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  • 1
    Book
    Book
    Mathematical statistical mechanics (1972)
    Princeton, N.J. :Princeton Univer. Press,
    Details
    Title: Mathematical statistical mechanics
    Author: Thompson, Colin J.
    Publisher: Princeton, N.J. :Princeton Univer. Press,
    Year of publication: 1972
    Pages: 278 S.
    Type of Medium: Book
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  • 2
    Electronic Resource
    Electronic Resource
    On limits to perception and pattern recognition (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 681-686 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459655
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Ising model in the high density limit (1974)
    Springer
    Communications in mathematical physics 36 (1974), S. 255-262 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01646198
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01608757
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Upper bounds for Ising model correlation functions (1971)
    Springer
    Communications in mathematical physics 24 (1971), S. 61-66 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01907034
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  • 6
    Electronic Resource
    Electronic Resource
    The high density limit for lattice spin models (1978)
    Springer
    Communications in mathematical physics 58 (1978), S. 131-138 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609416
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  • 7
    Electronic Resource
    Electronic Resource
    The classical limit ofn-vector spin models (1973)
    Springer
    Communications in mathematical physics 33 (1973), S. 53-60 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01645606
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Asymptotic properties of sequences of iterates of nonlinear transformations (1982)
    Springer
    Journal of statistical physics 27 (1982), S. 183-200 
    Details
    ISSN: 1572-9613
    Keywords: Maps on an interval ; iteration ; conjugacy ; universal ; fixed point function ; functional equation ; doubling transformation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract By considering functions defined on the unit interval with a single zero minimum and a single unit maximum we are led to a version of the doubling or universal transformation. The fixed point functions of this doubling transformation have certain invariance properties under conjugacy. These invariance properties lead to a widening of the concept of universality to power law conjugacy classes in which the Feigenbaum divergence parameter δ is a function only of the product of the powers with which iterating functions approach unity at the maximum and zero at the minimum. We also construct an effective method for computing the divergence parameter from iterates, and derivatives of iterates, generated by the appropriate fixed point function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01011746
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  • 9
    Electronic Resource
    Electronic Resource
    On mean field equations for spin glasses (1982)
    Springer
    Journal of statistical physics 27 (1982), S. 457-472 
    Details
    ISSN: 1572-9613
    Keywords: Cayley tree ; iteration ; fixed point ; spin glass ; Gaussian distribution ; local mean-field theory ; SK equations ; TAP equations.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01011086
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  • 10
    Electronic Resource
    Electronic Resource
    The spherical limit forn-vector correlations (1977)
    Springer
    Journal of statistical physics 17 (1977), S. 189-196 
    Details
    ISSN: 1572-9613
    Keywords: Spin correlations ; n-vector model ; spherical model ; n → ∞ limit
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We investigate then→ ∞ limit of then-vector model single-spin and pairspin correlation functions. In this limit we show that the correlation functions become those of the corresponding spherical model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01040101
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    Paper (German National Licenses)
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