Langevin equation in systems with also negative temperatures

TL;DR

This paper develops a generalized Langevin equation framework for systems with non-quadratic Hamiltonians, including negative temperature cases, and demonstrates its effectiveness through reconstruction from time-series data.

Contribution

It introduces a method to derive and reconstruct Langevin equations for systems with non-standard Hamiltonians, extending applicability to negative temperature regimes.

Findings

Langevin equations can be derived for systems with negative temperatures.

Reconstructed Langevin dynamics accurately reproduce statistical features.

Negative temperature systems do not exhibit pathological behavior with this approach.

Abstract

We discuss how to derive a Langevin equation (LE) in non standard systems, i.e. when the kinetic part of the Hamiltonian is not the usual quadratic function. This generalization allows to consider also cases with negative absolute temperature. We first give some phenomenological arguments suggesting the shape of the viscous drift, replacing the usual linear viscous damping, and its relation with the diffusion coefficient modulating the white noise term. As a second step, we implement a procedure to reconstruct the drift and the diffusion term of the LE from the time-series of the momentum of a heavy particle embedded in a large Hamiltonian system. The results of our reconstruction are in good agreement with the phenomenological arguments. Applying the method to systems with negative temperature, we can observe that also in this case there is a suitable Langevin equation, obtained with a precise protocol, able to reproduce in a proper way the statistical features of the slow variables. In other words, even in this context, systems with negative temperature do not show any pathology.