Communications in Mathematical Physics

, Volume 154, Issue 3, pp 603–611 | Cite as

A one-dimensional model with phase transition

  • Michel Mendès France
  • Gérald Tenenbaum


Two repellent particles are bound to occupy two among thek n +1 adjacent sites 0=x 0 (n) <x 1 (n) <...<x kn (n) =1, sayx q (n) ,x q+1 (n) . Define the Hamiltonian ℋ q (n) =−ln(x q+1 (n) −x q (n) ) and the partition function
We discuss the behaviour of the function
$$F(\beta ) = \mathop {\lim \sup }\limits_{n \to + \infty } \frac{{\ln Z(\beta , n)}}{{\ln k_n }},$$
closely related to the free energy. We prove that the smallest real zero ofF(β) is equal to the fractal dimension of the system and that this number, when less than one, is a critical value whereF is not analytic.


Neural Network Phase Transition Free Energy Statistical Physic Complex System 
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    Ramsey, N.F.: Thermodynamics and statistical mechanics at negative absolute temperatures. Phys. Rev.103, 20 (1956)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Michel Mendès France
    • 1
  • Gérald Tenenbaum
    • 2
  1. 1.Département de MathématiquesUniversité Bordeaux ITalence CedexFrance
  2. 2.Département de MathématiquesUniversité Nancy 1Vandoeuvre CedexFrance

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