Integrable multi-phase thermodynamic systems and Tsallis' composition rule

G De Nittis,P Lorenzoni,A Moro

doi:10.1088/1742-6596/482/1/012009

Abstract

We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N–phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.

Highlights

The mathematical description of a macroscopic physical system in thermodynamic equilibrium requires a suitable number of state functions together with their conjugated thermodynamic variables and a set of order parameters
We are interested in the description of a general thermodynamic system in equilibrium described by a Gibbs function Φ
We will look for a class of solutions to Maxwell’s relations (2) such that the order parameters satisfy a system of integrable equations of hydrodynamic type

Summary

Introduction

The mathematical description of a macroscopic physical system in thermodynamic equilibrium requires a suitable number of state functions together with their conjugated thermodynamic variables and a set of order parameters. We will look for a class of solutions to Maxwell’s relations (2) such that the order parameters satisfy a system of integrable equations of hydrodynamic type. In this case, we will show that the order parameters satisfy the system of N equations of state of the form. A particular class of such integrable hydrodynamic equations for the order parameters is associated with Tsallis’ type entropy [15, 16] that in the simple case of a two-phase system reads as. Solving the system (8) may be, in general, highly nontrivial, there exists special classes of systems for which the general solution is found by quadratures Such is the case of weakly non linear or linearly degenerate systems that are characterised by the condition. More details can be found in the papers [7, 8, 4]

Multi-phase equations of state

Tsallis-type entropies