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.
Highlights
In non-equilibrium statistical mechanics, dynamical properties of interacting many-body systems play as important role as non-equilibrium steady states
The motion of an individual particle in a system of interacting particles is a fundamental dynamical problem in statistical mechanics even when the entire system is in equilibrium, or in a non-equilibrium steady state
We studied the full statistics of the displacement of the tagged particle in single-file diffusion
Summary
In non-equilibrium statistical mechanics, dynamical properties of interacting many-body systems play as important role as non-equilibrium steady states. The only solvable case corresponds to Brownian particles where D(ρ) = 1 and σ(ρ) = 2ρ For this single-file system we deduce a closed form formula for the cumulant generating function and the associated large deviation function. We verify the MFT predictions in the particular case of Brownian particles by comparing with exact results which we derive using the microscopic dynamics: the large deviation functions coming from these independent methods match perfectly. The variance in the annealed and quenched cases differ by 2 This was observed for the symmetric random average process [72] and for impenetrable Brownian particles [42], and it remains generally valid for single-file diffusion.
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