Efficient numerical evaluation of thermodynamic quantities on infinite (semi-)classical chains

TL;DR

This paper introduces an efficient numerical method combining transfer operators and quadrature rules to accurately compute thermodynamic properties of one-dimensional classical systems, demonstrating exponential convergence for analytic kernels.

Contribution

The work develops a novel numerical approach that enhances the efficiency and accuracy of evaluating thermodynamic quantities in (semi-)classical chains, especially for analytic kernels.

Findings

Exponential convergence for analytic kernels.

Successful application to particle chains and nonlinear Schrödinger systems.

Improved computational efficiency in thermodynamic calculations.

Abstract

This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schr\"odinger equation and to a classical system on a cylindrical lattice.