Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: extracting information from transients

Single-file diffusion in spatially inhomogeneous systems

We study the effect of spatially varying potential and diffusivity on the dispersion of a tracer particle in single-file diffusion. Noninteracting particles in such a system exhibit normal diffusion at late times, which is characterized by an effective diffusion constant D_{eff}. Here we demonstrate the physically appealing result that the dispersion of single-file tracers in this system has the same long-time behavior as that for Brownian particles in a spatially homogeneous system with constant diffusivity D_{eff}. Our results are based on a late-time analysis of the Fokker-Planck equation, motivated by the mathematical theory of homogenization. The findings are confirmed by numerical simulations for both annealed and quenched initial conditions.

Universal Relation between Entropy and Kinetics

Relating thermodynamic and kinetic properties is a conceptual challenge with many practical benefits. Here, based on first principles, we derive a rigorous inequality relating the entropy and the dynamic propagator of particle configurations. It is universal and applicable to steady states arbitrarily far from thermodynamic equilibrium. Applying the general relation to diffusive dynamics yields a relation between the entropy and the (normal or anomalous) diffusion coefficient. The relation can be used to obtain useful bounds for the late-time diffusion coefficient from the calculated steady-state entropy or, conversely, to estimate the entropy based on measured diffusion coefficients. We demonstrate the validity and usefulness of the relation through several examples and discuss its broad range of applications, in particular, for systems far from equilibrium.

Interplay of adsorption and surface mobility in tracer diffusion in porous media

We model the diffusion of a tracer that interacts with the internal surface of a porous medium formed by a packing of solid spheres. The tracer executes a lattice random walk in which hops from surface to bulk sites and hops on the surface have small probabilities compared to hops from bulk sites; those probabilities are related to bulk and surface diffusion coefficients and to a desorption rate. A scaling approach distinguishes three regimes of steady state diffusion, which are confirmed by numerical simulations. If the product of desorption rate and sphere diameter is large, dominant bulk residence is observed and the diffusion coefficient is close to the bulk value. If that product is small and the surface mobility is low, the tracers are adsorbed most of the time but most hops are executed in the bulk. However, for high surface mobility, there is a nontrivial regime of dominant surface displacement, since the connectivity of solid walls allows the tracers to migrate to long distances while they are adsorbed. In this regime, we observe rounded tracer paths on the sphere walls, which are qualitatively similar to those of a recent experiment on polystyrene particle diffusion. The calculated average residence times are proportional to the bulk and surface densities of an equilibrium ensemble of noninteracting tracers, and the relation between those densities sets the adsorption isotherm. Simulations performed with initially uniform (nonequilibrium) distribution of tracers in the pores show other nontrivial results in cases of dominant surface residence: slow increase of the mean-square displacement at short times, since the tracer has not explored a homogeneous medium, and a remarkable slowdown between the first encounter with a solid wall and the first hop from that point. Relations between our results and other models of diffusion and adsorption in porous media are discussed.

Information Geometry of Spatially Periodic Stochastic Systems

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , with f - chosen to be particularly flat (locally cubic) at the equilibrium point x = 0 , and f + particularly flat at the unstable fixed point x = 1 . We numerically solve the Fokker-Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ , with μ in the range [ 0 , 1 ] . The strength D of the stochastic noise is in the range 10 - 4 - 10 - 6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L ∞ , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L ∞ as a function of initial position μ is qualitatively similar to the force, including the differences between f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , illustrating the value of information length as a useful diagnostic of the underlying force in the system.

Universal time-dependent dispersion properties for diffusion in a one-dimensional critically tilted potential

We consider the time-dependent dispersion properties of overdamped tracer particles diffusing in a one-dimensional periodic potential under the influence of an additional constant tilting force F. The system is studied in the region where the force is close to the critical value F_{c} at which the barriers separating neighboring potential wells disappear. We show that, when F crosses the critical value, the shape of the mean-square displacement (MSD) curves is strongly modified. We identify a diffusive regime at intermediate-time scales with an effective diffusion coefficient which is much larger than the late-time diffusion coefficient for F>F_{c}, whereas for F<F_{c} the late-time and intermediate-time diffusive regimes are indistinguishable. Explicit asymptotic regimes for the MSD curves are identified at all time scales.

Continuous-feed optical sorting of aerosol particles

We consider the problem of sorting, by size, spherical particles of order 100 nm radius. The scheme we analyze consists of a heterogeneous stream of spherical particles flowing at an oblique angle across an optical Gaussian mode standing wave. Sorting is achieved by the combined spatial and size dependencies of the optical force. Particles of all sizes enter the flow at a point, but exit at different locations depending on size. Exiting particles may be detected optically or separated for further processing. The scheme has the advantages of accommodating a high throughput, producing a continuous stream of continuously dispersed particles, and exhibiting excellent size resolution. We performed detailed Monte Carlo simulations of particle trajectories through the optical field under the influence of convective air flow. We also developed a method for deriving effective velocities and diffusion constants from the Fokker-Planck equation that can generate equivalent results much more quickly. With an optical wavelength of 1064 nm, polystyrene particles with radii in the neighborhood of 275 nm, for which the optical force vanishes, may be sorted with a resolution below 1 nm.

Kubo formulas for dispersion in heterogeneous periodic nonequilibrium systems

We consider the dispersion properties of tracer particles moving in nonequilibrium heterogeneous periodic media. The tracer motion is described by a Fokker-Planck equation with arbitrary spatially periodic (but constant in time) local diffusion tensors and drifts, eventually with the presence of obstacles. We derive a Kubo-like formula for the time-dependent effective diffusion tensor valid in any dimension. From this general formula, we derive expressions for the late time effective diffusion tensor and drift in these systems. In addition, we find an explicit formula for the late finite-time corrections to these transport coefficients. In one dimension, we give a closed analytical formula for the transport coefficients. The formulas derived here are very general and provide a straightforward method to compute the dispersion properties in arbitrary nonequilibrium periodic advection-diffusion systems.

Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities

We analyze the mean squared displacement of a Brownian particle in a medium with a spatially varying local diffusivity, which is assumed to be periodic. When the system is asymptotically diffusive, the mean-squared displacement, characterizing the dispersion in the system, is, at late times, a linear function of time. A Kubo-type formula is given for the mean-squared displacement, which allows the recovery of some known results for the effective diffusion constant D(e) in a direct way, but also allows an understanding of the asymptotic approach to the diffusive limit. In particular, as well as as computing the slope of a linear fit to the late-time mean-squared displacement, we find a formula for the constant where the fit intersects the y axis.