Work fluctuations in the active Ornstein–Uhlenbeck particle model

Abstract

We study the large deviations of the power injected by the active force for an active Ornstein–Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre–Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system’s dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation–dissipation theorem.

Work fluctuations for a confined Brownian particle: the role of initial conditions

“…A natural extension of this research will be to consider correlations in the initial condition or coloured thermal noise, such as the Ornstein-Uhlenbeck process [31,45] or the fractional Brownian motion [46]. Leaving the field of Gaussian processes, another possible extension will be to consider an anharmonic potential, which could disrupt the singularities of the rate function, and continuous-time random walks, for which large deviation principles are known in great generality [47,48].…”

Non-equilibrium thermodynamics of diffusion in fluctuating potentials

“…Nevertheless, many trapping potentials are well-approximated by parabolic potentials, thus we believe that our results will prove very useful for calculating the entropy production due to potential fluctuations in practice. Further, we focus here on diffusive motion in confining fluctuating potentials but our framework itself can be generalized to a more general class of models like random acceleration processes [43,[67][68][69] or active particles including RnT particles [19,[70][71][72], active Brownian particles [7,72,73] and AOUPs [74][75][76][77], which will be the subject of future work. Finally, we believe that our results provide a natural framework to study the stochastic thermodynamics of colloidal systems in optical traps [29,78,79].…”

Non-equilibrium thermodynamics of diffusion in fluctuating potentials

“…While we have derived exact results for the case where the potential is of quadratic form, the framework developed here can readily be extended to more complex confining potentials. Further, we focus here on diffusive motion in confining fluctuating potentials but our framework itself can be generalized to a more general class of models like random acceleration processes [43,[67][68][69] or active particles including run-and-tumble particles [19,[70][71][72], active Brownian particles [7,72,73] and active Ornstein-Uhlenbeck particles [74][75][76][77], which will be the subject of future work. Finally, we believe that our results provide a natural framework to study the stochastic thermodynamics of colloidal systems in optical traps [29,78,79].…”