Dynamics at the angle of repose: jamming, bistability, and collapse

When a sandpile relaxes under vibration, it is known that its measured angle of repose is bistable in a range of values bounded by a material-dependent maximal angle of stability; thus, at the same angle of repose, a sandpile can be stationary or avalanching, depending on its history. In the nearly jammed slow dynamical regime, sandpile collapse to a zero angle of repose can also occur, as a rare event. We claim here that fluctuations of {\it dilatancy} (or local density) are the key ingredient that can explain such varied phenomena. In this work, we model the dynamics of the angle of repose and of the density fluctuations, in the presence of external noise, by means of coupled stochastic equations. Among other things, we are able to describe sandpile collapse in terms of an activated process, where an effective temperature (related to the density as well as to the external vibration intensity) competes against the configurational barriers created by the density fluctuations.Comment: 15 pages, 1 figure. Minor changes and update

Nonconventional Large Deviations Theorems

We obtain large deviations theorems for nonconventional sums with underlying process being a Markov process satisfying the Doeblin condition or a dynamical system such as subshift of finite type or hyperbolic or expanding transformation

Trapping of a random walk by diffusing traps

We present a systematic analytical approach to the trapping of a random walk by a finite density rho of diffusing traps in arbitrary dimension d. We confirm the phenomenologically predicted e^{-c_d rho t^{d/2}} time decay of the survival probability, and compute the dimension dependent constant c_d to leading order within an eps=2-d expansion.Comment: 16 pages, to appear in J. Phys.

First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories

We investigate the dynamics of kinetically constrained models of glass formers by analysing the statistics of trajectories of the dynamics, or histories, using large deviation function methods. We show that, in general, these models exhibit a first-order dynamical transition between active and inactive dynamical phases. We argue that the dynamical heterogeneities displayed by these systems are a manifestation of dynamical first-order phase coexistence. In particular, we calculate dynamical large deviation functions, both analytically and numerically, for the Fredrickson-Andersen model, the East model, and constrained lattice gas models. We also show how large deviation functions can be obtained from a Landau-like theory for dynamical fluctuations. We discuss possibilities for similar dynamical phase-coexistence behaviour in other systems with heterogeneous dynamics.Comment: 29 pages, 7 figs, final versio

Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

The problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d. We consider the case where the traps are initially randomly distributed in space, with uniform density rho, and derive upper and lower bounds for the probability Q(t) (averaged over all particle and trap trajectories) that the particle survives up to time t. We show that, for 1<=d<2, the bounds converge asymptotically to give $Q(t) \sim exp(-\lambda_d t^{d/2})$ where $\lambda_d = (2/\pi d) sin(\pi d/2) (4\pi D)^{d/2} \rho$ and D is the diffusion constant of the traps, and that $Q(t) \sim exp(- 4\pi\rho D t/ln t)$ for d=2. For d>2 bounds can still be derived, but they no longer converge for large t. For 1<=d<=2, these asymptotic form are independent of the diffusion constant of the particle. The results are compared with simulation results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev. E v65 050101 (2002)] which is described in detail. Deviations from the predicted asymptotic forms are found to be large even for very small values of Q(t), indicating slowly decaying corrections whose form is consistent with the bounds. We also present results in d=1 for the case where the trap densities on either side of the particle are different. For this case we can still obtain exact bounds but they no longer converge.Comment: 13 pages, RevTeX4, 6 figures. Figures and references updated; equations corrected; discussion clarifie

Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling regime, focusing on the long time properties. By a saddle point analysis of the mode-coupling equations, we derive exact results for the correlation function in the long time limit - a limit which is hard to study using simulations. The correlation function at wavevector k in dimension d is found to behave asymptotically at time t as C(k,t)\simeq 1/k^{d+4-2z} (Btk^z)^{\gamma/z} e^{-(Btk^z)^{1/z}}, with \gamma=(d-1)/2, A a determined constant and B a scale factor.Comment: RevTex, 4 pages, 1 figur

Brownian Motions on Metric Graphs

Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.Comment: 43 pages, 7 figures. 2nd revision of our article 1102.4937: The introduction has been modified, several references were added. This article will appear in the special issue of Journal of Mathematical Physics celebrating Elliott Lieb's 80th birthda

On strong causal binomial approximation for stochastic processes

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In 1-dimension, the system models the zero temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d < 2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q + Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q) epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+ O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q)) ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.Comment: 12 pages, revtex, 5 figure

The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related waveComment: Latex fil