[Paper Review] Tagged particle in single-file diffusion

P. L. Krapivsky, Kirone Mallick+1 more|arXiv (Cornell University)|Jun 2, 2015

Stochastic processes and statistical mechanics2 references4 citations

TL;DR

This paper investigates the full statistical distribution of tagged particle displacement in single-file diffusion using macroscopic fluctuation theory (MFT). It derives the large deviation function for impenetrable Brownian particles and verifies it via exact microscopic dynamics, while also providing a perturbative formula for the variance and fourth cumulant in general single-file systems using transport coefficients.

ABSTRACT

Single-file diffusion is a one-dimensional interacting infinite-particle system in which the order of particles never changes. An intriguing feature of single-file diffusion is that the mean-square displacement of a tagged particle exhibits an anomalously slow sub-diffusive growth. We study the full statistics of the displacement using a macroscopic fluctuation theory. For the simplest single-file system of impenetrable Brownian particles we compute the large deviation function and provide an independent verification using an exact solution based on the microscopic dynamics. For an arbitrary single-file system, we apply perturbation techniques and derive an explicit formula for the variance in terms of the transport coefficients. The same method also allows us to compute the fourth cumulant of the tagged particle displacement for the symmetric exclusion process.

Motivation & Objective

• To understand the full probability distribution of tagged particle displacement in single-file systems, going beyond the mean-square displacement.
• To derive the large deviation function for the displacement in the simplest single-file model—impenetrable Brownian particles—using macroscopic fluctuation theory.
• To provide an independent verification of the large deviation function through an exact solution based on microscopic dynamics.
• To develop a perturbative framework for general single-file systems to compute the variance and higher cumulants in terms of transport coefficients.
• To explicitly compute the fourth cumulant of displacement for the symmetric exclusion process using the same method.

Proposed method

• Applies macroscopic fluctuation theory (MFT) to model the hydrodynamic fluctuations in single-file systems.
• Uses the large deviation principle to express the probability of tagged particle displacement in the form $ P(X_T / \\-sqrt{4T}} = x ) \asymp e^{-\sqrt{4T} \phi(x)} $, where $ \phi(x) $ is the large deviation function.
• Derives the large deviation function for impenetrable Brownian particles by solving the MFT action functional and verifying it with an exact solution based on the microscopic dynamics.
• Employs a perturbative expansion in the interaction strength to compute transport coefficients and their relation to cumulants of displacement.
• Uses the cumulant generating function $ \mu_T(\lambda) = \log \langle e^{\lambda X_T} \rangle $ to relate the cumulants to the large deviation function via Legendre transform.
• Applies combinatorial identities involving permutations and error functions to evaluate multi-particle path integrals in the exact solution.

Experimental results

Research questions

• RQ1 What is the full large deviation function for the displacement of a tagged particle in a system of impenetrable Brownian particles?
• RQ2 How does the variance of the tagged particle displacement scale in general single-file systems, and can it be expressed in terms of transport coefficients?
• RQ3 What is the fourth cumulant of the tagged particle displacement in the symmetric exclusion process?
• RQ4 Can the large deviation function derived via MFT be independently verified using an exact microscopic solution?
• RQ5 How do initial conditions affect the pre-factor of the $ \sqrt{T} $ scaling in the variance of displacement?

Key findings

• The large deviation function $ \phi(x) $ for impenetrable Brownian particles is derived using macroscopic fluctuation theory and confirmed by an exact solution based on the microscopic dynamics.
• For general single-file systems, the variance of the tagged particle displacement scales as $ \sqrt{T} $, with the pre-factor expressible in terms of transport coefficients via a perturbative expansion.
• The fourth cumulant of the displacement in the symmetric exclusion process is computed explicitly using the same perturbative framework.
• The full statistics of displacement, including higher cumulants, scale as $ \sqrt{T} $, consistent with sub-diffusive behavior.
• The large deviation function $ \phi(x) $ is related to the cumulant generating function via a Legendre transform, enabling full statistical inference.
• Combinatorial identities involving permutations and error functions are used to rigorously evaluate the exact solution, confirming the MFT result.