# Derivation of a Langevin equation in a system with multiple scales: the case of negative temperatures

## TL;DR

This paper develops a rigorous method to derive Langevin equations for systems with multiple scales, including cases where the bath has negative temperatures, revealing potential renormalization and spatial dependencies.

## Contribution

It generalizes previous coarse-graining methods to include negative temperature baths with potential renormalization and spatially varying parameters.

## Key findings

- Derived Fokker-Planck equations match numerical simulations
- Identified phase transition behavior in spin systems at negative temperatures
- Extended coarse-graining techniques to complex multi-scale systems

## Abstract

We consider the problem of building a continuous stochastic model, i.e. a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the "bath" to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as found - for instance - in models and experiments with bounded effective kinetic energy. Here, we generalise previous studies by considering the case in which the coarse-graining leads to (i) a renormalisation of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show an excellent agrement with numerical simulations.