Diffusion in periodic, correlated random forcing landscapes

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically–extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent H ∈ ( 0 , 1 ) ?> . While the periodicity ensures that the ultimate long–time behavior is diffusive, the generalized Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient D L : although one has the typical value D L typ ∼ exp ( − &bgr; L H ) ?> , we show via an exact analytical approach that the positive moments ( k > 0 ?> ) scale like ⟨ D L k ⟩ ∼ exp [ − c ′ ( k &bgr; L H ) 1 / ( 1 + H ) ] ?> , and the negative ones as ⟨ D L − k ⟩ ∼ exp ( a ′ ( k &bgr; L H ) 2 ) ?> , c ′ ?> and a ′ ?> being numerical constants and β the inverse temperature. These results demonstrate that D L is strongly non-self-averaging. We further show that the probability distribution of D L has a log–normal left tail and a highly singular, one–sided log–stable right tail reminiscent of a Lifshitz singularity.

• Publication year

  2014

• Venue

  Journal of Physics A: Mathematical and Theoretical

• Publication date

  2014-06-10

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1088/1751-8113/47/37/372001arXiv 1406.2612
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• The diffusion coefficient is strongly non-self-averaging, and its distribution has a log-normal left tail together with a highly singular one-sided log-stable right tail reminiscent of a Lifshitz singularity.

  Confidence 0.93

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• The negative moments of the diffusion coefficient grow as an exponential of (k β L^H)^2.

  Confidence 0.95

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• The positive moments of the diffusion coefficient obey a stretched-exponential scaling in k, β, and L^H, with exponent 1/(1+H).

  Confidence 0.95

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• Periodic extension preserves diffusive long-time transport while generating strong logarithmic confinement at intermediate times in the correlated random landscape.

  Confidence 0.96

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review

• brownian particle

  physical system

  A diffusing particle whose motion is studied in the random potential landscape.

  Aliases: particle

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• diffusion coefficient d_l

  quantity, transport coefficient

  The effective long-time diffusion coefficient associated with a particle moving in a period-L random landscape.

  Aliases: D_L, DL

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• fractional brownian motion

  stochastic process

  A Gaussian stochastic process used here as the source trajectory for the random potential, characterized by a Hurst exponent.

  Aliases: FBM

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• hurst exponent

  parameter

  The parameter H in (0,1) that controls the roughness and correlation structure of the fractional Brownian motion.

  Aliases: H

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• log-normal left tail

  distribution tail

  The low-value side of the diffusion-coefficient distribution, described here by a log-normal form.

  Aliases: left tail

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• one-sided log-stable right tail

  distribution tail

  The high-value side of the diffusion-coefficient distribution, described here by a singular one-sided log-stable form.

  Aliases: right tail

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review
• periodically extended quenched random potential

  potential, random medium

  A frozen random landscape built by repeating a finite fractional-Brownian trajectory with spatial period L.

  Aliases: periodic quenched random potential, periodically extended random forcing landscape

  박진우 (dztg5apj7m) extraction뀨 (7c402c1b98) reviewq (76h6bfydm6) reviewAK (4715169a40) review

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