Grand-canonical Thermodynamic Formalism via IFS: volume, temperature, gas pressure and grand-canonical topological pressure

Abstract

We consider here a dynamic model for a gas in which a variable number of particles $N\in {\mathbb{N}}_0:=\mathbb{N}\cup \{0\}$ can be located at a site. This point of view leads us to the grand-canonical framework and the need for a chemical potential. The dynamics is played by the shift acting on the set of sequences $\Omega :={\mathcal{A}}^{\mathbb{N}}$, where the alphabet is $\mathcal{A}:=\{1,2,...,r\}$. Introducing new variables like the number of particles $N$ and the chemical potential $\mu$, we adapt the concept of grand-canonical partition sum of thermodynamics of gases to a symbolic dynamical setting considering a Lipschitz family of potentials , $A_N:\Omega \to \mathbb{R}$. Our main results will be obtained from adapting well-known properties of the Thermodynamic Formalism for IFS with weights to our setting. In this direction, we introduce the grand-canonical-Ruelle operator: ${\mathcal{L}}_{\beta ,\mu }(f)=g$, when, $\beta >0,\mu <0,$ and \medskip $g(x)={\mathcal{L}}_{\beta ,\mu }(f)(x)={\sum }_{N\in {\mathbb{N}}_0}{e}^{\beta \mu N}{\sum }_{j\in \mathcal{A}}{e}^{-\beta A_N(jx)}f(jx).$ \medskip We show the existence of the main eigenvalue, an associated eigenfunction, and an eigenprobability for ${\mathcal{L}}_{\beta ,\mu }^{\ast }$. We can show the analytic dependence of the eigenvalue on the grand-canonical potential. Considering the concept of entropy for holonomic probabilities on $\Omega \times {\mathcal{A}}^{{\mathbb{N}}_0}$, we relate these items with the variational problem of maximizing grand-canonical pressure. In another direction, in the appendix, we briefly digress on a possible interpretation of the concept of topological pressure as related to the gas pressure of gas thermodynamics.

References in corpus (1)

• Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts