Statistical Mechanics of Dynamical Systems with Topological Phase Transitions

X iv :c on dm at /0 51 12 31 v1 [ co nd -m at .s ta tm ec h] 9 N ov 2 00 5 STATISTICAL MECHANICS OF DYNAMICAL SYSTEMS WITH TOPOLOGICAL PHASE TRANSITIONS AJAY PATWARDHAN Physics Department, St Xavier’s college, Mumbai Visitor, Institute of Mathematical Sciences, Chennai ABSTRACT Dynamical system properties give rise to effects in Statistical mechanics. Topological index changes can be the basis for phase transitions. The Euler characteristic is a versatile topological invariant that can be evaluated for model systems. These recent developments in the foundations of Statistical Mechanics, that are giving new results, provide insight into the Statistical thermodynamics of small N systems; such as molecular and spin clusters. This paper uses model systems to give a basis for redefining partition functions in classical statistical mechanics. It includes the properties of dynamical systems namely , KAM Torii, singular points and chaotic regions. The equipotential surfaces and the Morse and Euler index for it are defined. The conditions for the topology change in configuration space, and its effect on the partition function and the ensemble average quantities is found. The justification for topological phase transitions and their thermodynamic interpretation are discussed. 1.INTRODUCTION Since the classic work of Boltzmann, Gibbs , Poincare , and Birkhoff the relation between the axiomatic foundation of Statistical Mechanics and the properties of dynamical systems have been investigated by many persons. The Ergodic hypothesis in phase space for small N number of particles has to incorporate the conclusions of the KolmogorovArnoldMoser theorem. For dynamical systems that are hyperbolic, the Kolmogorov entropy has a role to play. In systems with both integrability and chaos on subspaces of phase space, the axioms for ergodicity and canonical ensemble partition functions need a fresh approach. For systems with symmetry and symmetry breaking, the ensemble averages are redefined. The singularities of the Hamiltonian vector field give rise to topological indices on phase space. Their role in the foundations of statistical mechanics is recently being explored by Cohen and others. References [1 to 16]. The work of Ruelle, Sinai , Leibowitz, Ford, Gallavotti, Cohen and others in making the framework of statistical mechanics consistent with the developments in classical and quantum mechanics of non linear dynamical systems has led to the resolution of some of the old questions and opened new possibilities for a more general Statistical mechanics. To obtain a complete theory inclusive of all the effects is still a distant goal. In an earlier arxiv eprint the author had given modifications to the partition functions by introducing the Kolmogorov entropy, the Casimir invariants, and the Euler number for micro partitions. In this paper the topological phase transitions introduced by Cohen and others are developed into a framework for Statistical Mechanics of Dynamical systems.