The effective potential and transshipment in thermodynamic formalism at temperature zero

TL;DR

This paper investigates the effective potential in thermodynamic formalism at zero temperature, establishing existence and uniqueness, and explores its asymptotic behavior as temperature approaches zero, linking it to transshipment problems.

Contribution

It introduces a novel framework for analyzing the effective potential at zero temperature and connects it with ergodic transshipment problems, providing new insights into thermodynamic formalism.

Findings

Existence and uniqueness of the effective potential and associated probabilities.

Asymptotic behavior of the effective potential as temperature approaches zero.

Connection between zero-temperature limit and ergodic transshipment problem.

Abstract

Denote the points in {1,2,..,r}^{Z}= {1,2,..,r}^{N} x {1,2,..,r}^{N} by ({y}^*, {x}). Given a Lipschitz continuous observable A: {1,2,..,r}^{Z} \to {R} , we define the map {G}^+: {H}\to {H} by {G}^+(\phi)({y}^*) = \sup_{\mu \in {M}_\sigma} [\int_{\{1,2,..,r\}^{N}} ( A({y}^*, {x}) + \phi({x})) d\mu({x}) + h_\mu(\sigma) ], where: \sigma is the left shift map acting on {1,2,..,r}^{N}; {M}_\sigma denotes the set of \sigma-invariant Borel probabilities; h_\mu(\sigma) indicates the Kolmogorov-Sinai entropy; {H } is the Banach space of Lipschitz real-valued functions on {1,2,..,r}^{N}. We show there exist a unique \phi^+ \in {H } and a unique \lambda^+\in {R} such that {G}^+ (\phi^+) = \phi^+ + \lambda^+. We say that \phi^+ is the effective potential associated to A. This also defines a family of $\sigma$-invariant Borel probabilities \mu_{{y}^*} on {1,2,..,r}^{N}, indexed by the points {y}^* \in {1,2,..,r}^{N}. Finally, for A fixed and for variable positive real values \beta, we consider the same problem for the Lipschitz observable \beta A. We investigate then the asymptotic limit when \beta\to \infty of the effective potential (which depends now on \beta) as well as the above family of probabilities. We relate the limit objects with an ergodic version of Kantorovich transshipment problem. In statistical mechanics \beta \propto 1/T, where T is the absolute temperature. In this way, we are also analyzing the problem related to the effective potential at temperature zero.