Effusion of stochastic processes on a line

Abstract

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

arXiv comment19 pages, 2 figues

References in corpus (24)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Langevin Equation for the Density of a System of Interacting Langevin Processes
• Current fluctuations in non-equilibrium diffusive systems: an additivity principle
• Fluctuations in Stationary non Equilibrium States
• Macroscopic current fluctuations in stochastic lattice gases
• Persistence exponents for fluctuating interfaces
• Non equilibrium current fluctuations in stochastic lattice gases
• Universal cumulants of the current in diffusive systems on a ring
• Large Deviations in Single File Diffusion
• Current Fluctuations in One Dimensional Diffusive Systems with a Step Initial Density Profile
• Fluctuations of Current in Non-Stationary Diffusive Lattice Gases
• The Lasting Effect of Initial Conditions on Single File Diffusion
• Current fluctuations in non-interacting run-and-tumble particles in one-dimension
• Large deviation function of a tracer position in single file diffusion
• Tagged Particle Correlations in the Asymmetric Simple Exclusion Process: Finite Size Effects
• Exact Tagged Particle Correlations in the Random Average Process
• Universal Extremal Statistics in a Freely Expanding Jepsen Gas
• Current Fluctuations in the exclusion process and Bethe Ansatz
• On the Brownian gas: a field theory with a Poissonian ground state
• Role of initial conditions in $1D$ diffusive systems: compressibility, hyperuniformity and long-term memory
• Intrinsic fractional noise in nanopores: The effect of reservoirs
• Dynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption
• Tagged particle in single-file diffusion with arbitrary initial conditions
• Large deviations of a tracer position in the dense and the dilute limits of a single-file diffusion

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