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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">|</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where m &gt; 0, and employ the optimal fluctuation method (OFM) to determine the large- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> tails of the steady-state probability distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> tails of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> up to a dimensionless factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>H</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is the Hurst exponent. We determine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and autocovariance. The form of the tails of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> yields exact conditions, in terms of H and m , for the particle confinement in the potential. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <