Classical Representation of a Quantum System at Equilibrium: Theory

TL;DR

This paper develops a theoretical framework to represent quantum systems at equilibrium with classical models, enabling the use of classical methods for strongly coupled quantum systems.

Contribution

It introduces a formal method to construct classical analogs of quantum systems using grand canonical ensembles and effective parameters, facilitating classical simulations of quantum properties.

Findings

Application to ideal Fermi gas demonstrates the method.

Weak coupling pair potential derived for quantum systems.

Framework enables classical simulation of strongly coupled quantum systems.

Abstract

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities, and the shell structure of harmonically bound charges.