Amoebas of complex hypersurfaces in statistical thermodynamics

TL;DR

This paper explores the application of amoebas of complex hypersurfaces to statistical thermodynamics, developing multidimensional thermodynamic concepts and analyzing energy distribution in quantum ensembles.

Contribution

It extends the concept of amoebas to transcendental hypersurfaces and develops a multidimensional Darwin-Fowler method for quantum thermodynamics.

Findings

Description of the domain of admissible average energies.

Development of multidimensional temperature and thermodynamic forms.

Analysis of thermodynamic limit conditions.

Abstract

The amoeba of a complex hypersurface is its image under a logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble. The spectrum ${\epsilon_k}\subset \mathbb{Z}^n$ of the ensemble is assumed to be multidimensional; this leads us to the notions of a multidimensional temperature and a vector of differential thermodynamic forms. Strictly speaking, in the paper we develop the multidimensional Darwin and Fowler method and give the description of the domain of admissible average values of energy for which the thermodynamic limit exists.