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Depinning transition for a one-dimensional conditioned Langevin process in the weak-noise limit | | Nicolás Tizón-Escamilla
; Vivien Lecomte
; Eric Bertin
; | | Date: |
5 Jun 2020 | | Abstract: | We consider a particle with a Langevin dynamics driven by a uniform
non-conservative force, in a one-dimensional potential with periodic boundary
conditions. The dynamics is assumed to be biased, at trajectory level, by the
time-integral of a generalized particle current, thus leading to a conditioned
Langevin process. We investigate, in the weak-noise limit, the phase diagram
spanned by the physical driving force and the bias parameter defining the
conditioned process. We focus in particular on the depinning transition in this
two-dimensional phase diagram. In the absence of trajectory bias, the depinning
transition as a function of the force is characterized by the standard exponent
$frac{1}{2}$. We show that for any non-zero bias, the depinning transition is
characterized by an inverse logarithmic behavior as a function of either the
bias or the force, close to the critical lines. We also report a scaling
exponent $frac{1}{3}$ for the current when considering the depinning
transition in terms of the bias, fixing the non-conservative force to its
critical value in the absence of bias. Then, focusing on the time-integrated
particle current, we study the thermal rounding effects in the zero-current
phase when the tilted potential exhibits a local minimum. We derive in this
case the Arrhenius scaling, in the small noise limit, of both the particle
current and the scaled cumulant generating function. This derivation of the
Arrhenius scaling relies on the determination of the left eigenvector of the
biased Fokker-Planck operator, to exponential order in the low-noise limit. An
effective Poissonian statistics of the integrated current emerges in this
limit. | | Source: | arXiv, 2006.3539 | | Services: | Forum | Review | PDF | Favorites | |
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