L. Acedo

An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Ficks law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

Reaction front in an A+B-->C reaction-subdiffusion process.

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

A square-well model for the structural and thermodynamic properties of simple colloidal systems

A model for the radial distribution function g(r) of a square-well fluid of variable width previously proposed [Yuste and Santos, J. Chem. Phys. 101, 2355 (1994)] is revisited and simplified. The model provides an explicit expression for the Laplace transform of rg(r), the coefficients being given as explicit functions of the density, the temperature, and the interaction range. In the limits corresponding to hard spheres and sticky hard spheres, the model reduces to the analytical solutions of the Percus–Yevick equation for those potentials. The results can be useful to describe in a fully analytical way the structural and thermodynamic behavior of colloidal suspensions modeled as hard-core particles with a short-range attraction. Comparison with computer simulation data shows a general good agreement, even for relatively wide wells.

Some exact results for the trapping of subdiffusive particles in one dimension

We study a generalization of the standard trapping problem of random walk theory in which particles move subdiffusively on a one-dimensional lattice. We consider the cases in which the lattice is filled with a one-sided and a two-sided random distribution of static absorbing traps with concentration c. The survival probability Φ(t) that the random walker is not trapped by time t is obtained exactly in both versions of the problem through a fractional diffusion approach. Comparison with simulation results is made.

Heat capacity of square-well fluids of variable width

We have obtained by Monte Carlo NVT simulations the constant-volume excess heat capacity of square-well fluids for several temperatures, densities and potential widths. Heat capacity is a thermodynamic property much more sensitive to the accuracy of a theory than other thermodynamic quantities, such as the compressibility factor. This is illustrated by comparing the reported simulation data for the heat capacity with the theoretical predictions given by the Barker-Henderson perturbation theory as well as with those given by a non-perturbative theoretical model based on Baxters solution of the Percus-Yevick integral equation for sticky hard spheres. Both theories give accurate predictions for the equation of state. By contrast, it is found that the Barker-Henderson theory strongly underestimates the excess heat capacity for low to moderate temperatures, whereas a much better agreement between theory and simulation is achieved with the non-perturbative theoretical model, particularly for small well widths, although the accuracy of the latter worsens for high densities and low temperatures, as the well width increases.

The penetrable-sphere fluid in the high-temperature, high-density limit

Abstract We consider a fluid of d -dimensional spherical particles interacting via a pair potential φ ( r ) which takes a finite value ϵ if the two spheres are overlapped ( r σ ) and 0 otherwise. This penetrable-sphere model has been proposed to describe the effective interaction of micelles in a solvent. We derive the structural and thermodynamic functions in the limit where the reduced temperature k B T / ϵ and density ρσ d tend to infinity, their ratio being kept finite. The fluid exhibits a spinodal instability at a certain maximum scaled density where the correlation length diverges and a crystalline phase appears, even in the one-dimensional model. By using a simple free-volume theory for the solid phase of the model, the fluid–solid phase transition is located.

Order statistics for d-dimensional diffusion processes.

We present results for the ordered sequence of first-passage times of arrival of N random walkers at a boundary in Euclidean spaces of d dimensions.

Number of distinct sites visited byNrandom walkers on a Euclidean lattice

The evaluation of the average number S_N(t) of distinct sites visited up to time t by N independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for large N, S_N(t) \approx \hat S_N(t) (1-\Delta), where \hat S_N(t) is the volume of a hypersphere of radius (4Dt \ln N)^{1/2}, \Delta={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N, d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the dimension and time. The first three terms of these series are calculated explicitly and the resulting expressions are compared with other approximations and with simulation results for dimensions 1, 2, and 3. Some implications of these results on the geometry of the set of visited sites are discussed.

On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile Ux(y)=ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f(r,v)=f(V), with V≡v−U(r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K(θ)∝lim∈→0∈−2δ(θ−∈), where θ is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker–Planck operator. We have found analytically that for shear rates larger than a certain threshold value ath≃0.3520ν (where ν is an average collision frequency and ath/ν is the real root of the cubic equation 64x3+16x2+12x−9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f(V;a)∼|V|−4−σ(a)Φ(ϕ;a), where ϕ≡tan Vy/Vx and the angular distribution function Φ(ϕ;a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition Φ(ϕ;a)=Φ(ϕ+π;a) allows one to obtain the exponent σ(a) as a function of the shear rate. It diverges when a→ath and tends to a minimum value σmin≃1.252 in the limit a→∞. As a consequence of this power-law decay for a>ath, all the velocity moments of a degree equal to or larger than 2+σ(a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~ϕ(a), which rotates from ~ϕ=−π/4,3π/4 when a→ath to ~ϕ=0,π in the limit a→∞.

Multiparticle trapping problem in the half-line

A variation of Rosenstocks trapping model in which N independent random walkers are all initially placed upon a site of a one-dimensional lattice in the presence of a one-sided random distribution (with probability c) of absorbing traps is investigated. The probability (survival probability) ΦN(t) that no random walker is trapped by time t for N⪢1 is calculated by using the extended Rosenstock approximation. This requires the evaluation of the moments of the number SN(t) of distinct sites visited in a given direction up to time t by N independent random walkers. The Rosenstock approximation improves when N increases, working well in the range Dtln2(1−c)⪡lnN, D being the diffusion constant. The moments of the time (lifetime) before any trapping event occurs are calculated asymptotically, too. The agreement with numerical results is excellent.