Clustering in a Self-Gravitating One-Dimensional Gas at Zero Temperature

We study a system of gravitationally interacting sticky particles. At the initial time, we have n particles, each with mass 1/n and momentum 0, independently spread on [0, 1] according to the uniform law. Due to the confining of the system, all particles merge into a single cluster after a finite time. We give the asymptotic laws of the time of the last collision and of the time of the kth collision, when n. We prove also that clusters of size k appear at time n ^{–1/2(k–1)}. We then investigate the system at a fixed time t<1. We show that the biggest cluster has size of order logn, whereas a typical cluster is of finite size.     

Noise Induced Dissipation in Lebesgue-Measure Preserving Maps on d-Dimensional Torus

We consider dissipative systems resulting from the Gaussian and alpha-stable noise perturbations of measure-preserving maps on the d dimensional torus. We study the dissipation time scale and its physical implications as the noise level vanishes. We show that nonergodic maps give rise to an O(1/) dissipation time whereas ergodic toral automorphisms, including cat maps and their d-dimensional generalizations, have an O(ln(1/)) dissipation time with a constant related to the minimal, dimensionally averaged entropy among the automorphism's irreducible blocks. Our approach reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity, Diophantine approximation and arithmetic progression. We show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also discuss the implication of the dissipation time in kinematic dynamo.     

On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to ^–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor .     

Effect of Phase-Sensitive Reservoir on the Decoherence of Pair-Cat Coherent States

We study the decoherence of a superposition of four coherent states under the action of a phase sensitive reservoir. We verify that the decoherence times _k,l, k,l=1,2,3,4, between any two coherent states of the superposition can be controlled through the reservoir parameters. The decoherence time between two components of any pair, for instance _1,2 or _3,4, can be significantly increased, compared with the decoherence time when the state is acted by a thermal reservoir. However, this occurs at the expense of decreasing the decoherence time between the ``cat states" (1,2) and (3,4). This can be useful in quantum computation.     

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time, _K, and compare it with the corresponding time, _K ⁽⁰⁾ , for the potential system which has the same stationary distribution. We show that _K/ _K ⁽⁰⁾ depends only on the properties of the drift force close to the saddle-point.The authors would like to dedicate this work to their colleagues Y. Orlov, R. Nazarian, and V. Brailovski.     

Phase transition for absorbed Brownian motion with drift

We study one-dimensional Brownian motion with constant drift toward the origin and initial distribution concentrated in the strictly positive real line. We say that at the first time the process hits the origin, it is absorbed. We study the asymptotic behavior, ast, ofm _t , the conditional distribution at time zero of the process conditioned on survival up to timet and on the process having a fixed value at timet. We find that there is a phase transition in the decay rate of the initial condition. For fast decay rate (subcritical case)m _t is localized, in the critical casem _t is located around , and for slow rates (supercritical case)m _t is located aroundt. The critical rate is given by the decay of the minimal quasistationary distribution of this process. We also study in each case the asymptotic distribution of the process, scaled by , conditioned as before. We prove that in the subcritical case this distribution is a Brownian excursion. In the critical case it is a Brownian bridge attaining 0 for the first time at time 1, with some initial distribution. In the supercritical case, after centering around the expected value—which is of the order oft—we show that this process converges to a Brownian bridge arriving at 0 at time 1 and with a Gaussian initial distribution.     

First passage time densities for random walk spans

A general expression is derived for the Laplace transform of the probability density of the first passage time for the span of a symmetric continuous-time random walk to reach levelS. We show that when the mean time between steps is finite, the mean first passage time toS is proportional toS ₂. When the pausing time density is asymptotic to a stable density we show that the first passage density is also asymptotically stable. Finally when the jump distribution of the random walk has the asymptotic formp(j)A/|j| ₊₁, 0 < < 2 it is shown that the mean first passage time toS goes likeS .     

Pair Production in a Uniform Electric Field

We study the Klein-Gordon field coupled with an external uniform vector potential. We compute pair production in a finite time t using the semiclassical approximation, and show that, after the interaction of the Klein-Gordon field with the external potential, when 0 the average number of produced pairs is zero. There is agreement with the classical limit because the classical limit involves no production of pairs. We compared our results with those of Schwinger. Finally we saw that the random variable N(t)= net number of pairs produced at time t is in the semiclassical limit a stochastic Poisson process.     

Time Flow in a Noncommutative Regime

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on depending on the state . The state defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair is a noncommutative probabilistic space and can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.     

Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints

We study a new Monte Carlo algorithm for generating self-avoiding walks with variable length (controlled by a fugacity) and fixed endpoints. The algorithm is a hybrid of local (BFACF) and nonlocal (cut-and-paste) moves. We find that the critical slowing-down, measured in units of computer time, is reduced compared to the pure BFACF algorithm: _CPU N^2.3 versus N^3.0. We also prove some rigorous bounds on the autocorrelation time for these and related Monte Carlo algorithms.     

Some new results on the two-dimensional kinetic Ising model in the phase coexistence region

We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL with open boundary conditions, in the absence of an external field and at large inverse temperature . We prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap. As a consequence we are able to show that there exists a new time scalet _even, much smaller than the global relaxation timet _rel, such that, with large probability, any initial configuration first relaxes to one of the two phases in a time scale of ordert _even and only after a time scale of the order oft _rel does it reach the final equilibrium by jumping, via a large deviation, to the opposite phase. It also follows that, with large probability, the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time.     

Dynamic critical behavior of a Swendsen-Wang-Type algorithm for the Ashkin-Teller model

We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (_int. const xC _H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio _int./C _H appears to tend to infinity either as a logarithm or as a small power (0.05p0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.     

Probabilistic Estimates for the Two-Dimensional Stochastic Navier–Stokes Equations

We consider the Navier–Stokes equation on a two-dimensional torus with a random force, white noise in time, and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long-time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as R.     

Critical behavior of the two-dimensional first passage time

We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probabilityp and 1–p, respectively. We prove that as the zero-time bonds approach the percolation thresholdp _c, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeys(p) ¦p–p _c¦^–v, then the first passage time constant satisfies(p)¦p–p _c¦^v. At p_c, where it has been asserted that the first passage time from 0 tox scales as ¦x¦ to a power with 0<<1, we show that the passage times grow like log ¦x¦, i.e., the fluid spreads exponentially rapidly.     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

A stochastic model of deposition processes with nucleation

We study an interacting particle system on a one-dimensional infinite lattice and one-dimensional lattices with a periodic boundary. In this system, each site of the lattice may be either empty or occupied and initially all the lattice sites are empty. The evolution of the system is defined as follows: an empty site waits an exponential time with mean 1 and becomes occupied, and an occupied site becomes empty at a time which is distributed exponentially with mean _k, wherek is the number of occupied neighboring sites of this site in the current state of the system. We show that the mean number of the occupied sites of the lattice, considered as a function of time, may possess a convex part. A sufficient condition for this is that ₀ is large and _k,k1, are small. The studied system has been proposed recently as a mathematical model of certain deposition processes, in particular those which exhibit nucleation caused by lateral attractive interaction between the deposited molecules. Our research was motivated by the observation that the density of deposited molecules contains a convex part, over some time interval, if the attractive forces are strong, while this density is a concave function of time if these forces are weak or absent. Our result agrees with this observation.     

The surfboard Schrödinger equations

We study the large time behavior of solutions of time dependent Schrödinger equationsiu/t=–(1/2)u+t V(x/t)u with bounded potentialV(x). We show that (1) if>–1, all solutions are asymptotically free ast, (2) if–1 a solution becomes asymptotically free if and only if it has the momentum support outside of suppV for large time, (3) if –1 <0 all solutions are still asymptotically modified free ast and that (4) if 0 <2, for each local minimumx ₀ ofV(x), there exist solutions which are asymptotically Gaussians centered atx=tx ₀ and spreading slowly ast.     

Time-Dependent Vacuum Energy Induced by {\mathcal{D}}–Particle Recoil

We consider cosmology in the framework of a material reference system of particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as 1/t ² for large times t. If this energy density is dominant, the Universe expands with a scale factor R(t)t ². We show that this possibility is compatible with recent observational constraints from high–redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology.     

Large-density fluctuations for the one-dimensional supercritical contact process

We consider the one-dimensional supercritical contact process. LetT _v be the first time the process reaches a densityq larger than the equilibrium one in the region [1N]. We prove that, starting from equilibrium,T _N /E(T _N ) converges to an exponential random time of mean one. In this way we extend previous results of Lebowitz and Schonmann.