Amoebas of Complex Hypersurfaces in Statistical Thermodynamics

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 {ε k } ⊂ 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.

Kinematic singularities of Feynman integrals and principal A-determinants

“…To the author's knowledge, more rigorous results about such a potential relation are missing. However, there are further connections known between tropical toric geometry and statistical thermodynamics as presented in [50,69]. In any case, a fundamental understanding of such a relation could be very inspiring for a physical point of view.…”

Analytic combinatorics in several variables

“…This and its applications to statistical physics, for example, make up the content of M. Passare, Pochekutov, and A. Tsikh (2011). Because the theory of amoebas of analytic hypersurfaces is still being formed, we mostly avoid its use.…”

Continuation of multiple power series in a sectorial domain

“…In the case of multivariate power series many similar problems remain open. Relevance of these questions is motivated by both the internal logic of multidimensional complex analysis and applications in mathematical physics, for example in quantum field theory and thermodynamics .…”