Abstract

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels—single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files—comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Highlights

  • We summarise different results on the diffusion of a tracer particle in lattice gases of hardcore particles with stochastic dynamics, which are confined to narrow channels—singlefiles, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters

  • In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels

  • Interactions between a biased tracer particle and a confined crowded environment produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems

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Summary

Unbiased tracer diffusion

It was understood for a long time that diffusion of a TP in a dynamical background formed by other interacting and mobile (randomly moving) particles is coupled in a non-trivial way to the evolution of the environment itself (see, e.g. [65–67]). It is often more probable that the TP will return back to this location, than keep on going away from it where its motion will be hindered by other particles This ‘anti-persistence’ of the TP motion and the circumstance whether it will be ‘permitted’ by the environment particles to return back depends essentially on how fast the environment can rearrange itself and close the void. This depends on plenty of physical factors—the density of the environment, the particle-particle interactions, the temperature and the viscosity of the embedding solvent, if any. The TP diffusion coefficient is expected to acquire some dependence on all the aforementioned parameters

Single-files
Comb-like structures
Unbounded lattices with d 2
Biased tracer diffusion
Density profiles of the environment particles
Mean velocity of the tracer particle
Variance of the tracer particle displacement
Biased tracer diffusion in narrow channels
Mean velocity of a biased tracer particle
Transient mean velocity
Terminal mean velocity
General force-velocity relation
Variance of the biased tracer particle displacement: from super-diffusion to giant diffusion
Summary and outlook
Full Text
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