Waiting time distribution

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

This expresses the waiting time distribution in terms of the sequence of distribution functions f of the random set. 

Remark. The following alternative derivation provides further insight. The random set of dots on the time axis can be visualized as an ensemble of numerous individual sample sets. From this ensemble extract the subensemble of those sample sets that have one dot between ra — dra and ta. This subensemble represents again a random set of dots. The quantities belonging to this new random set will be distinguished by a tilde. The required waiting time distribution w(9 ta) is the same as the quantity w(0 ta), i.e., the analog of (6.1) applied to the subensemble and with ta substituted for t0. [Note the semicolon in (6.1) versus the bar in (6.7) ]... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

The waiting time distribution can be analyzed in a simple model, assuming each channel obeys a simple exponential decay law for its switching between the open and closed states. Consider a single channel and let Po(t) and Pi (t) be the probability that the channel is open and closed, respectively, at a certain point t in time. Then Po and Pi evolve according to ... 

Note that in the limit q —> 0 of vanishing latency time, we obtain w(s) —> fo(s), as is to be expected. The only remaining problem then is to extract the waiting time distribution w(t) from its Laplace transform u>(s), a task complicated by the appearance of exponential factors in g< in the denominator in Eq. (22). 

In the special case where there are only two sites, the CTRW procedure, supplemented by the Poisson assumption of Eq. (69), yields the Pauli master equation of Eq. (2). This means that the Pauli master equation is compatible with a random picture, where a particle with erratic motion jumps back and forth from the one to the other state, with a condition of persistence expressed by the exponential waiting time distribution of Eq. (69). Recent fast technological advances are allowing us to observe in mesoscopic systems analogous intermittent properties, with distinct nonexponential distribution of waiting times. This is the reason why in this section we focus our attention on how to derive a v(/(t) with a non-Poisson character. 

The connection between experimental and theoretical waiting time distribution, established by Eq. (73) allows us to discuss the master equation of Section II from within the CTRW perspective. In Section IV we have studied the motion of a random walker jumping from one site to another of an infinitely extended lattice. However, the formalism adopted can be applied also to the case of only two sites, 1) and 2), which correspond to the physical condition... 

In fact, the waiting function of sojourn times in one site must be identified, according to the definitions of this subsection, with the experimental waiting time distribution. On the other hand, using Eq. (73), we obtain... 

In practice, to generate the time sequence x,, which plays a fundamental role for the dynamic approach to complexity discussed in this review, we do not run either the Manneville map or its idealized version. We draw the numbers y0 from a uniform distribution on the interval I, and we convert each of them into the corresponding x using Eq. (89). This turns out to be a procedure much faster than running a map, either the Manneville map or its idealized version. However, these maps are important, since they are an attractive example of dynamic model behind the waiting time distribution of Eq. (92). 

It is evident that the results of Section IV establishes a nice connection between the GME and the condition of subdiffusion explored with success in the last few years [43]. In fact, the more extended the sojourn time of a random walker on a given site, the slower the resulting diffusion process. Thus, if the exponential waiting time distribution of Eq. (69) is replaced by a slower decay, subdiffusion, namely a diffusion slower than ordinary diffusion, can emerge. 

The Levy walk is physically more plausible than the Levy flight. How to derive the Levy walk from a Liouville approach of the kind described in Section III Here, we illustrate a path explored some years ago, to establish a connection between GME and this kind of superdiffusion [49,50]. We assume that there exists a waiting time distribution v /(x), prescribed, for instance, by the dynamic model illustrated in Section V. This function corresponds to a distribution of uncorrelated times. We can imagine the ideal experiment of creating the sequence x,, by drawing in succession the numbers of this distribution. Then we create the fluctuating velocity E,(f), according to the procedure illustrated in Section V. 

Levy diffusion is a Markov process corresponding to the conditions established by the ordinary random walk approach with the random walker making jumps at regular time values. To explain why the GME, with the assumption of Eq. (112), yields Levy diffusion, we notice [50] that the waiting time distribution is converted into a transition probability n(x) through... 

The second term is responsible for the fluctuations of and the symbol R denotes the set of variables necessary to assign to the variable S, the proper intermittent properties. This is a crucial assumption. The model might rest, for instance, on a double-well potential, within which the variable E, moves, virtually attaining only the values corresponding to the bottoms of the two wells. The crucial issue is to make the distributions of time of sojourn at the bottom of these two wells distinctly non-Poisson and renewal at the same time. Here we limit ourselves to assuming that the theoretical waiting time distribution i(t) has the form of Eq. (92) and that /exp(f) is related to it via Eq. (73). In the specific case that we are here describing, a convenient form for the projection operator P is... 

It is now well understood that this is not an approximation rather it is a way to force an equation with infinite memory to become compatible with Levy diffusion. The assumption (152) makes it possible for us to get rid of the time convolution nature of the generalized diffusion equation (133). At the same time, this key relation replaces the correlation function (t) with its second-order derivative and, as a consequence of Eq. (147), with the waiting time distribution /( ). This fact is very important. In fact, any Liouville-like approach makes the correlation function 3F(f) enter into play. The CTRW is a perspective resting on trajectories and consequently on /(f). Establishing a connection between the two pictures implies the conversion of 4> (f) into j/(r), or vice versa. Here, this conversion has been realized paying the price of altering the physics of the generalized diffusion equation (133). 

The adoption of Eq. (195) yields an equilibrium distribution that is similar to the WS statistics, with the main difference, however, that the inverse power law is truncated by two peaks, at = W/y and = —W/y. Note that the Levy walk noise i (f) is generated according to the renewal prescriptions of Section VI that is, we use the waiting time distribution /(f) of Eq. (92) and, according to the... 

Let us imagine now that the same experiment is done, rather than at time t = 0, at a later time t = ta > 0, so that for any sequence we have to record the time at which the first failure occurs. The probability that the lamp under observation was turned on exactly at time t = ta is virtually zero. The lamp was turned on earlier, and, if ta is large, it is probable that many lamps have been switched on and have failed, prior to the unknown time at which the lamp under observation has been switched on. Thus we realize our histogram recording duration times that are shorter than the real ones. Nevertheless, because we are using the convenient normalization procedure, it is not impossible that the resulting waiting time distribution that we should denote as v may obey... 

This prediction has been confirmed by the results of Refs. 123 and 124. In fact, the numerical result of Ref. 123 indicates that the waiting time distribution of Eq. (274) has an exponential truncation, this being an effect of the tunneling from the boundary between chaotic sea and accelerator island, back to the chaotic sea. The authors of Refs. 31 and 122 argue that the quantum induced recovery of ordinary diffusion is followed by a corresponding localization process. 

In the case when either the barrier intensity Q [150] or temperature T [151] are slowly modulated, the resulting waiting time distribution becomes a superposition of infinitely many exponentials. At least since the important work of Shlesinger and Hughes [152], and probably earlier, it is known that a superposition of infinitely many exponentially decaying functions can generate an inverse power law. 

In conclusion, according to the modulation theory we write the waiting time distribution /(f) under the following form ... 

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