# The validity of the no-pumping theorem in systems with finite-range interactions between particles

## TL;DR

This paper investigates the conditions under which the no-pumping theorem holds in systems with finite-range interactions, showing that certain state-dependent interactions can violate the theorem, unlike finite-range models that satisfy it.

## Contribution

It identifies the qualitative property of interactions that determines the validity of the no-pumping theorem in multi-particle stochastic systems.

## Key findings

- Finite-range interactions satisfy the no-pumping theorem.
- State-dependent branching fractions can lead to violations.
- The results clarify the role of interaction range and dependence in directed motion.

## Abstract

The no-pumping theorem states that seemingly natural driving cycles of stochastic machines fail to generate directed motion. Initially derived for single particle systems, the no-pumping theorem was recently extended to many-particle systems with zero-range interactions. Interestingly, it is known that the theorem is violated by systems with exclusion interactions. These two paradigmatic interactions differ by two qualitative aspects: the range of interactions, and the dependence of branching fractions on the state of the system. In this work two different models are studied in order to identify the qualitative property of the interaction that leads to breakdown of no-pumping. A model with finite-range interaction is shown analytically to satisfy no-pumping. In contrast, a model in which the interaction affects the probabilities of reaching different sites, given that a particle is making a transition, is shown numerically to violate the no-pumping theorem. The results suggest that systems with interactions that lead to state-dependent branching fractions do not satisfy the no-pumping theorem.