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  • 1
    Digitale Medien
    Digitale Medien
    The classical limit ofn-vector spin models (1973)
    Springer
    Communications in mathematical physics 33 (1973), S. 53-60 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract It is proved that the free energy of a system ofn-dimensional spins with Kac type potential is equal, in the infinite range zero strength limit, to the free energy of the corresponding Curie-Weiss system in which every spin interacts equally with every other spin.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01645606
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  • 2
    Digitale Medien
    Digitale Medien
    Ising model in the high density limit (1974)
    Springer
    Communications in mathematical physics 36 (1974), S. 255-262 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract It is proved that the free energy per spin in the thermodynamic limit of an Ising model on a lattice with coordination numberz approaches the classical Curie-Weiss free energy in the limitz→∞. The infinite spacial dimension limit of nearest neighbour lattice models is a special case of this result.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01646198
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  • 3
    Digitale Medien
    Digitale Medien
    The anisotropic Heisenberg model in the long-range interaction limit (1975)
    Springer
    Communications in mathematical physics 41 (1975), S. 191-201 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We consider the general spin quantum mechanical anisotropic Heisenberg model on av-dimensional lattice with potential γ v ϱ(γ|r|) of Kac type. In the limit γ→0+ (after the thermodynamic limit) it is shown that the free energy is equivalent to the corresponding Curie-Weiss or mean-field expression.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01608757
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  • 4
    Digitale Medien
    Digitale Medien
    The high density limit for lattice spin models (1978)
    Springer
    Communications in mathematical physics 58 (1978), S. 131-138 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract Then-vector, spherical and quantum spin models are considered on a regular lattice with co-ordination numberq. In the limitq→∞ it is proved algebraically that the free energies are given by the corresponding Curie-Weiss or mean-field expressions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01609416
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  • 5
    Digitale Medien
    Digitale Medien
    Upper bounds for Ising model correlation functions (1971)
    Springer
    Communications in mathematical physics 24 (1971), S. 61-66 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract A Griffiths correlation inequality for Ising ferromagnets is refined and is used to obtain improved upper bounds for critical temperatures. It is shown that, for non-negative external fields, the mean field magnetization is an upper bound for the magnetization of Ising ferromagnets.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01907034
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  • 6
    Digitale Medien
    Digitale Medien
    On limits to perception and pattern recognition (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 681-686 
    Details
    ISSN: 1522-9602
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Biologie , Mathematik
    Notizen: Abstract We propose certain general conditions that we believe are reasonable for any pattern recognition algorithm. We find that these conditions give rise to paradoxical identification. The algorithms are incapable of distinguishing composite patterns and must be able to distinguish patterns at an atomistic level.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02459655
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  • 7
    Digitale Medien
    Digitale Medien
    Absence of long-range order in one-dimensional spin systems (1981)
    Springer
    Journal of statistical physics 25 (1981), S. 669-678 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Ising model ; plane rotator model ; inequalities ; long-range order
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract For a one-dimensional Ising model with interaction energy $$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i〈 j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$ it is proved that there is no long-range order at any temperature when $$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01022361
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  • 8
    Digitale Medien
    Digitale Medien
    Local properties of an Ising model on a Cayley tree (1982)
    Springer
    Journal of statistical physics 27 (1982), S. 441-456 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Ising model ; Cayley tree ; phase transition ; iteration ; fixed point ; bifurcation ; ferromagnetic ; antiferromagnetic.
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract The Ising model on a Cayley tree displays a peculiar (continuous order) phase transition with zero long-range order at all finite temperatures. When one studies expection values of spins far removed from the surface (which contains a finite fraction of the total number of spins in the thermodynamic limit), however, one obtains the so-called Bethe approximation. Here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface. In the ferromagnetic case the local magnetization is a fixed point of the iterative transformation, while in the antiferromagnetic case the fixed point bifurcates to a two-cycle of the transformation (for low temperatures and fields) giving rise to local sublattice magnetizations. In both cases, local thermodynamical properties are obtained by integration.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01011085
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  • 9
    Digitale Medien
    Digitale Medien
    Modulated phases and the mean-field theory of magnetism with competing interactions (1988)
    Springer
    Journal of statistical physics 51 (1988), S. 519-536 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Mean-field theory ; variational principle ; modulated phase ; ANNNI model
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract The mean-field theory of an Ising magnet with infinitely weak, infinitely long-range potentials of arbitrary sign is presented in terms of a variational principle for the magnetization. Previous studies of the theory have revealed paramagnetic, ferromagnetic, and modulated phases. For a particular choice of potential, which is an obvious continuous version of the between-plane ANNNI model interaction, exact solutions of the stationary condition implied by the variational principle are obtained. This leads us to formulate a trial magnetization to well describe the modulated phase in general. To illustrate the utility of the trial magnetization, both analytic and numerical calculations are performed, which determine the wavenumber in certain portions of the modulated phase for the above-mentioned potential.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01028470
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  • 10
    Digitale Medien
    Digitale Medien
    On mean field equations for spin glasses (1982)
    Springer
    Journal of statistical physics 27 (1982), S. 457-472 
    Details
    ISSN: 1572-9613
    Schlagwort(e): Cayley tree ; iteration ; fixed point ; spin glass ; Gaussian distribution ; local mean-field theory ; SK equations ; TAP equations.
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Physik
    Notizen: Abstract In this paper we study rigorously the random Ising model on a Cayley tree in the limit of infinite coordination numberz → 8. An iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice. In this way we obtain local properties of the model in the (thermodynamic) limit of an infinite number of shells. When the coupling constants are independent Gaussian random variables the SK expressions emerge as stable fixed points of our scheme and provide a valid local mean-field theory of spin glasses in which negative local entropy (at low temperatures) while perfectly possible mathematically may still perhaps be physically undesirable. Finally we examine the TAP equations and show that if the average over bond disorder and the limitz → 8 are actually performed, one recovers our iterative scheme and hence the SK equations also in the thermodynamic limit.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01011086
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