Strong Correlations Between Fluctuations and Response in Aging Transport

Once the problem of ensemble averaging is removed, correlations between the response of a single molecule to an external driving field $F$, with the history of fluctuations of the particle, become detectable. Exact analytical theory for the continuous time random walk and numerical simulations for the quenched trap model give the behaviors of the correlation between fluctuations of the displacement in the aging period $(0,t_a)$, and the response to bias switched on at time $t_a$. In particular in the dynamical phase where the models exhibit aging we find finite correlations even in the asymptotic limit $t_a \to \infty$, while in the non-aging phase the correlations are zero in the same limit. Linear response theory gives a simple relation between these correlations and the fractional diffusion coefficient.Comment: 5 page

Stable Equilibrium Based on L\'evy Statistics: A Linear Boltzmann Equation Approach

To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass $M$, undergoing elastic collisions with ideal gas particles of mass $m$, in the Rayleigh limit $m<<M$. The probability density function (PDF) of the gas particle velocity is $f(\tilde{v}_m)$. Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle $W_{eq}(V_M)$. Depending on asymptotic properties of $f(\tilde{v}_m)$ we find that $W_{eq}(V_M)$ is either the Maxwell velocity distribution or a L\'evy distribution. In particular our results yield a generalized Maxwell distribution based on L\'evy statistics using two approaches. In the first a thermodynamic argument is used, imposing on the dynamics the condition that equilibrium properties of the heavy tracer particle be independent of the coupling $\epsilon=m/M$ to the gas particles, similar to what is found for a Brownian particle in a fluid. This approach leads to a generalized temperature concept. In the second approach it is assumed that bath particles velocity PDF scales with an energy scale, i.e. the (nearly) ordinary temperature, as found in standard statistical mechanics. The two approaches yield different types of L\'evy equilibrium which merge into a unique solution only for the Maxwell--Boltzmann case. Thus, relation between thermodynamics and statistical mechanics becomes non-trivial for the power law case. Finally, the relation of the kinetic model to fractional Fokker--Planck equations is discussed

CTRW Pathways to the Fractional Diffusion Equation

The foundations of the fractional diffusion equation are investigated based on coupled and decoupled continuous time random walks (CTRW). For this aim we find an exact solution of the decoupled CTRW, in terms of an infinite sum of stable probability densities. This exact solution is then used to understand the meaning and domain of validity of the fractional diffusion equation. An interesting behavior is discussed for coupled memories (i.e., L\'evy walks). The moments of the random walk exhibit strong anomalous diffusion, indicating (in a naive way) the breakdown of simple scaling behavior and hence of the fractional approximation. Still the Green function $P(x,t)$ is described well by the fractional diffusion equation, in the long time limit.Comment: 11 pages, 4 figure

1/fβ1/f^{\beta} noise for scale-invariant processes: How long you wait matters

We study the power spectrum which is estimated from a nonstationary signal. In particular we examine the case when the signal is observed in a measurement time window $[t_w,t_w+t_m]$, namely the observation started after a waiting time $t_w$, and $t_m$ is the measurement duration. We introduce a generalized aging Wiener-Khinchin theorem which relates between the spectrum and the time- and ensemble-averaged correlation function for arbitrary $t_m$ and $t_w$. Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging $1/f^{\beta}$ noise. We illustrate our general results with two-state renewal models with sojourn times' distributions having a broad tail

Nonergodisity of a time series obeying L\'evy statistics

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a L\'evy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.Comment: 15 pages, 10 figure