arXiv subjects

S. B. Yuste

Publications and source records attributed to S. B. Yuste.

58 records · Page 4Linked to original sources

arXiv2000

I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket

We address the problem of evaluating the number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all initially placed on one site of a deterministic fractal lattice. For a wide class of fractals, of which the Sierpinski gasket is a typical example, we propose that, after the short-time compact regime and for large N, $S_N(t) \approx \hat{S}_N(t) (1-Δ)$, where $\hat{S}_N(t)$ is the number of sites inside a hypersphere of radius $R [\ln (N)/c]^{1/ u}$, R is the root-mean-square displacement of a single random walker, and u and c determine how fast $1-Γ_t({\bf r})$ (the probability that site ${\bf r}$ has been visited by a single random walker by time t) decays for large values of r/R: $1-Γ_t({\bf r})\sim \exp[-c(r/R)^u]$. For the deterministic fractals considered in this paper, $ u =d_w/(d_w-1)$, $d_w$ being the random walk dimension. The corrective term $Δ$ is expressed as a series in $\ln^{-n}(N) \ln^m \ln (N)$ (with $n\geq 1$ and $0\leq m\leq n$), which is given explicitly up to n=2. Numerical simulations on the Sierpinski gasket show reasonable agreement with the analytical expressions. The corrective term $Δ$ contributes substantially to the final value of $S_N(t)$ even for relatively large values of N.

cond-mat.stat-mech↗

arXiv2000

II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate

The average number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity $S_N(t)$ is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or topological space. This allows us to adapt the formalism developed for Euclidean and deterministic fractal lattices to the chemical language, and an asymptotic series for $S_N(t)$ analogous to that found for the non-disordered media is proposed here. The main term is equal to the number of sites (volume) inside a ``hypersphere'' in the chemical space of radius $L [\ln (N)/c]^{1/v}$ where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast $1-Γ_t(\ell)$ (the probability that a given site at chemical distance $\ell$ from the origin is visited by a single random walker by time t) decays for large values of $\ell/L$: $1-Γ_t(\ell)\sim \exp[-c(\ell/L)^v]$. The parameters appearing in the first two asymptotic terms of $S_N(t)$ are estimated by numerical simulation for the two-dimensional percolation cluster at criticality. The corresponding theoretical predictions are compared with simulation data, and the agreement is found to be very good.

cond-mat.stat-mech↗

arXiv2000

Direct correlation functions and bridge functions in additive hard-sphere mixtures

A method to obtain (approximate) analytical expressions for the radial distribution functions in a multicomponent mixture of additive hard spheres that was recently introduced is used to obtain the direct correlation functions and bridge functions in these systems. This method, which yields results practically equivalent to the Generalized Mean Spherical Approximation and includes thermodynamic consistency, is an alternative to the usual integral equation approaches and requires as input only the contact values of the radial distribution functions and the isothermal compressibility. Calculations of the bridge functions for a binary mixture using the Boubl\'ık-Mansoori-Carnahan-Starling-Leland equation of state are compared to parallel results obtained from the solution of the Percus-Yevick equation. We find that the conjecture recently proposed by Guzmán and del R\'ıo (1998, {\em Molec. Phys.}, {\bf 95}, 645) stating that the zeros of the bridge functions occur approximately at the same value of the shifted distance for all pairs of interactions is at odds with our results. Moreover, in the case of disparate sizes, even the Percus-Yevick bridge functions do not have this property. It is also found that the bridge functions are not necessarily non-positive.

cond-mat.stat-mech↗

arXiv2000

Number of distinct sites visited by N random 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-Δ), where \hat S_N(t) is the volume of a hypersphere of radius (4Dt \ln N)^{1/2}, Δ={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.

cond-mat.stat-mech↗