اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدModeling long-range memory with stationary Markovian processes
158
0
0.0
(
0
)
تأليف
Salvatore Miccich`e
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we give explicit examples of power-law correlated stationary Markovian processes y(t) where the stationary pdf shows tails which are gaussian or exponential. These processes are obtained by simply performing a coordinate transformation of a specific power-law correlated additive process x(t), already known in the literature, whose pdf shows power-law tails 1/x^a. We give analytical and numerical evidence that although the new processes (i) are Markovian and (ii) have gaussian or exponential tails their autocorrelation function still shows a power-law decay <y(t) y(t+T)>=1/T^b where b grows with a with a law which is compatible with b=a/2-c, where c is a numerical constant. When a<2(1+c) the process y(t), although Markovian, is long-range correlated. Our results help in clarifying that even in the context of Markovian processes long-range dependencies are not necessarily associated to the occurrence of extreme events. Moreover, our results can be relevant in the modeling of complex systems with long memory. In fact, we provide simple processes associated to Langevin equations thus showing that long-memory effects can be modeled in the context of continuous time stationary Markovian processes.
قيم البحث
اقرأ أيضاً
A class of non-local contact processes is introduced and studied using mean-field approximation and numerical simulations. In these processes particles are created at a rate which decays algebraically with the distance from the nearest particle. It i
s found that the transition into the absorbing state is continuous and is characterized by continuously varying critical exponents. This model differs from the previously studied non-local directed percolation model, where particles are created by unrestricted Levy flights. It is motivated by recent studies of non-equilibrium wetting indicating that this type of non-local processes play a role in the unbinding transition. Other non-local processes which have been suggested to exist within the context of wetting are considered as well.
The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaus
sian stationary processes in discrete time $n$. Few results are known for the persistence $P_0(n)$ in discrete time, except the large time behavior which is characterized by the nontrivial constant $theta$ through $P_0(n)sim theta^n$. Using a modified version of the Independent Interval Approximation (IIA) that we developed before, we are able to calculate $P_0(n)$ analytically in $z$-transform space in terms of the autocorrelation function $A(n)$. If $A(n)to0$ as $ntoinfty$, we extract $theta$ numerically, while if $A(n)=0$, for finite $n>N$, we find $theta$ exactly (within the IIA). We apply our results to three special cases: the nearest neighbor-correlated first order moving average process where $A(n)=0$ for $ n>1$, the double exponential-correlated second order autoregressive process where $A(n)=c_1lambda_1^n+c_2lambda_2^n$, and power law-correlated variables where $A(n)sim n^{-mu}$. Apart from the power-law case when $mu<5$, we find excellent agreement with simulations.
We propose a method to calculate the large deviations of current fluctuations in a class of stochastic particle systems with history-dependent rates. Long-range temporal correlations are seen to alter the speed of the large deviation function in anal
ogy with long-range spatial correlations in equilibrium systems. We give some illuminating examples and discuss the applicability of the Gallavotti-Cohen fluctuation theorem.
We present a treatment of non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of fractional master equation, as different memory functions in the Generalized Kolmo
gorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and for upto frst order expansion of stretched exponential function for very small $tau rightarrow 0$. A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.
We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state $sigma_i(t) in {0,1}$ of a cell $i$ does not only depend on the states in its local neighborhood at time $t-1$, but also on the memory of its own past s
tates $sigma_i(t-2), sigma_i(t-3),...,sigma_i(t-tau),...$. We assume that the weight of this memory decays proportionally to $tau^{-alpha}$, with $alpha ge 0$ (the limit $alpha to infty$ corresponds to the usual CA). Since the memory function is summable for $alpha>1$ and nonsummable for $0le alpha le 1$, we expect pronounced %qualitative and quantitative changes of the dynamical behavior near $alpha=1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance $H$ of initially close trajectories. We typically expect the asymptotic behavior $H(t) propto t^{1/(1-q)}$, where $q$ is the entropic index associated with nonextensive statistical mechanics. In all cases, the function $q(alpha)$ exhibits a sensible change at $alpha simeq 1$. We focus on the class II rules 61, 99 and 111. For rule 61, $q = 0$ for $0 le alpha le alpha_c simeq 1.3$, and $q<0$ for $alpha> alpha_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size $N$ indicate that the range of the power-law regime for $H(t)$ typically diverges $propto N^z$ with $0 le z le 1$. Similar studies have been carried out for other rules, e.g., the famous universal computer rule 110.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
