances ; tides ; capture probability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We have investigated the role of the 200 yr period discovered by Vienne and Duriez (1992) on the tidal evolution of the Mimas–Tethys system through the 2:4 ii′ present resonance. Three terms are found to generate this period. We present a perturbed‐pendulum model in which these terms bring about a perturbation to the ideal ii′ resonance pendulum, which is in a direct ratio to the eccentricity e′ of Tethys. Although e′ is now very small, it is shown that this quantity could have been much greater in the past. We also show, thanks to this model, that these terms may have brought about a stochastic layer of noticeable width at the time of capture in the ii′ resonance, with the consequence that the possible values of the inclination i of Mimas before capture range from 0.4° to 0.6° (these uncertainties arise from the present uncertainties on e′). The role of each one of the three terms is examined in the appearance of chaos. A capture into the 1/1 secondary resonance (between the libration period of the primary ii′ resonance and the period of about 200 yr) is found possible. It means that the system could have experienced several captures in the primary resonance, instead of a single one, and that i could have been, with this assumption, much lower than 0.4°. A probability of capture into this secondary resonance as a function of the eccentricity of Tethys on encounter is derived, using Malhotra's method (Malhotra, 1990). Allan's values of i = 0.42° and e′ ≈ 0 (Allan, 1969) are therefore called into question, and taking e′ ≠ 0 is shown to be absolutely necessary if we want to understand the phenomena at work in the Mimas–Tethys system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008314007365
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  • 5
    Electronic Resource
    Electronic Resource
    Triple collisions in the one-dimensional three-body problem (2000)
    Springer
    Celestial mechanics and dynamical astronomy 76 (2000), S. 23-34 
    Details
    ISSN: 1572-9478
    Keywords: one dimensional three-body problem ; triple collision ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values. Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time, that is, orbits which are finite both in space and time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008397420958
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  • 6
    Electronic Resource
    Electronic Resource
    Binary and triple collisions causing instability in the free-fall three-body problem (2000)
    Springer
    Celestial mechanics and dynamical astronomy 76 (2000), S. 187-214 
    Details
    ISSN: 1572-9478
    Keywords: three-body problem ; triple collision ; binary collision ; escape ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Dominant factors for escape after the first triple-encounter are searched for in the three-body problem with zero initial velocities and equal masses. By a global numerical survey on the whole initial-value space, it is found that not only a triple-collision orbit but also a particular family of binary-collision orbits exist in the set of escape orbits. This observation is justified from various viewpoints. Binary-collision orbits experiencing close triple-encounter turn out to be close to isosceles orbits after the encounter and hence lead to escape. Except for a few cases, binary-collision orbits of near-isosceles slingshot also escape.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008305928852
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  • 7
    Electronic Resource
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    Synthetic Proper Elements for Outer Main Belt Asteroids (2000)
    Springer
    Celestial mechanics and dynamical astronomy 78 (2000), S. 17-46 
    Details
    ISSN: 1572-9478
    Keywords: asteroids ; proper elements ; chaos ; families
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract For the orbits with low to moderate inclination and eccentricity, in the asteroid main belt, the analytically computed proper elements are accurate to a level very close to the best result achievable by any analytical theory. This fundamental limitation results from the infinite web of resonances and because of the occurrence of chaotic motions. Still, there are some regions of the belt in which these proper elements are of degraded accuracy, thus preventing a reliable definition of asteroid families and detailed studies of the dynamical structure. We have used a different method to compute asteroid proper elements, following the approach introduced in the LONGSTOP project to describe the secular dynamics of the major outer planets. By applying purely numerical techniques, we produced so-called ‘synthetic’ proper elements for a catalog of 10,256 asteroids with osculating semimajor axes between 2.5 and 4.0 AU. The procedure consisted of simultaneous integration of asteroid and planetary orbits for 2 Myr, with online filtering of the short-periodic perturbations. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series. For each asteroid we have also tested the accuracy and stability in time of the proper elements, and estimated the maximum Lyapunov Characteristic Exponent to monitor the chaotic behaviors. This provided information on the reliability of the data for each orbit, in particular allowing to select 1,852 cases for an extended integration (10 Myr) of the orbits showing instability. The results indicate that for more than half of the cases the proper elements have a time stability improved by more than a factor 3 with respect to the elements computed by the previous analytical theory. But of course there are also unstable cases for which the proper elements are less accurate and reliable, the extreme examples being 23 orbits exhibiting hyperbolic escape from the solar system. This form of escape from the asteroid belt could be responsible for a significant mass loss over the age of the solar system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011187405509
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  • 8
    Electronic Resource
    Electronic Resource
    Chaotic transitions in resonant asteroidal dynamics (1996)
    Springer
    Celestial mechanics and dynamical astronomy 64 (1996), S. 93-105 
    Details
    ISSN: 1572-9478
    Keywords: asteroids ; resonance ; chaos ; Kirkwood gaps
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The utilization of chaotic dynamics approaches allowed the identification of many modes of motion in resonant asteroidal dynamics. As these dynamical systems are not integrable, the motion modes are not separated and one orbit may transit from one mode to another. In some cases, as in the \31 resonance, these transitions may lead, in a relatively short time scale, to eccentricities so high that the asteroid may approach the Sun and be destroyed. In the \21 and \32 resonances these transitions are much slower and only indirect estimations of the time which is needed for a generic asteroid to leave the resonance are possible. It may reach hundreds of million years in the more robust regions of the \21 resonance and a time of the order of billions of years in those of the \32 resonance. These values are consistent with the observed depletion of the \21 resonance (only a few asteroids known while almost 60 asteroids are known in the \32 resonance).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051608
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  • 9
    Electronic Resource
    Electronic Resource
    The two-body problem with radiation pressure in a rotating reference frame (1995)
    Springer
    Celestial mechanics and dynamical astronomy 61 (1995), S. 287-313 
    Details
    ISSN: 1572-9478
    Keywords: Two-body problem ; radiation pressure ; horseshoes ; chaos ; three-body problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study a perturbed Newtonian two-body problem, in which the perturbation is due to a force field of constant magnitude but rotating direction. By considering this system as a perturbation of the non-rotating case a Melnikov-type analysis allows us to show the existence of horseshoes in the level sets of the Hamiltonian and the subsequent sensitive dependence on initial conditions and non-integrability. We discuss the consequences of these results for a particular planar restricted three-body problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051898
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  • 10
    Electronic Resource
    Electronic Resource
    Chaotic dynamics of planet‐encountering bodies (1998)
    Springer
    Celestial mechanics and dynamical astronomy 70 (1998), S. 181-200 
    Details
    ISSN: 1572-9478
    Keywords: chaos ; dynamics ; comets ; NEAs ; Lyapunov exponent
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of two families of minor inner solar system bodies that suffer frequent close encounters with the planets is analyzed. These families are: Jupiter family comets (JF comets) and Near Earth Asteroids (NEAs). The motion of these objects has been considered to be chaotic in a short time scale,and the close encounters are supposed to be the cause of the fast chaos. For a better understanding of the chaotic behavior we have computed Lyapunov Characteristic Exponents (LCEs) for all the observed members of both populations. LCEs are a quantitative measure of the exponential divergence of initially close orbits. We have observed that most members of the two families show a concentration of Lyapunov times (inverse of LCE) around 50–100yr. The concentration is more pronounced for JF comets than for NEAs, among which a lesser spread is observed for those that actually cross the Earth's orbit (mean perihelion distance q 〈 1.05 AU). It is also observed that a general correspondence exists between Lyapunov times and the time between consecutive encounters. A simple model is introduced to describe the basic characteristics of the dynamical evolution. This model considers an impulsive approach, where the particles evolve unperturbedly between encounters and suffer ‘kicks’ in semimajor axis at the encounters. It also reproduces successfully the short Lyapunov times observed in the numerical integrations and is able to estimate the dynamical lifetimes of comets during a stay in the Jupiter family in correspondence with previous estimates. It has been demonstrated with the model that the encounters with the largest effect on the exponential growth of the distance between initially nearby orbits are neither the infrequent deep encounters, nor the frequent and far ones; instead, the intermediate approaches have the most relevant contribution to the error growth. Such encounters are at a distance a few times the radius of the Hill's sphere of the planet (e.g. 3). An even simpler model allows us to get analytical estimates of the Lyapunov times in good agreement with the values coming from the model above and the numerical integrations. The predictability of the medium‐term evolution and the hazard posed to the Earth by those objects are analysed in the Discussion section.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008331422678
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  • 11
    Electronic Resource
    Electronic Resource
    Experimental Investigation of Stability Enhancement in Semiconductor Lasers with Optical Feedback (1998)
    Springer
    Optical review 5 (1998), S. 280-284 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; chaos ; relaxation oscillation ; linear stability analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The stability enhancement of laser output power for the change of external cavity position in semiconductor lasers with optical feedback is observed by experiment. The relaxation oscillation frequency which plays an important role in the dynamics of the nonlinear system is also investigated as a function of the external cavity length. The period of the stability enhancement along the position of the external cavity is exactly coincident with the length corresponding to the relaxation oscillation frequency of the solitary laser. The experimental results are compared with theoretical and excellent coincidence between the two is found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-998-0280-9
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  • 12
    Electronic Resource
    Electronic Resource
    Dynamics of Semiconductor Lasers with Optical Feedback from Photorefractive Phase Conjugate Mirror (1999)
    Springer
    Optical review 6 (1999), S. 359-364 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; photorefractive optical feedback ; instability ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of semiconductor lasers with photorefractive phase conjugate optical feedback are experimentally studied. Photorefractive fringes considered here are rather static compared with time fluctuations of laser output power. Therefore, it is expected that a semiconductor laser with photorefractive feedback shows similar dynamics to those with conventional optical feedback. We examine relaxation oscillation and external cavity modes of laser output power in the presence of photorefractive phase conjugate feedback. It is proved that the dynamics of photorefractive phase conjugate feedback are fundamentally the same as those of conventional optical feedback.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-999-0359-y
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  • 13
    Electronic Resource
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    Low-Frequency Fluctuation and Frequency-Locking in Semiconductor Lasers with Long External Cavity Feedback (1999)
    Springer
    Optical review 6 (1999), S. 399-401 
    Details
    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; low frequency fluctuation ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The properties of low-frequency fluctuations in semiconductor lasers with optical feedback from a long external cavity are experimentally studied. Frequency-locking of the laser light output to the injection current modulation is observed when the modulation frequency approaches the external cavity mode. The modulation frequency for the successful frequency-locking is always less than the external cavity mode frequency and the locking domains as a function of the modulation amplitude is asymmetric with respect to the frequency detuning.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10043-999-0399-3
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  • 14
    Electronic Resource
    Electronic Resource
    Intermittent behaviour of volcanic tremor at Mt. Etna (1996)
    Springer
    Pure and applied geophysics 147 (1996), S. 729-744 
    Details
    ISSN: 1420-9136
    Keywords: Volcanic tremor ; intermittency ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Harmonic tremor is widely studied and modelled in a very narrow frequency band (1–5 Hz) which represents the eigenfrequencies of a resonator assumed as the source of the phenomenon. Minimal effort was dedicated towards understanding its behaviour in larger temporal scales. Here we characterise the dynamic behaviour of volcanic tremor while evaluating the complete spectrum of the generalised dimension of the phase space. The starting time series constitutes the tremor amplitude picked every 10 minutes. The choice of this lag time is made on the basis of a qualitative analysis of the properties of the tremor. The results show intermittent behaviour of the dynamics which requires an 8-dimensional map to be completely described. An interesting result is that the maximum clustering of point density in phase space occurs in a monodimensional space which implies a periodicity sometimes observed experimentally. An appropriate predictive model needs more constraints on the nature of the eight variables involved in the process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01089699
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  • 15
    Electronic Resource
    Electronic Resource
    The Ising Model with Boundaries: Integrability and Functional Relations (1998)
    Springer
    Letters in mathematical physics 46 (1998), S. 207-218 
    Details
    ISSN: 1573-0530
    Keywords: Ising model ; boundary conditions ; Boltzmann weights.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this Letter, we analyse the boundary conditions of the planar Ising model and determine the boundary Boltzmann weights in terms of bulk Boltzmann weights. The commutativity of the transfer matrices and their functional relations are shown.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007523720496
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  • 16
    Electronic Resource
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    On the extremality of the disordered state for the Ising model on the Bethe lattice (1996)
    Springer
    Letters in mathematical physics 37 (1996), S. 137-143 
    Details
    ISSN: 1573-0530
    Keywords: 82B20 ; 82B26 ; 82B43 ; Bethe lattice ; FK representation ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(β c SG = 1/√k.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00416016
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  • 17
    Electronic Resource
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    Partition function zeros for aperiodic systems (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 285-297 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; Lee-Yang zeros ; edge singularities ; nonperiodic systems ; phase transitions ; gap labeling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02183349
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  • 18
    Electronic Resource
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    Some numerical results on the block spin transformation for the 2D Ising model at the critical point (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 731-757 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; renormalization group ; finite-size conditions ; critical point
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the block spin transformation for the 2D Ising model at the critical temperatureT c . We consider the model with the constraint that the total spin in each block is zero. An old argument by Cassandro and Gallavotti strongly supports the Gibbsianness of the transformed measure, provided that such model has a critical temperatureT′ c lower thanT c . After describing a possible rigorous approach to the problem, we present numerical evidence that indeedT′ c 〈T c and study the Dobrushin-Shlosman uniqueness condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02183686
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  • 19
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    Comparison and disjoint-occurrence inequalities for random-cluster models (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 1311-1324 
    Details
    ISSN: 1572-9613
    Keywords: Random-cluster model ; Ising model ; Potts model ; comparison inequality ; BK inequality ; FKG inequality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed elsewhere in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical pointp c (q) of the random-cluster model with cluster-weighting factorq (≥1) is strictly monotone inq. Our second result is a “BK inequality” for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.
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    URL: http://dx.doi.org/10.1007/BF02180133
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  • 20
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    Corrections to the critical temperature in 2D Ising systems with Kac potentials (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 1131-1138 
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    ISSN: 1572-9613
    Keywords: Kac potential ; Ising model ; critical fluctuations ; Euclidean field theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider ad=2 Ising system with a Kac potential whose mean-field critical temperature is 1. Calling γ〉0 the Kac parameter, we prove that there existsc *〉0 so that the true inverse critical temperature βcr(γ) 〉 1 +by 2 log γ-1, for anyb〈c * and γ correspondingly small. We also show that if γ→0 andb→c *, suitably, then the correlation functions (normalized and rescaled) converge to those of a non-Gaussian Euclidean field theory.
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    URL: http://dx.doi.org/10.1007/BF02183705
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  • 21
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    Anomalous biennial oscillations in a Fisher equation with a discretized verhulst term (1995)
    Springer
    Journal of statistical physics 81 (1995), S. 761-775 
    Details
    ISSN: 1572-9613
    Keywords: Logistic map ; diffusion ; Fisher equation ; chaos ; oscillations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179256
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  • 22
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    The computational complexity of generating random fractals (1996)
    Springer
    Journal of statistical physics 82 (1996), S. 1299-1326 
    Details
    ISSN: 1572-9613
    Keywords: Cluster algorithms ; computational complexity ; diffusion-limited aggregation ; Ising model ; Metropolis algorithm ; P-completeness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion-limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation, which can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02183384
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  • 23
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    Dynamic critical behavior of a Swendsen-Wang-Type algorithm for the Ashkin-Teller model (1996)
    Springer
    Journal of statistical physics 85 (1996), S. 297-361 
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    ISSN: 1572-9613
    Keywords: Ashkin-Teller model ; Ising model ; Potts model ; Monte Carlo ; dynamical critical behavior ; cluster algorithm ; Swedsen-Wang algorithm ; Li-Sokal bound ; critical slowing down ; autocorrelation time, fitting correlated data
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (τint.δ≥ const xC H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio τint.δ/C H appears to tend to infinity either as a logarithm or as a small power (0.05≲p≲0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.
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    Absence of renormalization group pathologies near the critical temperature. Two examples (1996)
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    Journal of statistical physics 85 (1996), S. 607-637 
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    ISSN: 1572-9613
    Keywords: Ising model ; renormalization group pathologies ; Dobrushin uniqueness theorem ; completely analytic potentials
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    Notes: Abstract We consider real-space renormalization group transformations for Ising-type systems which are formally defined by $$\exp \left[ { - H'(\sigma ')} \right] = \sum\limits_\sigma {T(\sigma ,\sigma ')} \exp \left[ { - H(\sigma )} \right]$$ whereT(σ, σ′) is a probability kernel, i.e., ∑σ′ T(σ,σ′) = 1 for every configuration σ. For each choice of the block spin configuration σ′, let σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional toT(σ, σ′) exp[−H(σ)]. We give a condition which is sufficient to imply that the renormalized HamiltonianH′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacingb = 2 on the square lattice with β 〈 1.36β c and the Kadanoff transformation with parameterp on the trian gular lattice in a subset of the β,p plane that includes values of β greater than β c .
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    Almost sure quasilocality fails for the random-cluster model on a tree (1996)
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    Journal of statistical physics 84 (1996), S. 1351-1361 
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    ISSN: 1572-9613
    Keywords: Random-cluster model ; quasilocality ; almost sure quasilocality ; tree ; Gibbs measure ; Ising model
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    Notes: Abstract We study the random-cluster model on a homogeneous tree, and show that the following three conditions are equivalent for a random-cluster measure: quasilocality, almost sure quasilocality, and the almost sure nonexistence of infinite clusters. As a consequence of this, we find that the plus measure for the Ising model on a tree at sufficiently low temperatures can be mapped, via a local stochastic transformation, into a measure which fails to be almost surely quasilocal.
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    Kolmogorov–Sinai Entropy, Lyapunov Exponents, and Mean Free Time in Billiard Systems (1997)
    Springer
    Journal of statistical physics 88 (1997), S. 807-824 
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    ISSN: 1572-9613
    Keywords: Billiards ; Kolmogorov–Sinai entropy ; Lyapunov exponents ; ergodic theory ; chaos ; numerical experiments
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    Notes: Abstract We perform new experiments on the Kolmogorov–Sinai entropy, Lyapunov exponents, and the mean free time in billiards. We study their dependence on the geometry of the scatterers made up of two interpenetrating square lattices, each one with circular scatterers with different radius. We find, in particular, that the above quantities are continuous functions of the ratio of the scatterer radius. However, it seems that their derivative is discontinuous around the radius ratio which separates the diffusive and nondiffusive types of geometries.
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    Ernst Ising—Physicist and Teacher (1997)
    Springer
    Journal of statistical physics 88 (1997), S. 991-995 
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    Keywords: History of statistical physics ; Ising model
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    Notes: Abstract The Ising model is one of the standard models in statistical physics. Since 1969 more than 13800 publications using this model have appeared. In 1997 Ernst Ising celebrated his 97th birthday. Some biographical notes and milestones of the development of the Ising model are given.
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    Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness (1997)
    Springer
    Journal of statistical physics 88 (1997), S. 795-805 
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    ISSN: 1572-9613
    Keywords: Cellular automata ; Ising model ; voting models ; single sin-flip dynamics ; computational complexity ; parallel computation ; P-completeness
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    Notes: Abstract We study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood. These are equivalent, for a restricted set of initial conditions, to nonzero probability transitions in single spin-flip dynamics of the Ising model at zero temperature. We show that in three or more dimensions these systems can simulate Boolean circuits of AND and OR gates, and are therefore P-complete. That is, predicting their state t time-steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer. Therefore, unless a widely believed conjecture in computer science is false, it is impossible even with parallel computation to predict majority-vote cellular automata, or zero-temperature single spin-flip Ising dynamics, qualitatively faster than by explicit simulation.
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    Finite-Volume Glauber Dynamics in a Small Magnetic Field (1998)
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    Journal of statistical physics 90 (1998), S. 1015-1035 
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    ISSN: 1572-9613
    Keywords: Ising model ; mixing conditions ; Basuev region ; boundary conditions ; Glauber dynamics ; exponential relaxation ; spectral gap
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    Notes: Abstract We consider Glauber dynamics on a finite cube in d-dimensional lattice (d≥2), which is associated with basic Ising model at temperature T=1/β≪1 under a magnetic field h 〉 0. We prove that if the “effective magnetic field” is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (β, h) is so near the phase transition line, (β, h); β c 〈 β&h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.
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    Chaotic Behavior in Shell Models and Shell Maps (1998)
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    Journal of statistical physics 93 (1998), S. 833-842 
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    ISSN: 1572-9613
    Keywords: Shell models ; turbulence ; chaos ; Lyapunov exponent ; conserved quantities ; shell maps
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    Notes: Abstract We study the chaotic behavior of the GOY shell model by measuring the variation of the maximal Lyapunov exponent with the parameter ε which determines the nature of the second invariant (the generalized “helicity” invariant). After a Hopf bifurcation, we observe a critical point at ε c ∼0.38704 above which the maximal Lyapunov exponent grows nearly linearly. For high values of ε the evolution becomes regular again, which can be explained by a simple analytic argument. A model with few shells shows two transitions. To simplify the model substantially we introduce a shell map which exhibits similar properties as the GOY model.
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    Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening (1999)
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    Journal of statistical physics 95 (1999), S. 867-902 
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    Keywords: kinetics of phase transitions ; domain coarsening ; asymptotic behavior ; self-similarity ; stability ; chaos
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    Notes: Abstract The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.
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    Cassie's Law and Concavity of Wall Tension with Respect to Disorder (2000)
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    Journal of statistical physics 98 (2000), S. 1115-1134 
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    Keywords: Ising model ; correlation inequalities ; surface tension ; disorder
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    Notes: Abstract For the semiinfinite Ising model with quenched boundary disorder, we prove concavity inequalities for the difference of wall tensions associated with the minus and plus phases. These inequalities generalize phenomenological equalitiesknown as Cassie's law.
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    The Onsager formula, the fisher-hartwig conjecture, and their influence on research into Toeplitz operators (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 575-584 
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    ISSN: 1572-9613
    Keywords: Ising model ; correlation functions ; spontaneous magnetization ; Toeplitz determinants ; Szegö limit theorems ; Toeplitz operators
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    Notes: Abstract This is not primarily a paper about applications of mathematics to statistical physics, but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory. The problem alluded to is the computation of the spontaneous magnetizationM o (T) of the two-dimensional Ising model with nearest-neighbor interactions, whose solution for temperaturesT below the Curie pointT c was given by the famous formula of Lars Onsager in 1948. The theory grown out of this formula is the edifice of Toeplitz determinants, matrices, and operators.
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    Relaxation times in a finite Ising system with random impurities (1995)
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    Journal of statistical physics 78 (1995), S. 893-916 
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    Keywords: Dynamic critical phenomena ; disordered spin systems ; Ising model ; finite size ; scaling
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    Notes: Abstract Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM 2(t). In addition, universal cumulant ratios are calculated to second order in ε1/4, where ε=4−d andd〈4 denotes the spatial dimension.
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    Instability of renormalization-group pathologies under decimation (1995)
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    Journal of statistical physics 79 (1995), S. 25-42 
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    Keywords: Renormalization group ; decimation ; non-Gibbsianness ; Ising model
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    Notes: Abstract We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.
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    Ergodicity, ensembles, irreversibility in Boltzmann and beyond (1995)
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    Journal of statistical physics 78 (1995), S. 1571-1589 
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    Keywords: Boltzmann ; ergodicity ; irreversibility ; Ruelle principle ; SRB measures ; chaos ; nonequilibrium
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    Notes: Abstract The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.
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    A remark on papers by pixton and oliveira: Genericity of symplectic diffeomorphisms ofS 2 with positive topological entropy (1995)
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    Journal of statistical physics 80 (1995), S. 481-485 
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    Keywords: Genericity ; chaos ; topological entropy
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    Notes: Abstract We prove the existence of an open and dense subset of mapsfεDiff ω ∞ (S2) which have positive topological entropy. It follows that these maps have infinitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem
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    sl(N) Onsager's algebra and integrability (1996)
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    Journal of statistical physics 82 (1996), S. 87-113 
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    Keywords: Onsager's algbera ; loop algebras ; Ising model ; integrability
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    Notes: Abstract We define ansl(N) analog of Onsager's algebra through a finite set of relations that generalize the Dolan-Grady defining relations for the original Onsager's algebra. This infinite-dimensional Lie algebra is shown to be isomorphic to a fixed-point subalgebra ofsl(N) loop algebra with respect to a certain involution. As the consequence of the generalized Dolan-Grady relations a Hamiltonian linear in the generators ofsl(N) Onsager's algebra is shown to posses an infinite number of mutually commuting integrals of motion.
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    Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice (1996)
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    Journal of statistical physics 82 (1996), S. 1099-1112 
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    Keywords: Ising model with long-range interactions ; Sierpiński-gasket lattice ; correlation functions ; chaos
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    Notes: Abstract A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ 1=Q,l=1,2,..., on the Sierpiński-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM k-spin correlation functions, whereM k=3 k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.
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    Some new results on the two-dimensional kinetic Ising model in the phase coexistence region (1996)
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    Journal of statistical physics 84 (1996), S. 655-696 
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    Keywords: Ising model ; Glauber dynamics ; relaxation time
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    Notes: Abstract We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL with open boundary conditions, in the absence of an external field and at large inverse temperature β. We prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap. As a consequence we are able to show that there exists a new time scalet even, much smaller than the global relaxation timet rel, such that, with large probability, any initial configuration first relaxes to one of the two “phases” in a time scale of ordert even and only after a time scale of the order oft rel does it reach the final equilibrium by jumping, via a large deviation, to the opposite phase. It also follows that, with large probability, the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time.
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    Ergodic theorems arising in correlation dimension estimation (1996)
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    Journal of statistical physics 85 (1996), S. 25-40 
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    Keywords: Glivenko-Cantelli theorem ; fractal ; almost sure convergence ; moment estimators ; least square estimators ; dynamical systems ; chaos
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    Notes: Abstract The Grassberger-Procaccia (GP) empirical spatial correlation integral, which plays an important role in dimension estimation, is the proportion of pairs of points in a segment of an orbit of lengthn, of a dynamical system defined on a metric space, which are no more than a distancer apart. It is used as an estimator of the GP spatial correlation integral, which is the probability that two points sampled independently from an invariant measure of the system are no more than a distancer apart. It has recently been proven, for the case of an ergodic dynamical system defined on a separable metric spaceythat the GP empirical correlation integral converges a.s. to the GP correlation integral at continuity points of the latter asn→∞. It is shown here that for ergodic systems defined on ℜd with the “max” metric the convergence is uniform inr. Further, a simplified proof based on weak convergence arguments of the result in separable spaces is given. Finally, the Glivenko-Cantelli theorem is used to obtain ergodic theorems for both the moment estimators and least square estimators of correlation dimension.
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    Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model (1997)
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    Journal of statistical physics 86 (1997), S. 149-164 
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    Keywords: Ising model ; Gibbs measures ; large-deviation principle
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    Notes: Abstract In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube (Λ(B/h) ash↘0 for variousB. We find a critical balance at a valueB 0 ofB.
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    Thermodynamic formalism and localization in Lorentz gases and hopping models (1997)
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    Journal of statistical physics 87 (1997), S. 1253-1271 
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    Keywords: Lorentz lattice gases ; chaos ; thermodynamic formalism ; random walks ; localization transition
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    Notes: Abstract The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure ψ(β). The inverse-temperature-like variable β allows one to scan the structure of the probability distributin in the dynamic phase space. This formalism is applied here to a lorentz lattice gas. where a particle moving on a lattice of sizeL d collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at β=1. The low- and high-β branches correspond to localization of trajectories on respectively the “most chaotic” (highest density) region and the “most deterministic” (lowest density) region, i.e. ψ(β) is completely controlled by rare fluctuations in the distribution of scatterers on the lattice. and it dose not carry and information on the global structure of the static disorder. As β approaches unity from either side, a localization-delocalization transition leads to a state where trajectories are extended and carry information on transprot properties. At finiteL the narrow region around β=1 where the trajectories are extended scales as (InL)−2. where α depends on the sign of 1−β, ifd〉1, and as (L InL)−1 ifd=1. This result appears to be general for diffusive systems with static disorder, such as random walks in random environments or for the continuous Lorentz gas. Other models of random walks on disordered lattices, showing the same phenomenon, are discussed.
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    Metastability in the Two-Dimensional Ising Model with Free Boundary Conditions (1998)
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    Journal of statistical physics 90 (1998), S. 211-226 
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    Keywords: Ising model ; stochastic dynamics ; metastability ; nucleation
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    Notes: Abstract We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit T → 0, h → 0 and via Monte Carlo simulations at fixed values of T and h and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.
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    Improved Peierls Argument for High-Dimensional Ising Models (1998)
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    Journal of statistical physics 90 (1998), S. 1051-1059 
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    Keywords: Ising model ; Peierls contour ; low-temperature expansion ; high dimension
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    Notes: Abstract We consider the low-temperature expansion for the Ising model on $$\mathbb{Z}^d ,d \geqslant 2$$ , with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than Cd(log d)−1, which is the correct order in d.
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    Finite Range of Large Perturbations in Hamiltonian Dynamics (1998)
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    Journal of statistical physics 92 (1998), S. 909-972 
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    Keywords: Perturbation theory ; Hamiltonian dynamics ; wave–particle interaction: transport properties ; chaos ; plasma turbulence
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    Notes: Abstract The dynamics defined by the Hamiltonian $$H = p^2 /2 + A\sum\nolimits_{m = - M}^M {\cos (q - mt + \varphi m)}$$ , where the φ m are fixed random phases, is investigated for large values of A, and for $$M \gg A^{2/3}$$ . For a given P * and for $$\Delta \upsilon \geqslant A^{2/3}$$ , this Hamiltonian is transformed through a rigorous perturbative treatment into a Hamiltonian where the sum of all the nonresonant terms, having a Q dependence of the kind cos(kQ − nt + φ m) with $$|n/k - P^ * | 〉 \Delta \upsilon$$ , is a random variable whose r.m.s. with respect to the φ m is exponentially small in the parameter $$\varepsilon = A/\Delta \upsilon ^{3/2}$$ . Using this result, a rationale is provided showing that the statistical properties of the dynamics defined by H, and of the reduced dynamics including at each time t only the terms in H such that $$|m - p(t)| \leqslant \alpha A^{2/3}$$ , can be made arbitrarily close by increasing α. For practical purposes α close to 5 is enough, as confirmed numerically. The reduced dynamics being nondeterministic, it is thus analytically shown, without using the random-phase approximation, that the statistical properties of a chaotic Hamiltonian dynamics can be made arbitrarily close to that of a stochastic dynamics. An appropriate rescaling of momentum and time shows that the statistical properties of the dynamics defined by H can be considered as independent of A, on a finite time interval, for A large. The way these results could generalize to a wider class of Hamiltonians is indicated.
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    The O(n) Loop Model on the 3–12 Lattice (1998)
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    Journal of statistical physics 92 (1998), S. 1203-1208 
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    Keywords: Loop model ; criticality ; universality ; 3–12 lattice ; self-avoiding walk ; connective constant ; Ising model
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    Notes: Abstract The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3–12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n = 0 this gives the recently found exact value μ = 1.711041... for the connective constant of self-avoiding walks on the 3–12 lattice. The exact critical points are recovered for the Ising model on the 3–12 lattice and the dual asanoha lattice at n = 1.
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    URL: http://dx.doi.org/10.1023/A:1023065215233
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    On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice (1998)
    Springer
    Journal of statistical physics 93 (1998), S. 33-78 
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    ISSN: 1572-9613
    Keywords: Random external field ; Ising model ; Gibbs states ; ground states ; Bethe lattice ; residual entropy ; dipole configurations ; Griffiths singularities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξ x = ±α and the temperature T≥0. We give a description of a part of the phase diagram of this model in the T−α plane, where we are able to construct limiting Gibbs states and ground states. By comparison with the model with a constant external field we show that for all realizations ξ = {ξ x = ±α} of the external random field: (i) the Gibbs state is unique for T 〉 T c (k ≥ 2 and any α) or for α 〉 3 (k = 2 and any T); (ii) the ±-phases coexist in the domain {T 〈 T c, α ≤ H F(T)}, where T c is the critical temperature and H F(T) is the critical external field in the ferromagnetic Ising model on the Bethe lattice with a constant external field. Then we prove that for almost all ξ: (iii) the ±-phases coexist in a larger domain {T 〈 T c, α ≤H F(T) + ε(T)}, where ε(T)〉0; and (iv) the Gibbs state is unique for 3≥α≥2 at any T. We show that the residual entropy at T = 0 is positive for 3≥α≥2, and we give a constructive description of ground states, by so-called dipole configurations.
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    An Equilibrium Lattice Model of Wetting on Rough Substrates (1999)
    Springer
    Journal of statistical physics 94 (1999), S. 299-320 
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    ISSN: 1572-9613
    Keywords: wetting ; surface tension ; rough surfaces ; Wenzel's law ; semi-infinite systems ; Ising model ; cluster expansions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider a semi-infinite 3-dimensional Ising system with a rough wall to describe the effect of the roughness r of the substrate on wetting. We show that the difference of wall free energies Δτ(r)=τ AW(r)−τ BW(r) of the two phases behaves like Δτ(r)∼rΔτ(1), where r=1 characterizes a purely flat surface, confirming at low enough temperature and small roughness the validity of Wenzel's law, cos θ(r)≈r cos θ(1), which relates the contact angle θ of a sessile droplet to the roughness of the substrate
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    The Ising Model on a Quenched Ensemble of c=−5 Gravity Graphs (1999)
    Springer
    Journal of statistical physics 94 (1999), S. 321-345 
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    ISSN: 1572-9613
    Keywords: dynamical triangulations ; quenched disorder ; Ising model
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    Topics: Physics
    Notes: Abstract We study with Monte Carlo methods an ensemble of c=−5 gravity graphs, generated by coupling a conformal field theory with central charge c=−5 to two-dimensional quantum gravity. We measure the fractal properties of the ensemble, such as the string susceptibility exponent γ s and the intrinsic fractal dimension d H. We find γ s=−1.5(1) and d H=3.36(4), in reasonable agreement with theoretical predictions. In addition, we study the critical behavior of an Ising model on a quenched ensemble of the c=−5 graphs and show that it agrees, within numerical accuracy, with theoretical predictions for the critical behavior of an Ising model coupled dynamically to two-dimensional quantum gravity, with a total central charge of the matter sector c=−5.
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    Statistical Entropy of a Lattice-Gas Model: Multiparticle Correlation Expansion (1999)
    Springer
    Journal of statistical physics 96 (1999), S. 135-167 
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    ISSN: 1572-9613
    Keywords: statistical entropy ; multiparticle correlations ; cumulant expansion ; lattice gases ; Ising model
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    Topics: Physics
    Notes: Abstract A formula expressing the statistical entropy of a lattice-gas model as a multiparticle correlation expansion is derived in the grand-canonical and in the canonical ensembles. The differences from the analogous expansion in the continuum case are elucidated. The Ising model in one dimension is discussed as a case study.
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    Renormalization-Group Transformations Under Strong Mixing Conditions: Gibbsianness and Convergence of Renormalized Interactions (1999)
    Springer
    Journal of statistical physics 97 (1999), S. 831-915 
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    ISSN: 1572-9613
    Keywords: renormalization group ; Gibbsianness ; finite-size conditions ; complete analyticity ; strong mixing ; equivalence of ensembles ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite-range lattice gases, when suitable strong mixing conditions are satisfied. Using a block decimation procedure, cluster expansion, and detailed comparison between statistical ensembles, we are able to prove Gibbsianness and convergence to a trivial (i.e., Gaussian and product) fixed point. Our results apply to the 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.
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    Universality and Conformal Invariance for the Ising Model in Domains with Boundary (2000)
    Springer
    Journal of statistical physics 98 (2000), S. 131-244 
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    ISSN: 1572-9613
    Keywords: Ising model ; boundary conditions ; renormalization group ; free boson ; conformal invariance ; critical phenomena
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    Topics: Physics
    Notes: Abstract The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of nonformal invariance and universality are established numerically.
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    Monte Carlo Transition Dynamics and Variance Reduction (2000)
    Springer
    Journal of statistical physics 98 (2000), S. 321-345 
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    ISSN: 1572-9613
    Keywords: statistical mechanics ; variance reduction ; Monte Carlo algorithms ; Metropolis algorithm ; statistical estimators ; Ising model ; histogram methods ; transition probabilities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract For Metropolis Monte Carlo simulations in statistical physics, efficient, easy- to-implement, and unbiased statistical estimators of thermodynamic properties are based on the transition dynamics. Using an Ising model example, we demonstrate (problem-specific) variance reductions compared to conventional histogram estimators. A proof of variance reduction in a microstate limit is presented.
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    A Comparison Between Broad Histogram and Multicanonical Methods (2000)
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    Journal of statistical physics 99 (2000), S. 691-705 
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    ISSN: 1572-9613
    Keywords: Monte Carlo methods ; Ising model ; computational physics
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    Notes: Abstract We discuss the conceptual differences between the broad histogram (BHM) and reweighting methods in general, and particularly the so-called multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depend only on the good statistics taken inside each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be adopted. As a numerical test, we have used both BHM and MUCA in order to obtain the spectral energy degeneracy of the Ising model in 4×4×4 and 32×32 lattices, for which exact results are known. We discuss why BHM gives more accurate results than MUCA, even using the same Markovian sequence of states. In addition, such an advantage increases for larger systems.
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    A new dromion solution of the Davey-Stewartson equation (1996)
    Springer
    Il nuovo cimento della Società Italiana di Fisica 18 (1996), S. 765-770 
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    ISSN: 0392-6737
    Keywords: Solitons ; chaos ; Solitons ; BGK modes
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    Topics: Physics
    Notes: Summary A new dromion solution is obtained for a (2+1)-dimensional integrable model: the Davey-Stewartson equation. Some interesting questions which emerge in the procedure of getting the solution are also discussed.
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    URL: http://dx.doi.org/10.1007/BF02457363
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    Relaxation Oscillation Enhancement and Coherence Collapse in Semiconductor Lasers with Optical Feedback (1999)
    Springer
    Optical review 6 (1999), S. 365-371 
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    ISSN: 1349-9432
    Keywords: semiconductor laser ; optical feedback ; chaos ; relaxation oscillation ; coherence collapse
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    Topics: Physics
    Notes: Abstract The dynamic characteristics of a semiconductor laser with optical feedback are strongly dependent on the injection current and the reflectivity and position of the external feedback reflector. We investigated the relaxation oscillation enhancement and coherence collapse state of the laser oscillation based on the laser rate equations. It is well known that laser output power jumps with increase of the injection current due to external mode transition. But here for the first we time demonstrate the existence of a chaotic scenario within successive laser power jumps. The results calculated by numerical simulations based on the rate equations are compared with those of the experiments and good coincidence between them is found.
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    URL: http://dx.doi.org/10.1007/s10043-999-0365-0
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    Parametric open-plus-closed-loop control of chaos in continuous dynamical systems (1997)
    Springer
    Acta mechanica solida Sinica 10 (1997), S. 316-321 
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    ISSN: 0894-9166
    Keywords: continuous dynamical system ; chaos ; parametric open-plus-closed-loop control ; the Lorenz model
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract This paper presents a parametric open-plus-closed-loop control approach to controlling chaos in continuous dynamical systems. As an example, chaos in the Lorenz model is controlled to demonstrate its application. Finally, the relations between the parametric open-plus-closed-loop control and the former control methods, such as the open-plus-closed-loop control and the parametric entrainment control, are discussed.
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    Global bifurcation and chaos in a van der Pol-Duffing-Mathiue system with a single-well potential oscillator (1997)
    Springer
    Acta mechanica solida Sinica 10 (1997), S. 262-275 
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    ISSN: 0894-9166
    Keywords: global bifurcation ; chaos ; nonlinear vibration ; basin ; fractal
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract The global bifurcation and chaos are investigated in this paper for a van der Pol-Duffing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The autonomous system corresponding to the system under discussion is analytically studied to draw all global bifurcation diagrams in every parameter space. These diagrams are called basic bifurcation ones. Then fixing parameter in every space and taking the parametrically excited amplitude as a bifurcation parameter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of numerical methods. The results are sufficient to show that the system has distinct dynamic behavior. Finally, the properties of the basins of attraction are observed and the appearance of fractal basin boundaries heralding the onset of a loss of structural integrity is noted in order to consider how to control the extent and the rate of the erosion in the next paper.
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    On hydrodynamic instabilities, chaos and phase transition (1996)
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    Acta mechanica Sinica 12 (1996), S. 1-14 
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    ISSN: 1614-3116
    Keywords: ellipticity ; hydrodynamic instabilities ; chaos ; phase transition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Ellipticity as the underlying mechanism for instabilities of physical systems is highlighted in the study of model nonlinear evolution equations with dissipation and the study of phase transition in Van der Waals fluid. Interesting results include spiky solutions, chaotic behavior in the context of partial differential equations, as well as the nucleation process due to ellipticity in phase transition.
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    Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator (1995)
    Springer
    Acta mechanica Sinica 11 (1995), S. 357-372 
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    ISSN: 1614-3116
    Keywords: semi-analytical and semi-numerical method ; global bifurcations ; chaos ; van del Pol-Duffing-Mathieu system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods.
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    Chaos of liquid surface waves in a vessel under vertical excitation with slowly modulated amplitude (1997)
    Springer
    Acta mechanica Sinica 13 (1997), S. 106-112 
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    ISSN: 1614-3116
    Keywords: faraday experiment ; chaos ; resonant interaction
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Free surface waves in a cylinder of liquid under vertical excitation with slowly modulated amplitude are investigated in the current paper. It is shown by both theoretical analysis and numerical simulation that chaos may occur even for a single mode with modulation which can be used to explain Gollub and Meyer's experiment. The implied resonant mechanism accounting for this phenomenon is further elucidated.
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    Experimental investigation on the chaotic phenomena in the wake of a natural thermal convection flow (2000)
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    Acta mechanica Sinica 16 (2000), S. 121-127 
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    ISSN: 1614-3116
    Keywords: thermal convection flow ; transition ; chaos ; electron beam fluorescence technique
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract Chaotic phenomena in the wake of thermal convection flow fields above a heating flat plate were investigated experimentally. A newly developed electron beam fluorescence technique (EBF) was used to simultaneously measure density fluctuation at 7 points in a cross section above the plate. Correlation dimensions, intermittence coefficients, Fourier spectrum have been obtained for different Grashof numbers. Spatial distribution of correlation dimensions are presented. The experimental result shows that there is a certain relationship between the density fluctuation and theGr number. And time-spacial characteristic of chaos evolution is also given.
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    Steady state motions of shallow arch under periodic force with 1∶2 internal resonance on the plane of physical parameters (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 625-635 
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    ISSN: 1573-2754
    Keywords: shallow arch ; internal resonance ; steady state motion ; bifurcation ; chaos
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1∶2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to the types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end.
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    The study on the chaotic motion of a nonlinear dynamic system (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 830-836 
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    ISSN: 1573-2754
    Keywords: chaos ; Melnikov method ; Poincaré map ; phase portrait ; time-displacement diagram ; O343.5
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract In this paper, the system of the forced vibration $$\ddot T - \lambda _1 T + \lambda _2 T^2 + \lambda _3 T^3 = \varepsilon \left( {g\cos \omega t - \varepsilon '\dot T} \right)$$ is discussed, which contains square and cubic items. The critical condition that the system enters chaotic states is given by the Melnikov method. By Poincaré map, phase portrait and time-displacement history diagram, whether the chaos occurs is determined.
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    On a Physically Realistic Fast Dynamo (1998)
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    Studia geophysica et geodaetica 42 (1998), S. 335-342 
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    ISSN: 1573-1626
    Keywords: fast dynamo ; rotating convection ; chaos ; MHD
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Geosciences , Physics
    Notes: Abstract As a step towards a physically realistic model of a fast dynamo, we study numerically a kinematic dynamo driven by convection in a rapidly rotating cylindrical annulus. Convection maintains the quasi-geostrophic balance whilst developing more complicated time-dependence as the Rayleigh number is increased. We incorporate the effects of Ekman suction and investigate dynamo action resulting from a chaotic flow obtained in this manner. We examine the growth rate as a function of magnetic Prandtl number Pm, which is proportional to the magnetic Reynolds number. Even for the largest value of Pm considered, a clearly identifiable asymptotic behaviour is not established. Nevertheless the available evidence strongly suggests a fast dynamo process.
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    Control of the Lorenz chaos by the exact linearization (1998)
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    Applied mathematics and mechanics 19 (1998), S. 67-73 
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    ISSN: 1573-2754
    Keywords: Reyleigh number ; Lorenz system ; chaos
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    Notes: Abstract Controlling chaos in the Lorenz system with a controllable Rayleigh number is investigated by the state space exact linearization method. Based on proving the exact linearizability, the nonlinear feedback is utilized to design the transformation changing the original chaotic system into a linear controllable one so that the control is realized. Numerical examples of control are presented.
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    Anomalous dynamics response of nonlinear elastic bar (2000)
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    Applied mathematics and mechanics 21 (2000), S. 1008-1015 
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    ISSN: 1573-2754
    Keywords: subharmonic bifurcation ; heteroclinic orbit ; chaos ; Melnikov function ; O34
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract The dynamics behaviour of tension bar with periodic tension velocity was presented. Melnikov method was used to study the dynamic system. The results show that material nonlinear may result in anomalous dynamics response. The subharmonic bifurcation and chaos may occur in the determined system when the tension velocity exceeds the critical value.
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    The double mode model of the chaotic motion for a large deflection plate (1999)
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    Applied mathematics and mechanics 20 (1999), S. 360-364 
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    ISSN: 1573-2754
    Keywords: buckled plate ; chaos ; Poincaré section
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Mathematics , Physics
    Notes: Abstract The primary aim of this paper is to study the chaotic motion of a large deflection plate. Considered here is a buckled plate, which is simply supported and subjected to a lateral harmonic excitation. At first, the partial differential equation governing the transverse vibration of the plate is derived. Then, by means of the Galerkin approach, the partial differential equation is simplified into a set of two ordinary differential equations. It is proved that the double mode model is identical with the single mode model. The Melnikov method is used to give the approximate excitation thresholds for the occurrence of the chaotic vibration. Finally numerical computation is carried out.
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    Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish (1997)
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    Journal of computational neuroscience 4 (1997), S. 335-347 
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    ISSN: 1573-6873
    Keywords: electroreceptor ; catfish ; chaos ; unstable orbit ; periodic orbit ; sensory oscillator ; noise ; random process
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Medicine , Physics
    Notes: Abstract We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.
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    Analysis of seismic dynamical systems (1998)
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    Journal of seismology 2 (1998), S. 159-171 
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    ISSN: 1573-157X
    Keywords: chaos ; dynamical systems ; Hurst exponent ; stochasticprocess ; time series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract A procedure is presented for the analysis of complex stationary time series for which the Fourier power spectra reveals broadband noise or broadened pulses. We first determine the Hurst exponent from which we may know whether the time series under study is mainly random or if the data points present correlations. If the data are correlated, a chaotic analysis will reveal whether they may be interpreted as a low dimensional nonlinear system (defined by a low correlation dimension and a finite and positive Kolmogorov entropy and largest positive Lyapunov exponent) or as a stochastic process. We have studied three kind of temporal series: inter-event time series of infrasonic pulses recorded at Stromboli volcano, and, S-coda waves and microseisms, that have been recorded at the eastern Pyrenees. Results show that microseisms and Coda waves can be modeled as a low dimensional deterministic system, Correlation dimensions 2.3, 3.2, respectively. At the contrary infrasonic has resulted stochastic. This chaotic character can be attributed to the medium properties. Coda waves with scattering through a fractal distribution of scatters or to multiple reflection inside resonators (for example sedimentary basins) and microseisms as a propagation of wave guide of variable cross section which have the same temporal characteristics as a nonlinear forced oscillator.
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    Source characterisation of low frequency events at Stromboli and Vulcano Islands (Isole Eolie Italy) (1999)
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    Journal of seismology 3 (1999), S. 393-408 
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    ISSN: 1573-157X
    Keywords: cepstrum ; chaos ; correlation dimension ; nonlinear oscillators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract The discussion about the source of low frequency events generally wrecorded on active volcanoes is still open and needs deeper understanding of the phenomena involved in their generation. Most of the models view such a phenomenon as a source effect (oscillation of volcanic fluids in conduits or cracks), although a different explanation as a path or site effect exists. In the present paper we analysed 26 seismic signals recorded at Vulcano and 60 at Stromboli in order to put some constraints on the functional shape of the recorded signals. They evidence all the characteristics of the low frequency events. The spectral analysis reveals sharp peaks in the range 0.5–4 Hz, while the cepstra suggest that the signals are composed by a two-sine kind function. This suggestion is confirmed by the two-dimensional projections of the reconstructed phase space. The correlation dimension of the attractor is very close to 3 for both the volcanoes confirming the existence of a toroidal structure in the phase space. This implies that we can suggest as a model for the low frequency events source a physical mechanism very similar to a Duffing oscillator or any other quasiperiodic one. In particular in the cases of the analysed volcanoes we recognise two independent oscillations.
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    URL: http://dx.doi.org/10.1023/A:1009842411153
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    Chaotic dynamics of periodically forced spheroids in simple shear flow with potential application to particle separation (1995)
    Springer
    Rheologica acta 34 (1995), S. 504-511 
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    ISSN: 1435-1528
    Keywords: Periodically forced spheroids ; chaos ; particle separation
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    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract In this paper, we consider the technologically important problem of periodically forced spheroids in simple shear flow and demonstrate the existence of chaotic parametric regimes. The approach used by Strand (1989) (for the Strong Brownian limit) is inappropriate in the chaotic regimes corresponding to the weak Brownian limit. Our results also indicate a strong dependence of the solutions obtained on the aspect ratio of the spheroids. This strong dependence on the aspect ratio may be utilized to separate particles from a suspension of particles having different shapes but similar sizes.
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    URL: http://dx.doi.org/10.1007/BF00396563
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    Turbulence Under The Microscope (2000)
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    Journal of biological physics 26 (2000), S. 77-83 
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    ISSN: 1573-0689
    Keywords: Bacterial movement ; chaos ; cryptic growth cell ; proteusmirabilis ; turbulence
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    Topics: Biology , Physics
    Notes: Abstract Our objective was to observe a new form of turbulence caused bybiological effects – biological micro-turbulence and explore itsprocess and controlling factors. The methods used were proteusmirabilis CGCs micro-cultured to render the occurrence of the specific movement on micro-organic suspension and its controllingfactors were determined by comparison with the control trials.The results showed that turbulence under the microscope was generally in a mass but partially regular. It was also confirmedthat the turbulence under the microscope exhibited hollow effect,temperature-dependent switching on of occurrence and self-controlof suspension quantity. It is clarified that this new form ofturbulence is a spontaneous and self-control process, which providesan experimental model with controllable conditions for studies ofturbulence and a new way for researches on the mechanism andphysiological functions of the flow of body liquid.
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    URL: http://dx.doi.org/10.1023/A:1005293915802
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    A simple generalization of wisdom's perturbative method (1996)
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    Celestial mechanics and dynamical astronomy 64 (1996), S. 243-260 
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    ISSN: 1572-9478
    Keywords: Perturbative methods ; chaos ; resonance ; adiabatic invariant ; asteroids
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    Topics: Physics
    Notes: Abstract A simple generalization of Wisdom's perturbative method, as originally proposed by Wisdom (1985), is obtained. Any number of resonant cosines can be handled and the method can also accommodate more involved disturbing functions. Averaged trajectories are easily obtained by drawing level curves of the action. Here, the method is first tested for simple models of 3:1 and 2:1 resonant problems. Comparisons with numerical integration and surface-section curves show very good agreements.
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    URL: http://dx.doi.org/10.1007/BF00728350
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    Entrainment Versus Chaos in a Model for a Circadian Oscillator Driven by Light-Dark Cycles (2000)
    Springer
    Journal of statistical physics 101 (2000), S. 649-663 
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    ISSN: 1572-9613
    Keywords: limit cycle oscillations ; chaos ; periodic forcing ; entrainment ; model ; circadian rhythms
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    Topics: Physics
    Notes: Abstract Circadian rhythms occur in nearly all living organisms with a period close to 24 h. These rhythms constitute an important class of biological oscillators which present the characteristic of being naturally subjected to forcing by light-dark (LD) cycles. In order to investigate the conditions in which such a forcing might lead to chaos, we consider a model for a circadian limit cycle oscillator and assess its dynamic behavior when a light-sensitive parameter is periodically forced by LD cycles. We determine as a function of the forcing period and of the amplitude of the light-induced changes in the light-sensitive parameter the occurrence of various modes of dynamic behavior such as quasi-periodicity, entrainment, period-doubling and chaos. The type of oscillatory behavior markedly depends on the forcing waveform; thus the domain of entrainment grows at the expense of the domain of chaos as the forcing function progressively goes from a square wave to a sine wave. Also studied is the dependence of the phase of periodic or aperiodic oscillations on the amplitude of the light-induced changes in the control parameter. The results are discussed with respect to the main physiological role of circadian rhythms which is to allow organisms to adapt to their periodically varying environment by entrainment to the natural LD cycle.
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    URL: http://dx.doi.org/10.1023/A:1026410121183
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    Microscopic Chaos and Diffusion (2000)
    Springer
    Journal of statistical physics 101 (2000), S. 775-817 
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    ISSN: 1572-9613
    Keywords: chaos ; diffusion ; Ehrenfest wind-tree model ; Lorentz gas ; statistical mechanics ; periodic orbits ; Brownian motion ; billiards ; time series analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We investigate the connections between microscopic chaos, defined on a dynamical level and arising from collisions between molecules, and diffusion, characterized by a mean square displacement proportional to the time. We use a number of models involving a single particle moving in two dimensions and colliding with fixed scatterers. We find that a number of microscopically nonchaotic models exhibit diffusion, and that the standard methods of chaotic time series analysis are ill suited to the problem of distinguishing between chaotic and nonchaotic microscopic dynamics. However, we show that periodic orbits play an important role in our models, in that their different properties in our chaotic and nonchaotic models can be used to distinguish them at the level of time series analysis, and in systems with absorbing boundaries. Our findings are relevant to experiments aimed at verifying the existence of chaoticity and related dynamical properties on a microscopic level in diffusive systems.
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    URL: http://dx.doi.org/10.1023/A:1026477605331
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    Solvable models in statistical mechanics, from Onsager onward (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 7-16 
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    ISSN: 1572-9613
    Keywords: Statistical mechanics ; lattice models ; Ising model ; solvable models ; integrable systems ; Yang-Baxter relations
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    Topics: Physics
    Notes: Abstract There is now a whole field in mathematical physics concerned with solvable models in statistical mechanics, field theory, and related areas. We indicate the influence that Onsager's solution of the planar Ising model has had, and continues to have, on this field.
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    URL: http://dx.doi.org/10.1007/BF02183337
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    Ising metamagnet under staggered field (1995)
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    Journal of statistical physics 81 (1995), S. 837-842 
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    ISSN: 1572-9613
    Keywords: Ising model ; staggered field ; metastable state
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    Topics: Physics
    Notes: Abstract We report a Monte Carlo simulation of the layered Ising antiferromagnet under an external magnetic field. We show that under a staggered field, there occurs a phase transition from a metastable state which follows a Vogel-Fulcher law. For a staggered intrasublattice interaction a similar situation occurs.
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    URL: http://dx.doi.org/10.1007/BF02179260
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    Testing for scaling in natural forms and observables (1995)
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    Journal of statistical physics 81 (1995), S. 869-880 
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    ISSN: 1572-9613
    Keywords: Self-similarity ; self-affinity ; fractals ; scaling ; chaos
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    Topics: Physics
    Notes: Abstract The general procedure of calculating fractal dimensions or other exponents is based on estimating some quantity as a function of scale and on assessing whether or not this function is a power law. This power law manifests itself in a log (quantity) versus log (scale) plot as a linear region (scaling). It has thus become the practice to estimate dimensions by the slope of some linear region in those log-log plots. When we are dealing with exact fractals (the Koch curve, for example) there are no problems. When, however, we are working with natural forms or observables, problems begin to emerge. In such cases the scaling region is subjectively estimated and often is only the result of the generic property of the quantity to increase monotonically or decrease monotonically as the scale goes to zero irrespective of the geometry of the object. Here we discuss these issues and suggest a procedure to deal with them.
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    URL: http://dx.doi.org/10.1007/BF02179296
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    Constrained variational problem with applications to the Ising model (1996)
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    Journal of statistical physics 83 (1996), S. 867-905 
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    ISSN: 1572-9613
    Keywords: Ising model ; Wulff shape ; large deviations ; boundary effects
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    Topics: Physics
    Notes: Abstract We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (−)-boundary conditions under a positive external field, the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB 0 (T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a “squeezed Wulff shape”. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.
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    URL: http://dx.doi.org/10.1007/BF02179548
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    One-dimensional random-field ising model: Gibbs states and structure of ground states (1996)
    Springer
    Journal of statistical physics 84 (1996), S. 1077-1093 
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    ISSN: 1572-9613
    Keywords: Gibbs states ; ground states ; residual entropy ; random field ; Ising model
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    Topics: Physics
    Notes: Abstract We consider the random Gibbs field formalism for the ferromagnetic ID dichotomous random-field Ising model as the simplest example of a quenched disordered system. We prove that for nonzero temperatures the Gibb state is unique for any realization of the external field. Then we prove that asT→0, the Gibbs state converges to a limit, a ground state, for almost all realizations of the external field. The ground state turns out to be a probability measure concentrated on an infinite set of configurations, and we give a constructive description of this measure.
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    URL: http://dx.doi.org/10.1007/BF02174129
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    Phase transitions in phosphatidylcholine foam bilayers (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 147-160 
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    ISSN: 1572-9613
    Keywords: Foam bilayer ; phase transition ; Ising model ; mean-field ; binding energy
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    Topics: Physics
    Notes: Abstract Foam bilayers from individual and mixed phosphatidylcholines are experimentally studied at different temperatures. Occurrence of a chain-melting phase transition in the foam bilayers is detected by two independent parameters—the critical concentrationC c for formation of foam bilayer and the foam bilayer thickness. The data forC c are discussed on the basis of the hole-nucleation theory, which applies the Ising model to foam bilayers and uses the mean-field approximation for interpretation of their stability. This allows the determination of the binding energy of a phospholipid molecule in gel and liquid-crystalline foam bilayers. New possibilities to relate the microscopic and macroscopic characteristics of foam bilayers are demonstrated.
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    URL: http://dx.doi.org/10.1007/BF02183343
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    Stacking order and disorder in layered K3Me(CN)6 compounds studied by diffuse X-ray scattering: A realization of the ANNNI model? (1995)
    Springer
    Journal of statistical physics 78 (1995), S. 135-145 
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    ISSN: 1572-9613
    Keywords: Polytypism ; Ising model ; Order-disorder ; X-ray scattering
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    Topics: Physics
    Notes: Abstract Layered crystalline materials like K3Me(CN)6 with Me=Cr, Mn, Fe, Co may often exist in various polytypic forms, due to a variety of choices of layer stacking modes. For cases where the interlayer constellations can be limited to only two energetically almost equivalent ways, the buildup of the crystal may be described by a spin-1/2 Ising-like model. For the system presently being studied one can rationalize the layer stacking to a four-valued choice (i.e., a 1D 4-state Potts case), or use an Ising-like two-sublattice model. Previous diffraction studies of K3Me(CN)6 indicated that two long-range ordered structures prevailed, an orthohombic one named MDO1, with one double layer per repetition unit, and a monoclinic one, MDO2, with two double-layer units. Our studies reveal a more complex situation: The Fe material is for the most part of the MDO2 type. But in addition, in some crystal samples, a hitherto unobserved phase also appears, with six double-layer repetition units, in fact a hybrid of MDO1 and MDO2. The Co material is for the most part of the MDO2 type, but contains in addition a considerable contribution of stacking disorder, as evidenced by the presence of diffuse X-ray scattering lines. The lines do, however, contain distinct maxima, indicating the presence of several layer stacking modes with preference of two, three, four, five, and seven double-layer correlations. The findings can be qualitatively discussed in terms of the ANNNI model.
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    A discrete variational problem related to Ising droplets at low temperatures (1995)
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    Journal of statistical physics 80 (1995), S. 103-123 
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    ISSN: 1572-9613
    Keywords: Discrete variational problem ; Ising model ; droplets ; metastability
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    Topics: Physics
    Notes: Abstract We consider a variational problem on thed-dimensional latticeZ d which has applications in the study of the meatastable behavior of the stochastic Ising model. The problem, an isoperimetric one, is to find what is the smallest area a finite subset ofZ d can have restricted to three classes of subsets ofZ d . If ϕ is one of these subsets, we define its volume as the number of points in it and its area as the number of pairs of points inZ d which are neighbors and such that only one of them belongs to ϕ.
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    URL: http://dx.doi.org/10.1007/BF02178355
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    Quantum transport and classical dynamics in open billiards (1996)
    Springer
    Journal of statistical physics 83 (1996), S. 203-214 
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    ISSN: 1572-9613
    Keywords: Quantum transport ; open billiards ; chaos ; Ericson fluctuation ; path-length spectrum
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    Topics: Physics
    Notes: Abstract We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.
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    Chaos for the Sierpinski Carpet (1997)
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    Journal of statistical physics 88 (1997), S. 979-984 
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    ISSN: 1572-9613
    Keywords: Fractal ; chaos ; Hausdorff dimension ; thermodynamic formalism ; strong-mixing ; outer measure
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    Topics: Physics
    Notes: Abstract We study the chaotic behavior of the Sierpinski carpet. It is proved that this dynamical system has a chaotic set whose Hausdorff dimension equals that of the Sierpinski carpet.
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    URL: http://dx.doi.org/10.1023/B:JOSS.0000015182.90436.5b
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    Renormalization group at criticality and complete analyticity of constrained models: A numerical study (1997)
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    Journal of statistical physics 86 (1997), S. 1117-1151 
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    ISSN: 1572-9613
    Keywords: Majority rule ; renormalization group ; non-Gibbsianness ; finite-size conditions ; complete analyticity ; Ising model
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    Notes: Abstract We study the majority rule transformation applied to the Gibbs measure for the 2D Ising model at the critical point. The aim is to show that the renormalized Hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman uniqueness (DSU) finite-size condition for the “constrained models” corresponding to different configurations of the “image” system. It is known that DSU implies, in our 2D case, complete analyticity from which, as recently shown by Haller and Kennedy. Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite-volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed the DSU condition is verified for a large enough volumeV for all constrained models.
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    Irregular Fluctuations in Uncoupled Map Lattices (1998)
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    Journal of statistical physics 90 (1998), S. 749-765 
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    ISSN: 1572-9613
    Keywords: Uncoupled logistic maps ; chaos ; fluctuations ; theoretical models ; computer simulations
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    Topics: Physics
    Notes: Abstract Analytic approximations for the spatial average and its variance are derived for a system of N uncoupled chaotic logistic maps with growth parameter r = 4. The arising nontrivial closure problem is investigated with various techniques related to the classical moment problem. A Lyapunov-like linear stability analysis is presented for the transient as well as for the fluctuation regime.
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    On Criticality for Competing Influences of Boundary and External Field in the Ising Model (1998)
    Springer
    Journal of statistical physics 92 (1998), S. 35-45 
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    ISSN: 1572-9613
    Keywords: Competing influences ; Ising model ; Gibbs measures
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    Notes: Abstract We continue a study of Schonmann (1994), Schonmann and Shlosman (1996), and Greenwood and Sun (1997) regarding the competing influences of boundary conditions and external field for the Ising model. We find a critical point B 0 in the competing influences for low temperature in dimension d 2A7E; 2.
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    Critical Exponents of the Diluted Ising Model between Dimensions 2 and 4 (1998)
    Springer
    Journal of statistical physics 92 (1998), S. 785-808 
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    Keywords: Critical phenomena ; diluted spin systems ; Ising model ; renormalization group
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    Notes: Abstract Within the massive field-theoretic renormalization-group approach the expressions for the β and γ functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 ≤ d 〈 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.
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    Ising-Model, Block-Spin Distributions by the Markov Property Method (1998)
    Springer
    Journal of statistical physics 93 (1998), S. 573-582 
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    ISSN: 1572-9613
    Keywords: Scaling ; Ising model ; Markov property ; critical phenomena ; parallel computational procedures
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    Notes: Abstract The idea that near the critical point each block of spins behaves just like a single big spin is investigated. The case where a diamond-shaped block of spins is embedded in a (small) sea of spins is studied. Use is made of the Markov property method to make exact computations of the various spin moments needed to test this hypothesis. The residual fluctuation about the mean value of the block spin is seen to tend to a finite fraction of the length of the mean block-spin. This result is in line with previous studies which used different types of boundary conditions.
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    Note on Two Theorems in Nonequilibrium Statistical Mechanics (1999)
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    Journal of statistical physics 96 (1999), S. 1343-1349 
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    ISSN: 1572-9613
    Keywords: chaos ; fluctuation theorem ; large deviations ; chaotic hypothesis ; nonequilibrium statistical mechanics ; time reversal
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    Notes: Abstract An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.
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    On the Ising Model with Strongly Anisotropic External Field (1999)
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    Journal of statistical physics 97 (1999), S. 87-144 
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    ISSN: 1572-9613
    Keywords: Ising model ; anisotropic field ; phase diagram ; cluster expansion
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    Notes: Abstract In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.
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    Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus (2000)
    Springer
    Journal of statistical physics 98 (2000), S. 551-588 
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    ISSN: 1572-9613
    Keywords: Ising model ; universal amplitude ratios ; conformal field theory ; torus ; finite-size scaling ; corrections to scaling ; Monte Carlo ; Swendsen–Wang algorithm ; cluster algorithm
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ★=lim L→∞ ξ(L)/L and the first four magnetization moment ratios V 2n =〈 $$M$$ 2n 〉/〈 $$M$$ 2〉 n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n . We confirm these predictions by a high-precision Monte Carlo simulation.
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    URL: http://dx.doi.org/10.1023/A:1018611122166
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    Distribution of Energies for the Two-Dimensional Ising Model (2000)
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    Journal of statistical physics 98 (2000), S. 1063-1073 
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    ISSN: 1572-9613
    Keywords: polygon statistics ; Ising model ; exact solution
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    Notes: Abstract We calculate the number of polygons with fixed total length drawn on a square lattice with periodic boundary conditions. In addition, we study the statistics of polygons with the number of horizontal and vertical links fixed separately. The analysis is performed via a mapping to the Ising model with isotropic and anisotropic interactions. We deal with the case of finite lattice sizes as well as the thermodynamic limit.
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    Low-temperature series for renormalized operators: The ferromagnetic square-lattice Ising model (1995)
    Springer
    Journal of statistical physics 80 (1995), S. 1309-1326 
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    ISSN: 1572-9613
    Keywords: Renormalization group ; position-space renormalization-group transformations ; Ising model ; low-temperature expansions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A method for computing low-temperature series for renormalized operators in the two-dimensional Ising model is proposed. These series are applied to the study of the properties of the truncated renormalized Hamiltonians when we start at very low temperature and zero field. The truncated Hamiltonians for majority rule, Kadanoff transformation, and decimation for 2×2 blocks depend on the how we approach the first-order phase-transition line. The renormalization group transformations are multivalued and discontinuous at this first-order transition line when restricted to some finite-dimensional interaction space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179872
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  • 98
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    Electronic Resource
    The correlation functions near intermittency in a one-dimensional Piecewise parabolic map (1996)
    Springer
    Journal of statistical physics 83 (1996), S. 1199-1210 
    Details
    ISSN: 1572-9613
    Keywords: Weak intermittency ; chaos ; phase transition ; correlation function ; scaling function ; crossover behavior ; critical slowing down
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Piecewise parabolic maps constitute a family of maps in the fully developed chaotic state and depending on a parameter that can be smoothly tuned to a weakly intermittent situation. Approximate analytic expressions are derived for the corresponding correlation functions. These expressions produce power-law decay at intermittency and a crossover from power-law decay to exponential decay below intermittency. It is shown that the scaling functions and the exponent of the power law depend on the kind of the correlations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179558
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  • 99
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    Electronic Resource
    Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model (1996)
    Springer
    Journal of statistical physics 84 (1996), S. 85-118 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; Markov chain ; transfer matrix ; Friedrichs model ; saddle-point method ; scattering theory ; T-matrix
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We find the asymptotic decrease of correlations 〈σ A +y ,σ B 〉,y∈Z v +1, |y|→∞, in the Ising model at high temperatures. For the case when monomialsσ A andσ B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension ν. For even monomialsσ A ,σ B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) ν=1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary ν. Here we use besides the saddle-point method, some arguments from scattering theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179578
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  • 100
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    Electronic Resource
    Ising models on the lattice Sierpinski gasket (1996)
    Springer
    Journal of statistical physics 84 (1996), S. 295-307 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; lattice Sierpinski gasket ; Dobrushin-Shlosmann mixing condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179588
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