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One-dimensional lattice random walks in a Gaussian random potential

Silvio Kalaj, Enzo Marinari, Gleb Oshanin, Luca Peliti

TL;DR

This work analyzes a particle performing continuous-time random walks on a 1D lattice with quenched Gaussian potentials, comparing three dynamical scenarios that modify how transition rates depend on local energies. By computing exact moments and limit distributions for five realization-dependent observables—steady-state current, resistance, splitting probability, mean first-passage time, and diffusion coefficient—the authors show that currents, resistances, and splitting probabilities are not self-averaging in the large-$N$ limit, with boundary fluctuations driving the non-self-averaging behavior. In contrast, the mean first-passage time and the diffusion coefficient become self-averaging as $N\to\infty$, though finite-$N$ samples exhibit strong fluctuations; typical values differ from disorder-averaged ones, revealing heavy-tailed statistics and boundary-dominated effects. The results, derived via exact expressions and Derrida-type diffusion formulas, illuminate how Gaussian disorder influences transport in disordered 1D channels and traps, with potential implications for diffusion in porous media and similar systems.

Abstract

We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on $U_x$: the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with $N$ sites: the probability current, its reciprocal (the resistance), the splitting probability, the mean first-passage time $T_N$, and the diffusion coefficient $D_N$ in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current, resistance, and splitting probability are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, $T_N$ and $D_N$ become self-averaging when $N \to \infty$, though they exhibit strong sample-to-sample fluctuations for finite $N$.

One-dimensional lattice random walks in a Gaussian random potential

TL;DR

This work analyzes a particle performing continuous-time random walks on a 1D lattice with quenched Gaussian potentials, comparing three dynamical scenarios that modify how transition rates depend on local energies. By computing exact moments and limit distributions for five realization-dependent observables—steady-state current, resistance, splitting probability, mean first-passage time, and diffusion coefficient—the authors show that currents, resistances, and splitting probabilities are not self-averaging in the large- limit, with boundary fluctuations driving the non-self-averaging behavior. In contrast, the mean first-passage time and the diffusion coefficient become self-averaging as , though finite- samples exhibit strong fluctuations; typical values differ from disorder-averaged ones, revealing heavy-tailed statistics and boundary-dominated effects. The results, derived via exact expressions and Derrida-type diffusion formulas, illuminate how Gaussian disorder influences transport in disordered 1D channels and traps, with potential implications for diffusion in porous media and similar systems.

Abstract

We study random walks evolving in continuous time on a one-dimensional lattice where each site hosts a quenched random potential . The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on : the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with sites: the probability current, its reciprocal (the resistance), the splitting probability, the mean first-passage time , and the diffusion coefficient in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current, resistance, and splitting probability are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, and become self-averaging when , though they exhibit strong sample-to-sample fluctuations for finite .

Paper Structure

This paper contains 19 sections, 130 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model and dynamical scenarios
  3. Scenario I: Random force-like model
  4. Scenario II: Random walk with randomized stepping-times
  5. Scenario III: Gaussian random trap model.
  6. The realization-dependent currents and the resistances
  7. The moments of resistances
  8. The limit distributions of the resistances and their typical values.
  9. Probability currents
  10. Sample-to-sample fluctuations of the resistances and currents
  11. The realization-dependent splitting probability
  12. The realization-dependent mean first-passage times
  13. The realization-dependent diffusion coefficient
  14. Model I: Random force-like model
  15. Model II: Random walks with randomized stepping-times
  16. ...and 4 more sections

Figures (8)

  • Figure 1: Four individual short trajectories for the Model I (left panel), Model II (middle panel) and Model III (right panel) for four values of the parameter $\beta \sigma$ (see the inset).
  • Figure 2: The relative variances $R_{\tau}$ of the resistances $\tau_N$ in the Models I (left panel), II (middle panel) and III (right panel) as functions of $\beta \sigma$. Solid curves are our analytical predictions in eqs. \ref{['Rtau']} to \ref{['Rtau3']} for $N = \infty$, while the symbols depict $R_{\tau}$ evaluated by a numerical averaging $\tau_N$ and $\tau^2_N$ for finite values of $N$ (see the insets).
  • Figure 3: The probability density functions of the reduced resistance $\overline{\tau}$ for the Models I (panel (a)), II (panel (b)) and III (panel (c)). The histrograms present the results of a numerical analysis of the random variable $\overline{\tau}$. The enveloping dashed (red) curve depicts our analytical predictions in eqs. \ref{['aa']}, \ref{['bb']} and \ref{['cc']}, respectively. Vertical dashed lines indicate the typical values of $\overline{\tau}$ (see eqs. \ref{['tautyp']} and \ref{['tautyp3']}).
  • Figure 4: The disorder-averaged versus typical currents (multiplied by $N$) through a finite interval with $N$ sites for the Models I (left panel), II (middle panel) and III (right panel). The black solid curves depict our analytical predictions for the averaged currents in eqs. \ref{['jNrm']} and \ref{['jNrm3']} (with $q = 1$). The red solid curves depict the typical currents in eqs. \ref{['typj']} and \ref{['typj3']}. Symbols present the results of a numerical averaging of $j_N$ in eqs. \ref{['jNr']} and the results for an exponentiated averaged $\ln j_N$ (see the inset).
  • Figure 5: Sample-to-sample fluctuations. Probability density function $P(\omega)$ in eq. \ref{['Pomega']} as function of $\omega$ for $\beta \sigma = 0.1$ (blue curve), $\beta \sigma = 1$ (magenta curve) and $\beta \sigma = 1.2$ (green curve).
  • ...and 3 more figures