Exclusion processes in higher dimensions: Stationary measures and convergence

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on $Z$. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on $Z^d$ and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on $Z^d$. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as $t\to\infty$ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Positive recurrence of reflecting Brownian motion in three dimensions

Consider a semimartingale reflecting Brownian motion (SRBM) $Z$ whose state space is the $d$-dimensional nonnegative orthant. The data for such a process are a drift vector $\theta$, a nonsingular $d\times d$ covariance matrix $\Sigma$, and a $d\times d$ reflection matrix $R$ that specifies the boundary behavior of $Z$. We say that $Z$ is positive recurrent, or stable, if the expected time to hit an arbitrary open neighborhood of the origin is finite for every starting state. In dimension $d=2$, necessary and sufficient conditions for stability are known, but fundamentally new phenomena arise in higher dimensions. Building on prior work by El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229--253, Math. Methods Oper. Res. 56 (2002) 243--258], we provide necessary and sufficient conditions for stability of SRBMs in three dimensions; to verify or refute these conditions is a simple computational task. As a byproduct, we find that the fluid-based criterion of Dupuis and Williams [Ann. Probab. 22 (1994) 680--702] is not only sufficient but also necessary for stability of SRBMs in three dimensions. That is, an SRBM in three dimensions is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. The problem of recurrence classification for SRBMs in four and higher dimensions remains open.Comment: Published in at http://dx.doi.org/10.1214/09-AAP631 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two site self consistent method for front propagation in reaction-diffusion system

We study front propagation in the reaction diffusion process $A\leftrightarrow2A$ on one dimensional lattice with hard core interaction between the particles. We propose a two site self consistent method (TSSCM) to make analytic estimates for the front velocity and are in excellent agreement with the simulation results for all parameter regimes. We expect that the simplicity of the method will allow one to use this technique for estimating the front velocity in other reaction diffusion processes as well.Comment: 6 figure

An ϵ\epsilon-expansion for Small-World Networks

I construct a well-defined expansion in $\epsilon=2-d$ for diffusion processes on small-world networks. The technique permits one to calculate the average over disorder of moments of the Green's function, and is used to calculate the average Green's function and fluctuations to first non-leading order in $\epsilon$, giving results which agree with numerics. This technique is also applicable to other problems of diffusion in random media.Comment: 7 pages Europhysics style, 3 figure

Water on Mars, With a Grain of Salt: Local Heat Anomalies Are Required for Basal Melting of Ice at the South Pole Today

Recent analysis of radar data from the Mars Express spacecraft has interpreted bright subsurface radar reflections as indicators of local liquid water at the base of the south polar layered deposits (SPLD). However, the physical and geological conditions required to produce melting at this location were not quantified. Here we use thermophysical models to constrain parameters necessary to generate liquid water beneath the SPLD. We show that no concentration of salt is sufficient to melt ice at the base of the SPLD in the present day under typical Martian conditions. Instead, a local enhancement in the geothermal heat flux of >72 mW/m(2) is required, even under the most favorable compositional considerations. This heat flow is most simply achieved via the presence of a subsurface magma chamber emplaced 100 s of kyr ago. Thus, if the liquid water interpretation of the observations is correct, magmatism on Mars may have been active extremely recently.6 month embargo; published online: 12 February 2019This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Symmetry and species segregation in diffusion-limited pair annihilation

We consider a system of q diffusing particle species A_1,A_2,...,A_q that are all equivalent under a symmetry operation. Pairs of particles may annihilate according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the symmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d > 2 mean-field theory predicts that the total particle density decays as n(t) ~ 1/t, provided the system remains spatially uniform. We determine the conditions on the matrix k under which there exists a critical segregation dimension d_{seg} below which this uniformity condition is violated; the symmetry between the species is then locally broken. We argue that in those cases the density decay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that when d_{seg} exists, its value can be expressed in terms of the ratio of the smallest to the largest eigenvalue of k. The existence of a conservation law (as in the special two-species annihilation A + B -> 0), although sufficient for segregation, is shown not to be a necessary condition for this phenomenon to occur. We work out specific examples and present Monte Carlo simulations compatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include

Exact Results for a Three-Body Reaction-Diffusion System

A system of particles hopping on a line, singly or as merged pairs, and annihilating in groups of three on encounters, is solved exactly for certain symmetrical initial conditions. The functional form of the density is nearly identical to that found in two-body annihilation, and both systems show non-mean-field, ~1/t**(1/2) instead of ~1/t, decrease of particle density for large times.Comment: 10 page

Particle Dynamics in a Mass-Conserving Coalescence Process

We consider a fully asymmetric one-dimensional model with mass-conserving coalescence. Particles of unit mass enter at one edge of the chain and coalescence while performing a biased random walk towards the other edge where they exit. The conserved particle mass acts as a passive scalar in the reaction process $A+A\to A$, and allows an exact mapping to a restricted ballistic surface deposition model for which exact results exist. In particular, the mass- mass correlation function is exactly known. These results complement earlier exact results for the $A+A\to A$ process without mass. We introduce a comprehensive scaling theory for this process. The exact anaytical and numerical results confirm its validity.Comment: 5 pages, 6 figure

Model of Cluster Growth and Phase Separation: Exact Results in One Dimension

We present exact results for a lattice model of cluster growth in 1D. The growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable-phase, +1, regions by overturning the unstable-phase, -1, spins with probability p. For cluster coarsening at phase coexistence, p=0, the conventional structure-factor scaling applies. In this limit our model falls in the class of diffusion-limited reactions A+A->inert. The +1 cluster size grows diffusively, ~t**(1/2), and the two-point correlation function obeys scaling. However, for p>0, i.e., for the dynamics of formation of stable phase from unstable phase, we find that structure-factor scaling breaks down; the length scale associated with the size of the growing +1 clusters reflects only the short-distance properties of the two-point correlations.Comment: 12 page