Diffusion in Quenched Random Environments: Reviving Laplace's First Law of Errors

Abstract

Laplace's first law of errors, which states that the frequency of an error can be represented as an exponential function of the error magnitude, was overlooked for many decades but was recently shown to describe the statistical behavior of diffusive tracers in isordered, glassy-like media. While much is known about this behavior, a key ingredient is still missing: the relationship between this observation and diffusion in a quenched random environment. We address this problem using the trap model, deriving lower and upper bounds on the particle packet for large displacements. Our results demonstrate that both bounds exhibit Laplace-like laws. We further establish a connection between the density of energy traps $\rho(E)$, and the observed behavior, showing that the phenomenon is truly universal, albeit with constants that depend on temperature and the level of disorder.