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1.
Summary We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isv
,
, a product measure with densities and to the left and right of the origin respectively, <, then there exists a (microscopic) shock for the system. A shock is a random positionX
t
such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim
t
t
–1(E(X
t
)2–(EX
t
)2) and findD=(p–q)(–)–1((1–)+(1–)) as conjectured by Spohn (1991). We show that in the scale
the position ofX
t
is determined by the initial distribution of particles in a region of length proportional tot. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and . This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration. 相似文献
2.
We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the tagged particle at time t depends on the initial configuration, through the number of empty sites in the interval [0,(p−q)αt] divided by α, on the hyperbolic time scale and on a longer time scale, namely N4/3. 相似文献
3.
4.
Tertuliano Franco Patrícia Gonçalves Adriana Neumann 《Stochastic Processes and their Applications》2013
We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn−β, with α>0, β∈[0,+∞] and n is the scaling parameter. Depending on the regime of β, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α, interpolating a fractional Brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero. 相似文献
5.
6.
7.
Summary. We consider asymmetric simple exclusion processes on the lattice Zopf;
d
in dimension d≥3. We denote by L the generator of the process, ∇ the lattice gradient, η the configuration, and w the current of the dynamics associated to the conserved quantity. We prove that the fluctuation–dissipation equation w=Lu+D∇η has a solution for some function u and some constant D identified to be the diffusion coefficient. Intuitively, Lu represents rapid fluctuation and this equation describes a decomposition of the current into fluctuation and gradient of
the density field, representing the dissipation. Using this result, we proved rigorously that the Green-Kubo formula converges
and it can be identified as the diffusion coefficient.
Received: 14 May 1996 / In revised form: 20 February 1997 相似文献
8.
Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process. 相似文献
9.
Yu Zhang 《Probability Theory and Related Fields》2006,136(2):298-320
We consider the first passage percolation model on Z
d
for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R
d
and
It is well known that if p < p
c
, there exists a compact shape B
d
⊂ R
d
such that for all
> 0,
t
B
d
(1 −
) ⊂ B(t) ⊂ tB
d
(1 +
) and G(t)(1 −
) ⊂ B(t) ⊂ G(t)(1 +
) eventually w.p.1. We denote the fluctuations of B(t) from tB
d
and G(t) by
In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB
d
) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the
paths themselves. This linearity is also independently interesting.
Research supported by NSF grant DMS-0405150 相似文献
10.
Chih-Chung Chang 《Probability Theory and Related Fields》1994,100(3):269-283
Summary The Boltzmann-Gibbs principle is known to be crucial in the study of the fluctuations of interacting particle systems. A new method is proposed in this paper which confirms this principle for models with gradient reversible dynamics in equilibrium. The method is simpler and can be applied to more general models than the conventional one which is developed by Brox et al. To illustrate the idea in more detail, we study the weakly asymmetric simple exclusion process of which the jump rates are slowly varying. As a consequence of the Boltzmann-Gibbs principle, the limit of the density fluctuation fields is identified as a generalized Ornstein-Uhlenbeck process. Finally, the extension to models with long range interactions is briefly discussed. 相似文献
11.
C. Bahadoran 《Probability Theory and Related Fields》1998,110(3):287-331
Summary. We prove hydrodynamical limit for spatially heterogeneous, asymmetric simple exclusion processes on Z
d
. The jump rate of particles depends on the macroscopic position x through some nonnegative, smooth velocity profile α(x). Hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form
To derive this result, we prove an alternative characterization of entropy solutions involving stationary solutions, and work
with macroscopically stationary states rather than the unknown stationary measures of the process. The method can be extended
to spatially heterogeneous, asymmetric misanthrope processes with slow birth and death.
Received: 11 November 1996/In revised form: 10 October 1997 相似文献
12.
Tertuliano Franco Patrícia Gonçalves Adriana Neumann 《Stochastic Processes and their Applications》2019,129(4):1413-1442
We consider a one-dimensional symmetric simple exclusion process in contact with slowed reservoirs: at the left (resp. right) boundary, particles are either created or removed at rates given by or (resp. or ) where and is a scaling parameter. We obtain the non-equilibrium fluctuations and from the latter we obtain also the non-equilibrium stationary fluctuations. 相似文献
13.
Sunder Sethuraman 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2007,43(2):215
Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given. 相似文献
14.
The Axelrod model is a spatial stochastic model for the dynamics of cultures which, similar to the voter model, includes social influence, but differs from the latter by also accounting for another social factor called homophily, the tendency to interact more frequently with individuals who are more similar. Each individual is characterized by its opinions about a finite number of cultural features, each of which can assume the same finite number of states. Pairs of adjacent individuals interact at a rate equal to the fraction of features they have in common, thus modeling homophily, which results in the interacting pair having one more cultural feature in common, thus modeling social influence. It has been conjectured based on numerical simulations that the one-dimensional Axelrod model clusters when the number of features exceeds the number of states per feature. In this article, we prove this conjecture for the two-state model with an arbitrary number of features. 相似文献
15.
Andreas Greven 《Acta Appl Math》1994,34(1-2):17-35
In this paper we consider the construction of couplings for Markovian evolutions on a state space of the formE
, with
(measurable) and a countable group (d for example). The evolutions we focus on are mainly systems of linearly interacting diffusions, withE compact. We explain and state properties of such couplings and show how they are used to obtain information on the behaviour of the evolution in finite time and as time tends to infinity. An important property of a coupling is to be a successful coupling. The latter concept is introduced here in the context of interacting systems, which is different from the classical concept for Markov chains or processes with state space d. The analysis of the question when a coupling is successful depends heavily on the structure of the interaction term and is investigated in detail. We formulate some open problems and conjectures.The paper puts in perspective the coupling statements appearing in the proofs of various results and is largely based on the works of Cox and Greven, Fleischmann and Greven, Dawson and Greven, Greven, and Cox, Greven and Shiga. 相似文献
16.
17.
Carl Graham 《Probability Theory and Related Fields》1995,101(3):291-302
Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity. 相似文献
18.
T. S. Mountford 《Probability Theory and Related Fields》1992,93(2):159-167
Summary We find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation. It is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line.Research partially supported by NSF DMS 9157461 and a grant from the Sloan Foundation 相似文献
19.
Hideki Tanemura 《Probability Theory and Related Fields》1996,104(3):399-426
Sumamry An infinite system of Skorohod type equations is studied. The unique solution of the system is obtained from a finite case by passing to the limit. It is a diffusion process describing a system of infinitely many Brownian hard balls and has a Gibbs state associated with the hard core pair potential as a reversible measure.On leave of, Department of Mathematics and Informatics, Faculty of Science Chiba University Chiba, 263 JapanSupported by Swiss National Foundation, contract Nr. 20-36305.92 相似文献