# Large deviations and optimal control forces for hard particles in one dimension

## TL;DR

This paper investigates large deviations in one-dimensional diffusing hard particle systems, analyzing effective forces for biased trajectory sampling and identifying distinct regimes with different interaction characteristics.

## Contribution

It introduces an optimal-control framework to study effective interactions and sampling improvements in large deviation regimes of one-dimensional hard particle systems.

## Key findings

- High activity leads to hyperuniform, long-ranged repulsive interactions.
- Low activity exhibits near-equilibrium behavior with long-ranged attractive forces.
- Far-from-equilibrium regime features macroscopic gaps and strongly size-dependent interactions.

## Abstract

We analyse large deviations of the dynamical activity in one-dimensional systems of diffusing hard particles. Using an optimal-control representation of the large-deviation problem, we analyse effective interaction forces which can be added to the system, to aid sampling of biased ensembles of trajectories. We find several distinct regimes, as a function of the activity and the system size: we present approximate analytical calculations that characterise the effective interactions in several of these regimes. For high activity the system is hyperuniform and the interactions are long-ranged and repulsive. For low activity, there is a near-equilibrium regime described by macroscopic fluctuation theory, characterised by long-ranged attractive forces. There is also a far-from-equilibrium regime in which one of the interparticle gaps becomes macroscopic and the interactions depend strongly on the size of this gap. We discuss the extent to which transition path sampling of these ensembles is improved by adding suitable control forces.