共查询到20条相似文献,搜索用时 0 毫秒
1.
Letw = {w(x)xZd} be a positive random field with i.i.d. distribution. Given its realization, letX
t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w–1(Xt). The processX={X
t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities
t
=(d/dt) E
(X
t
2
–M
–1
t and
t(w) = (d/dt)Ew(X
t
2
– M– 1t). Here Ew. is expectation overX at fixedw and E = Ew (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) limt td/2t= V2Md/2–3(d/2)d/2 and (2) limt td/4
st(w)= Zs weakly in path space, with {Zs:s>0} the Gaussian process with EZs=0 and EZrZs= V2Md/2–4(d)d/2 (r + s)–d/2. HereM and V2 are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since
, with
0 the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function. 相似文献
2.
Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed
with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast,
we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of
localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk,
we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for
all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement,
dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem
(the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking
features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive
regimes) is demonstrated. 相似文献
3.
Consider a simple random walk on
d
whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn
k
be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let
k
=E(n
k
)–q
–1. Our main result is a proof that (*) lim
k
k
d/2
k
= (1 –q)q
d/2 – 2(d/2)
d/2. Since it is known thatq
– 1
k
=E(n
1
n
k + 1 B) –E(n
1 B)E(n
k + 1 B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of
k
(1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of
k
for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of
k
cannot be faster than (*). 相似文献
4.
Michael Nauenberg 《Journal of statistical physics》1985,41(5-6):803-810
Applying scaling and universality arguments, the long-time behavior of the probability distribution for a random walk in a one-dimensional random medium satisfying Sinai's constraint is obtained analytically. The convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk. 相似文献
5.
David A. Croydon 《Journal of statistical physics》2009,136(2):349-372
We study the random walk X on the range of a simple random walk on ℤ
d
in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate
that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin. 相似文献
6.
Maury Bramson 《Journal of statistical physics》1991,62(3-4):863-875
We describe a family of random walks in random environment which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and are subdiffusive in any dimensiond<. The random environments have no potential ind>1. 相似文献
7.
J. K. E. Tunaley 《Journal of statistical physics》1974,11(5):397-408
Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed. 相似文献
8.
Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like
diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen
in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the
diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the
sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two
extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge
as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’
counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales
in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken
up in part 2. 相似文献
9.
We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and <0, where 0 is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system. 相似文献
10.
We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of
the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution
is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem
becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage
time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out. 相似文献
11.
The dynamical properties of one-dimensional random transverse Ising model (RTIM) with a double-Gaussian disorder is investigated by the recursion method. Based on the first twelve recurrences derived analytically, the spin autocorrelation function (SAF) and associated spectral density at high temperature were obtained numerically. Our results indicate that when the standard deviation σg (or OrB) of the exchange couplings Ji (or the random transverse fields Bi) is small, no long-time tail appears in the SAE The spin system undergoes a crossover from a central-peak behavior to a collectivemode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when σJ (or σB) is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large σJ or a disordered behavior for large σB. In this instance, SAFs exhibit a similar long-time tail, i.e., C(t) ~ t ^-2 for large t. Similar properties are obtained when Ji (or Bi) satisfy the double-exponential distribution or the double-uniform distribution. Besides, when both the standard deviations and the mean values of the exchange couplings are small, the effects of the Gaussian random bonds may drive the system undergo two crossovers from a triplet state to a doublet state, and then to a collective-mode state. 相似文献
12.
We construct diffusions in random velocity fields which present anomalous superdiffusive behavior. The mean square displacement can be made to have any power lawt
for 1<2. Higher moments and characteristic functions are also investigated. 相似文献
13.
14.
15.
We consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time. In spatial dimensionsd>2, we establish that the behavior is diffusive with probability one. The diffusion constant is not renormalized by the interaction. 相似文献
16.
We show that the random walk generated by a hierarchical Laplacian in d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions. 相似文献
17.
This paper considers the asymptotic distribution for the horizontal displacement of a random walk in a medium represented by a two-dimensional lattice, whose transitions are to nearest-neighbor sites, are symmetric in the horizontal and vertical directions, and depend on the column currently occupied. On either side of a change-point in the medium, the transition probabilities are assumed to obey an asymptotic density condition. The displacement, when suitably normalized, converges to a diffusion process of oscillating Brownian motion type. Various special cases are discussed. 相似文献
18.
We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. It was shown by Imbrie and Spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the Bernoulli distribution. Under the same assumption on the random environment as that of Imbrie and Spencer, we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment. More generally, we can prove that, if the random environment is bounded and if the supremum of the support of the distribution has a positive mass, then there is an integerd
0 such that in dimensions higher thand
0 the behavior of the random polymer is always diffusive. 相似文献
19.
Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (grant No. 476/82). 相似文献
20.
Pierre Vallois 《Physica A》2007,386(1):303-317
This paper considers a memory-based persistent counting random walk, based on a Markov memory of the last event. This persistent model is a different than the Weiss persistent random walk model however, leading thereby to different results. We point out to some preliminary result, in particular, we provide an explicit expression for the mean and the variance, both nonlinear in time, of the underlying memory-based persistent process and discuss the usefulness to some problems in insurance, finance and risk analysis. The motivation for the paper arose from the counting of events (whether rare or not) in insurance that presume that events are time independent and therefore based on the Poisson distribution for counting these events. 相似文献