**113**(2016) 60008

## Out-of-equilibrium fluctuations in stochastic long-range interacting systems

^{1} Max-Planck Institute for the Physics of Complex Systems - Nöthnitzer Straße 38, D-01187 Dresden, Germany ^{2} Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique - F-69342 Lyon, France ^{3} SISSA, INFN and ISC-CNR - Via Bonomea 265, I-34136 Trieste, Italy

Received: 26 December 2015

Accepted: 5 April 2016

For a many-particle system with long-range interactions and evolving under stochastic dynamics, we study for the first time the out-of-equilibrium fluctuations of the work done on the system by a time-dependent external force. For equilibrium initial conditions, the work distributions for a given protocol of variation of the force in time and the corresponding time-reversed protocol exhibit a remarkable scaling and a symmetry when expressed in terms of the average and the standard deviation of the work. The distributions of the work per particle predict, by virtue of the Crooks fluctuation theorem, the equilibrium free-energy density of the system. For a large number *N* of particles, the latter is in excellent agreement with the value computed by considering the Langevin dynamics of a single particle in a self-consistent mean field generated by its interaction with other particles. The agreement highlights the effective mean-field nature of the original many-particle dynamics for large *N*. For initial conditions in non-equilibrium stationary states (NESSs), we study the distribution of a quantity similar to dissipated work that satisfies the non-equilibrium generalization of the Clausius inequality, namely, the Hatano-Sasa equality, for transitions between NESSs. Besides illustrating the validity of the equality, we show that the distribution has exponential tails that decay differently on the left and on the right.

PACS: 05.70.Ln – Nonequilibrium and irreversible thermodynamics / 05.40.Ca – Noise

*© EPLA, 2016*