## Abstract

At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials V ( x ) ∼ | x | m , where m > 0, and employ the optimal fluctuation method (OFM) to determine the large- | x | tails of the steady-state probability distribution P ( x ) of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large- | x | tails of ln P ( x ) up to a dimensionless factor α ( H , m ) , where 0 < H < 1 is the Hurst exponent. We determine α ( H , m ) analytically (i) in the limits of H → 0 and H → 1, and (ii) for m = 2 and arbitrary H, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact P ( x ) and autocovariance. The form of the tails of P ( x ) yields exact conditions, in terms of H and m, for the particle confinement in the potential. For H ≠ 1 / 2 , the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for m = 2. To compute the optimal paths and the factor α ( H , m ) for arbitrary permissible H and m, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for H < 1 / 2 .

Original language | English |
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Article number | 445002 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 57 |

Issue number | 44 |

DOIs | |

State | Published - 22 Nov 2024 |

### Bibliographical note

Publisher Copyright:© 2024 The Author(s). Published by IOP Publishing Ltd.

## Keywords

- fractional Brownian motion
- fractional Gaussian noise
- fractional Ornstein-Uhlenbeck process
- nonequilibrium steady state
- optimal fluctuation method