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  • 1
    Electronic Resource
    Electronic Resource
    On the definition of entropy production, via examples (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 1528-1554 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which "the density" is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem and it is illustrated via a number of examples. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533195
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  • 2
    Electronic Resource
    Electronic Resource
    The interaction potential of the stationary measure of a high-noise spinflip process (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 3030-3038 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A continuous time spinflip process, started from an arbitrary state, is proven to yield a Gibbs state after any time t(approximately-greater-than)0 if the noise is high enough. This is done by approximating the process by discrete time processes. The relative energies of those Gibbs states are constructed explicitly via a high-noise cluster expansion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530061
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  • 3
    Electronic Resource
    Electronic Resource
    When is an interacting particle system ergodic? (1993)
    Springer
    Communications in mathematical physics 151 (1993), S. 447-466 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider stochastic flip dynamics for an infinite number of Ising spins on the lattice ℤ d . We find a sequence of constructive criteria for the system to be exponentially ergodic. The main idea is to approximate the continuous time process with discrete time processes (itsEuler polygon) and to use an improved version of previous results [MS] about constructive ergodicity of discrete time processes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02097021
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  • 4
    Electronic Resource
    Electronic Resource
    Percolation, Path Large Deviations and Weakly Gibbs States (2000)
    Springer
    Communications in mathematical physics 209 (2000), S. 517-545 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the Pirogov--Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths--Pearce--Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with relative energies well-defined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program: to give a Gibbsian description to non-Gibbsian states. A disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050029
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  • 5
    Electronic Resource
    Electronic Resource
    From PCA's to equilibrium systems and back (1989)
    Springer
    Communications in mathematical physics 125 (1989), S. 71-79 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Stationary measures for probabilistic cellular automata (PCA's) ind dimensions give rise to space-time histories whose statistics may naturally be described by Gibbs states ind+1 dimensions for an interaction energy ℋ obtained from the PCA. In this note we study the converse question: Do all Gibbs states for this ℋ correspond to statistical space-time histories for the PCA? Our main result states that the answer is yes, at least for translation invariant or periodic Gibbs states. Thus ergodicity questions for PCA's can, at least partially, be formulated as questions of uniqueness of Gibbs states.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01217769
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  • 6
    Electronic Resource
    Electronic Resource
    Relative Energies for Non-Gibbsian States (1997)
    Springer
    Communications in mathematical physics 189 (1997), S. 277-286 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We investigate the suggestion of R.L. Dobrushin, that for many examples of non-Gibbsian measures, a well-defined interaction potential can still be derived. Such an interaction cannot be absolutely summable, uniformly in all configurations. Rather there will exist a typical set of configurations on which the interaction decays sufficiently fast. We show how to construct an interaction, using quite elementary methods, for example of the restriction of the Ising pure phases to one layer of the lattice. We sketch a general setup, showing that there is an intimate relation between the problem treated here and the decay of correlations in certain disordered systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050201
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  • 7
    Electronic Resource
    Electronic Resource
    Ergodicity of probabilistic cellular automata: A constructive criterion (1991)
    Springer
    Communications in mathematical physics 135 (1991), S. 233-251 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give a sequence of criteria (of increasing complexity) for the exponential ergodicity of discrete time interacting particle systems. Each criterion involves estimating the dependence on initial conditions of the process on finite space-time volumes. It generalizes and improves the existing single site condition and is the analog of the Dobrushin-ShlosmanC v condition in equilibrium statistical mechanics. Our “dynamic” criterion may also be used to prove the uniqueness of Gibbs state in situations where theC v condition fails. As a converse we prove that if there is a certain form of convergence to the stationary measure faster thann −d , wheren is the time andd is the dimension of the lattice, then our condition holds for some space-time volumes and hence the convergence must be exponentially fast.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02098042
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  • 8
    Electronic Resource
    Electronic Resource
    Transformations of Gibbs measures (1998)
    Springer
    Probability theory and related fields 112 (1998), S. 121-147 
    Details
    ISSN: 1432-2064
    Keywords: Mathematics Subject Classification (1991): 60K35 ; 60G60 ; 82B20 ; 82B05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. We study local transformations of Gibbs measures. We establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies. General results are illustrated by simple examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050185
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  • 9
    Electronic Resource
    Electronic Resource
    Freezing transition in the Ising model without internal contours (1999)
    Springer
    Probability theory and related fields 115 (1999), S. 479-503 
    Details
    ISSN: 1432-2064
    Keywords: Mathematical Subject Classification (1991): 60G, 82B20, 82B31, 82B26
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. We consider the low temperature Ising model in a uniform magnetic field h 〉 0 with minus boundary conditions and conditioned on having no internal contours. This simple contour model defines a non-Gibbsian spin state. For large enough magnetic fields (h 〉: h c ) this state is concentrated on the single spin configuration of all spins up. For smaller values (h≤h c ), the spin state is non-trivial. At the critical point h c ≠ 0 the magnetization jumps discontinuously. Freezing provides also an example of a translation invariant weakly Gibbsian state which is not almost Gibbsian.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050246
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  • 10
    Electronic Resource
    Electronic Resource
    Positivity of Entropy Production (2000)
    Springer
    Journal of statistical physics 101 (2000), S. 3-15 
    Details
    ISSN: 1572-9613
    Keywords: entropy production ; nonequilibrium Gibbs states
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium, as it was defined in refs. 7 and 8. Non-zero entropy production is closely linked with violation of the detailed balance condition. This connection is rigorously obtained for spinflip dynamics. We remark that the positivity of entropy production depends on the choice of time-reversal transformation, hence on the choice of the dynamical variables in the system of interest.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026434726635
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