# No-pumping theorem for non-Arrhenius rates

## TL;DR

This paper extends the no-pumping theorem to non-Arrhenius rates, specifically destination and Fokker-Planck rates, showing conditions under which probability currents still nullify in periodically driven Markov systems.

## Contribution

It introduces a new mechanism for the no-pumping theorem applicable to symmetric external fields and destination rates, broadening its relevance beyond Arrhenius rates.

## Key findings

- No-pumping theorem applies to destination rates with symmetric external fields.
- Approximate no-pumping behavior observed for Fokker-Planck rates.
- Mechanism links local equilibrium master equations to probability current nullification.

## Abstract

The no-pumping theorem refers to a Markov system that holds the detailed balance, but is subject to a time-periodic external field. It states that the time-averaged probability currents nullify in the steady periodic (Floquet) state, provided that the Markov system holds the Arrhenius transition rates. This makes an analogy between features of steady periodic and equilibrium states, because in the latter situation all probability currents vanish explicitly. However, the assumption on the Arrhenius rates is fairly specific, and it need not be met in applications. Here a new mechanism is identified for the no-pumping theorem, which holds for symmetric time-periodic external fields and the so called destination rates. These rates are the ones that lead to the locally equilibrium form of the master equation, where dissipative effects are proportional to the difference between the actual probability and the equilibrium (Gibbsian) one. The mechanism also leads to an approximate no-pumping theorem for the Fokker-Planck rates that relate to the discrete-space Fokker-Planck equation.