Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

TL;DR

We address how fractional Brownian motion confined by scale-invariant potentials $V(x)=k|x|^m/m$ reaches a nonequilibrium steady state with non-Boltzmann tails. Using the optimal fluctuation method (OFM) and exact action functionals for the fractional Langevin equation, we determine the large-$|x|$ tails of the steady-state distribution and extract the tail coefficient $\alpha(H,m)$ across limits $H\to 0$, $H\to 1$, and for $m=2$ at arbitrary $H$, recovering known results for the fractional Ornstein–Uhlenbeck process. The analysis uncovers confinement criteria and reveals path irreversibility for $H\neq 1/2$, with reversibility restored only for $H=1/2$ or linear $m=2$ dynamics. A specialized numerical iteration algorithm handles the cusp singularities of optimal paths, enabling computation of $\alpha(H,m)$ for general $H$ and $m$ and providing a practical route to explore large-deviation statistics in non-Markovian confined systems.

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 $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ 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 $\ln \mathcal{P}(x)$ 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 $H\to 0$ and $H\to 1$, 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 $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential. For $H\neq 1/2$, 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 $H<1/2$.

Figures (6)

• Figure 1: Shown is the coefficient $\alpha(H,m)$ in Eq. (\ref{['P160']}) for $m=2$ (orange curve) and $m=5/2$ (blue points). The orange curve corresponds to Eq. (\ref{['alpham=2']}). The blue points for $H=0$,1/2 and 1 are plotted in accordance to Eq. (\ref{['threelimits']}), whereas the three points on the interval $0<H<1/2$, for $H=1/8$, $1/4$ and $3/8$, are plotted using numerical solutions for the optimal paths, described in Sec. \ref{['numnum']}, Eq. (\ref{['alphax(0)']}).
• Figure 2: Left panel: The optimal paths $x(t)$ for the quadratic potential $V(x)=x^2/2$ for several values of $H$, as described by Eq. (\ref{['xquad1']}). Right panel: the optimal realization of the fGn, $p(t)\equiv \dot{x}(t) +x(t)$ for the same set of $H$.
• Figure 3: The ratio of the forward and backward actions (\ref{['Ssub']}) vs. $H$ for the closed path (\ref{['closedpath']}). The parameters are $D=1/2$ and $k=X=1$.
• Figure 4: Numerical optimal path $x(t)$ for $m=5/2$ and $H=1/4$. The parameters $k$ amd $\lambda D$ are $k=1$ and $\lambda D=1/2$.
• Figure 5: Irreversibility of the optimal path shown in Fig. \ref{['fig3']}. Plotted are the functions $(x(t)-x(-t))/2$ and $(x(t)+x(-t))/2$.