# Motility-induced temperature difference in coexisting phases

## TL;DR

This paper demonstrates that active particles with inertia can sustain a persistent temperature difference between coexisting phases, challenging the traditional thermodynamic principle that phases in equilibrium share the same temperature.

## Contribution

It reveals that inertia in active particles enables stable temperature gradients across coexisting phases, a phenomenon not observed in standard models without inertia.

## Key findings

- Active particles with inertia exhibit phase separation with different kinetic temperatures.
- A persistent temperature gradient exists between coexisting phases due to inertia.
- This phenomenon is fundamentally beyond equilibrium thermodynamics.

## Abstract

In nature, objects which are in thermal contact with each other, usually approach the same temperature, unless a heat source (or sink) cherishes a persistent flow of heat. Accordingly, in a well-isolated apartment flat, most items are at a similar temperature. This is a general consequence of equilibrium thermodynamics, requiring coexisting phases to have identical temperatures. Opposing this generic situation, here we identify a system showing different temperatures in coexisting phases, which are separated from each other by a sharp and persistent temperature gradient. Thermodynamically, such a "hot" and a "cold" phase are allowed to coexist, as the system we consider comprises "active" particles which self-propel relative to their environment and are thus intrinsically out-of-equilibrium. Although these microparticles are well known to spontaneously phase-separate into a liquid- and a gas-like state, different kinetic temperatures in coexisting phases occur if and only if inertia is introduced, which is neglected in standard models describing active particles. Our results, therefore, exemplify a novel route to use active particles to create a self-sustained temperature gradient across coexisting phases, a phenomenon, which is fundamentally beyond equilibrium physics.