Statistical Mechanics of Dynamical Systems With Topological Phase Transitions.pdf

5 0 STATISTICAL MECHANICS OF DYNAMICAL SYSTEMS 0 2 WITH TOPOLOGICAL PHASE TRANSITIONS AJAY PATWARDHAN v o Physics Department, St Xavier’s college, Mumbai N Visitor, Institute of Mathematical Sciences, Chennai ABSTRACT 9 Dynamical system properties give rise to effects in Statistical mechanics. Topological index changes can be the basis for phase transitions. The Euler ] h characteristic is a versatile topological invariant that can be evaluated for model c systems. These recent developments in the foundations of Statistical Mechanics, e that are giving new results, provide insight into the Statistical thermodynamics m of small N systems; such as molecular and spin clusters. - t This paper uses model systems to give a basis for redefining partition func- a tions in classical statistical mechanics. It includes the properties of dynamical t s. systems namely , KAM Torii, singular points and chaotic regions. The equipo- t tential surfaces and the Morse and Euler index for it are defined. The conditions a m for the topology change in configuration space, and its effect on the partition - function and the ensemble average quantities is found. The justification for d topological phase transitions and their thermodynamic interpretation are dis- n cussed. o 1.INTRODUCTION c [ Since the classic work of