Continuous Time Limit Asymmetric Random Walk

Continuous-Time Random Walk

Considering the CTRW approximation of the charge carrier motion and the idea of a set of energy barriers exponentially distributed that control the jump frequency, the principal features of the anomalous transport may be accounted.

From: Supramolecular Photosensitive and Electroactive Materials , 2001

Continuous-Time Random Walk and Fractional Calculus

Hasan A. Fallahgoul , ... Frank J. Fabozzi , in Fractional Calculus and Fractional Processes with Applications to Financial Economics, 2017

Key points of the chapter

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The continuous-time random walk is an extension of the discrete random walk process.

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The difference between the continuous-time random walk and the discrete random walk processes is related to the waiting time to the next jump.

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In the continuous-time random walk process the waiting time random variables are independent and identically distributed.

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The probability density function for the continuous-time random walk process satisfies an integral equation.

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The probability density function for the continuous-time random walk process does not exist in closed-form, it can be obtained in an asymptotic form.

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Based on the correlation between waiting time random variables and jump variables, the continuous-time random walk process is divided into two cases: coupled, and uncoupled.

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Continuous-time random walk processes are used to model the dynamics of asset prices.

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The compound poisson process is special class of the continuous-time random walk processes where the distribution of the waiting time random variable is exponential.

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Diffusion Processes

Oliver C. Ibe , in Markov Processes for Stochastic Modeling (Second Edition), 2013

10.7.1 Fractional Diffusion and Continuous-Time Random Walk

Fractional diffusion equation can be derived from the continuous-time random walk (CTRW). Recall from Chapter 8 that CTRW is a random walk that permits intervals between successive walks to be independent and identically distributed. Thus, the walker starts at the point zero at time $T_0=0$ and waits until time $T_1$ when he makes a jump of size $x_1$ , which is not necessarily positive. The walker then waits until time $T_2$ when he makes another jump of size $x_2$ , and so on. The jump sizes $x_i$ are also assumed to be independent and identically distributed. Thus, we assume that the times $T_1,T_2,\dots$ are the instants when the walker makes jumps. The intervals ${\tau }_i=T_i-T_{i-1}$ , $i=1,2,\dots$ , are called the waiting times (or pausing times) and are assumed to be independent and identically distributed.

Let T denote the waiting time and let $X$ denote the jump size. Similarly, let $f_X(x)$ denote the PDF of $X$ , let $f_T(t)$ denote the PDF of T, and let $P(x,t)$ denote the probability that the position of the walker at time t is x, given that it was in position 0 at time $t=0$ ; that is,

(10.71) $P(x,t)=P[X(t)=x|X(0)=0]$

We consider an uncoupled CTRW in which the waiting time and the jump size are independent so that the master equation is given by:

(10.72) $P(x,t)=\delta (x)R_T(t)+{\int }_0^t{\int }_{-\infty }^{\infty }P(u,\tau )f_X(x-u)f_T(t-\tau )\mathrm{d}u\mathrm{d}\tau$

where $\delta (x)$ is the Dirac delta function and $R_T(t)=P[T>t]$ is called the survival probability, which is the probability that the waiting time when the process is in a given state is greater than t. The master equation states that the probability that $X(t)=x$ is equal to the probability that the process was in state 0 up to time t, plus the probability that the process was at some state u at time $\tau$ , where $0<\tau \le t$ , and within the waiting time $t-\tau$ , a jump of size $x-u$ took place. Note that

$\begin{array}{ll}\hfill R_T(t)& ={\int }_t^{\infty }{f}_T(v)\mathrm{d}v=1-{\int }_0^t{f}_T(v)\mathrm{d}v\hfill \\ \hfill f_T(t)& =-\frac{\mathrm{d}{R}_T(t)}{\mathrm{d}t}\hfill \end{array}$

When the waiting times are exponentially distributed such that $f_T(t)=\lambda {\mathrm{e}}^{-\lambda t}t>0$ , the survival probability is $R_T(t)={\mathrm{e}}^{-\lambda t}$ and satisfies the following relaxation ordinary differential equation:

(10.73) $\frac{\mathrm{d}}{\mathrm{d}t}{R}_T(t)=-\lambda R_T(t)t>0,R_T(0^{+})=1$

The simplest fractional generalization of Eq. (10.73) that gives rise to anomalous relaxation and power-law tails in the waiting time PDF can be written as follows:

(10.74) $\frac{{\mathrm{d}}^{\alpha }}{\mathrm{d}{t}^{\alpha }}{R}_T(t)=-R_T(t)t>0,0<\alpha <1;R_T(0^{+})=1$

where ${\mathrm{d}}^{\alpha }/d{t}^{\alpha }$ is the Caputo fractional derivative; that is,

(10.75) $\frac{{\mathrm{d}}^{\alpha }}{\mathrm{d}{t}^{\alpha }}{R}_T(t)=\frac{1}{\Gamma (1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^t\frac{R_T(u)}{{(t-u)}^{\alpha }}\mathrm{d}u-\frac{t^{-\alpha }}{\Gamma (1-\alpha )}{R}_T(0^{+})$

Taking the Laplace transform of Eq. (10.74) incorporating Eq. (10.75) we obtain

(10.76) $s^{\alpha }{\tilde{R}}_T(s)-s^{\alpha -1}{R}_T(0^{+})=-{\tilde{R}}_T(s)\Rightarrow {\tilde{R}}_T(s)=\frac{s^{\alpha -1}{R}_T(0^{+})}{1+s^{\alpha }}=\frac{s^{\alpha -1}}{1+s^{\alpha }}$

From Section 10.9.3 we have that the inverse Laplace transform of ${\tilde{R}}_T(s)$ is

(10.77) $R_T(t)=L^{-1}\left\{\frac{s^{\alpha -1}}{1+s^{\alpha }}\right\}=E_{\alpha }(-t^{\alpha })$

Thus, the corresponding PDF of the waiting time is

$f_T(t)=-\frac{\mathrm{d}{R}_T(t)}{\mathrm{d}t}=-\frac{\mathrm{d}{E}_{\alpha }(-t^{\alpha })}{\mathrm{d}t}=t^{\alpha -1}{E}_{\alpha ,\alpha }(-t^{\alpha })$

The Laplace transform is:

${\tilde{F}}_T(s)=\frac{1}{s^{\alpha }+1}0<\alpha \le 1$

The asymptotic behavior of the PDF of the waiting time is as follows:

$f_T(t)~\{\begin{array}{ll}{t}^{\alpha -1}\hfill & t\to 0\hfill \\ \hfill \frac{1}{t^{\alpha +1}}& t\to \infty \hfill \end{array}$

From this, we observe that for $0<\alpha <1$ at large t, the function $f_T(t)$ does not decay exponentially anymore; instead, it decays according to a power law. This means that because of the power-law asymptotic behavior of the process, it is no longer Markovian but of the long-memory type. Thus, the kind of diffusion that is associated with the CTRW depends on the distribution of the step increments. If the increments are small, we obtain normal diffusion. In this case, for jump sizes (or displacements) with finite variance ${\sigma }^2$ and waiting times with finite mean $\tau$ we have that

$\frac{\partial P(x,t)}{\partial t}=D\frac{{\partial }^{2}P(x,t)}{\partial x^2}$

with a diffusion coefficient $D={\sigma }^2/\tau$ . Assume that both $f_X(x)$ and $f_T(t)$ exhibit algebraic tail such that

$f_X(x)~|x{|}^{-(1+\beta )}\mathrm{and}{f}_T(t)~t^{-(1+\alpha )}$

for which ${\sigma }^2$ and $\tau$ are infinite. In this case we can derive a space-time fractional diffusion equation for the dynamics of $P(x,t)$ :

$\frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}=D_{\alpha ,\beta }\frac{{\partial }^{\beta }P(x,t)}{\partial |x{|}^{\beta }}$

where the constant $D_{\alpha ,\beta }$ is a generalized diffusion coefficient. Thus, the space-time fractional diffusion equation is obtained by replacing the first-order time derivative and second-order space derivative in the standard diffusion equation by a fractional derivative of order $\alpha$ and $\beta$ , respectively.

As stated earlier, if we limit the power-law distribution to the waiting time, we obtain the time-fractional diffusion equation of the form:

$\frac{{\partial }^{\alpha }P(x,t)}{\partial t^{\alpha }}=D_{\alpha }\frac{{\partial }^{2}P(x,t)}{\partial x^2}$

When $\alpha =1$ , we obtain the classical diffusion equation. Similarly, if only the jump size (or displacement) has power-law distribution, we obtain the space-fractional diffusion equation of the form:

$\frac{\partial P(x,t)}{\partial t}=D_{\beta }\frac{{\partial }^{\beta }P(x,t)}{\partial x^{\beta }}$

In this case, when $\beta =2$ , we obtain the classical diffusion equation. As stated earlier, the Caputo fractional derivative is generally used to solve FDEs because it allows traditional initial and boundary conditions to be included in the formulation in a standard way, whereas models based on other fractional derivatives may require the values of the fractional derivative terms at the initial time.

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Random Walk

Oliver C. Ibe , in Markov Processes for Stochastic Modeling (Second Edition), 2013

8.15.1 The Master Equation

The relationship between $P(x,t)$ and $f_{\mathrm{\Theta }T}(\theta ,t)$ is given by the following equation that is generally called the master equation of the CTRW:

(8.44) $P(x,t)=\delta (x)R(t)+{\int }_0^t{\int }_{-\infty }^{\infty }P(u,\tau )f_{\mathrm{\Theta }T}(x-u,t-\tau )\mathrm{d}u\mathrm{d}\tau$

where $\delta (x)$ is the Dirac delta function and $R(t)=P[T>t]=1-F_T(t)$ is called the survival probability, which is the probability that the waiting time when the process is in a given state is greater than t. The equation states that the probability that $X(t)=x$ is equal to the probability that the process was in state 0 up to time t, plus the probability that the process was at some state u at time $\tau$ , where $0<\tau \le t$ , and within the waiting time $t-\tau$ , a jump of size $x-u$ took place. Note that

$\begin{array}{cc}R(t)& ={\int }_t^{\infty }{f}_T(v)\mathrm{d}v=1-{\int }_0^t{f}_T(v)\mathrm{d}v\\ f_T(t)& =-\frac{\mathrm{d}R(t)}{\mathrm{d}t}\end{array}$

For the uncoupled CTRW, the master equation becomes

(8.45) $P(x,t)=\delta (x)R(t)+{\int }_0^t{\int }_{-\infty }^{\infty }P(u,\tau )f_{\mathrm{\Theta }}(x-u)f_T(t-\tau )\mathrm{d}u\mathrm{d}\tau$

Let the joint Fourier–Laplace transform of $P(x,t)$ be defined as follows:

(8.46) $P^{*}(w,s)={\int }_0^{\infty }{\mathrm{e}}^{-st}\mathrm{d}t{\int }_{-\infty }^{\infty }{\mathrm{e}}^{jwx}P(x,t)\mathrm{d}x$

Then the master equation is transformed into

$\begin{array}{cc}{P}^{*}(w,s)& ={\int }_0^{\infty }{\mathrm{e}}^{-st}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{jwx}[\delta (x)R(t)+{\int }_0^t{\int }_{-\infty }^{\infty }P(u,\tau )f_{\mathrm{\Theta }T}(x-u,t-\tau )\mathrm{d}u\mathrm{d}\tau ]\mathrm{d}x\mathrm{d}t\\ & =\tilde{R}(s)+P^{*}(w,s){\Phi }_{\mathrm{\Theta }T}(w,s)\end{array}$

where $\tilde{R}(s)$ is the Laplace transform of $R(t)$ . This gives

(8.47) $P^{*}(w,s)=\frac{\tilde{R}(s)}{1-{\Phi }_{\mathrm{\Theta }T}(w,s)}$

where ${\Phi }_{\mathrm{\Theta }T}(w,s)$ is the joint Fourier–Laplace transform of $f_{\mathrm{\Theta }T}(\theta ,\tau )$ . For an uncoupled CTRW, ${\Phi }_{\mathrm{\Theta }T}(w,s)={\Phi }_{\mathrm{\Theta }}(w){\Phi }_T(s)$ . Thus,

(8.48) $P^{*}(w,s)=\frac{\tilde{R}(s)}{1-{\Phi }_{\mathrm{\Theta }}(w)\Phi {}_T(s)}$

Thus, for the uncoupled CTRW we have that

(8.49) $P^{*}(w,s)=\frac{\tilde{R}(s)}{1-{\Phi }_{\mathrm{\Theta }}(w){\Phi }_T(s)}=\frac{1-\Phi (s)T}{s[1-{\Phi }_{\mathrm{\Theta }}(w){\Phi }_T(s)]}$

Because $|{\Phi }_{\mathrm{\Theta }}(w)|<1$ if $w\ne 0$ and $|\Phi {}_T(s\}|<1$ if $s\ne 0$ , we can rewrite Eq. (8.48) as follows:

(8.50) $P^{*}(w,s)=\tilde{R}(s)\sum _{n=0}^{\infty }{[{\Phi }_{\mathrm{\Theta }}(w)\Phi {}_T(s)]}^n$

Taking the inverse Fourier and Laplace transforms we obtain:

(8.51) $P(x,t)=\sum _{n=0}^{\infty }P(n,t)f_{\mathrm{\Theta }}^{(n)}(x)$

where $P(n,t)$ is the probability that n jumps occur up to time t, and $f_{\mathrm{\Theta }}^{(n)}(x)$ is the n-fold convolution of the number of jumps. $P(n,t)$ is given by

$P(n,t)={\int }_0^t{f}_T^{(n)}(t-u)R(u)\mathrm{d}u$

where $f_T^{(n)}(t)$ is the n-fold convolution of the PDF of the waiting time.

Consider the special case where T is exponentially distributed with a mean of $1/\lambda$ , that is,

$f_T(t)=\lambda {\mathrm{e}}^{-\lambda t}t\ge 0$

This means that

$R_T(t)=e^{-\lambda t}t\ge 0$

In this case, we have that ${\Phi }_T(s)=\lambda /(s+\lambda )$ . Thus, for the uncoupled CTRW we obtain

(8.52) $P^{*}(w,s)=\frac{1-\lambda /(s+\lambda )}{s[1-\lambda {\Phi }_{\mathrm{\Theta }}(w)/(s+\lambda )]}=\frac{1}{s+\lambda -\lambda {\Phi }_{\mathrm{\Theta }}(w)}$

Taking the inverse Laplace transform for this special case we obtain

(8.53) $P(w,t)=\exp \{-\lambda t[1-{\Phi }_{\mathrm{\Theta }}(w)\left]\right\}$

Note that for the special case when $N(t)$ is a Poisson process, the CTRW becomes a compound Poisson process and the analysis is simplified as the characteristic function of $X(t)$ becomes

(8.54) ${\Phi }_{X(t)}(w)=\exp \{-\lambda t[1-{\Phi }_{\mathrm{\Theta }}(w)\left]\right\}=P(w,t)$

which is not surprising because the fact that $N(t)$ is a Poisson process implies that T is exponentially distributed. The inverse Laplace transform of $P^{*}(w,s)$ for the uncoupled system is given by

(8.55) $P(w,t)={\Phi }_{X(t)}(w)=G_{N(t)}({\Phi }_{\mathrm{\Theta }}(w))$

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Applications of Fractional Processes

Hasan A. Fallahgoul , ... Frank J. Fabozzi , in Fractional Calculus and Fractional Processes with Applications to Financial Economics, 2017

8.4 Order Arrival Processes

An important area where fractional processes find application is the modeling of the order arrival process. This topic has become ever more interesting with the diffusion of high- and ultrahigh-frequency data and with high-frequency trading. The availability of data has enabled the empirical study of the econometrics of high frequency data while high frequency trading has provided a powerful economic motivation for their study.

Ultrahigh-frequency data are tick-by-tick data. As orders and trading occur at random times, tick-by-tick data are not classical time series but point processes. The representation of tick-by-tick data includes a representation of the orderarrival process and a representation of the magnitude of the transaction.

Scalas et al. (2000) and Mainardi et al. (2000) were the first to apply tick-by-tick data to the formalism of Continuous-Time Random Walks (CTRW) developed in Montroll and Weiss. A CTRW is a random walk where both the length of the time between two steps and the magnitude of each step are random. Call t_i, x_i the time and magnitude of the i-th transaction, call τ_i   = t _{i  +   1}  − t_i, ξ_i   = x _{i  +   1}  − x_i the waiting times and the magnitude of jumps, and call φ (ξ, τ) the joint density of jumps and waiting times and p (x, t) the joint probability that the diffusing quantity be at position x at time t. Montroll and Weiss (1965) demonstrated that the Laplace-Fourier transform of p has the following form:

$\tilde{p}\left(\kappa ,s\right)=\frac{1-\tilde{\psi }\left(s\right)}{s}\frac{1}{1-\tilde{\phi }\left(\kappa ,s\right)}$

where $\tilde{p}\left(\kappa ,s\right)$ is the Laplace-Fourier tranform of $p\left(x,t\right);\tilde{\psi }\left(s\right)$ is the Laplace transform of the waiting time pdf $\psi \left(\tau \right)={\displaystyle \int \phi \left(\xi ,\tau \right)d\xi }$ .

Scalas et al. (2000) demonstrated that in the hydrodynamic limit (i.e., long waiting times and long jumps), assuming that:

$\tilde{\phi }\left(\kappa ,s\right)=1-s^{\beta }-\left|\kappa \right|{}^{\alpha },0<\beta \le 1,0<\alpha \le 2$

the density p (x, t) obeys the following fractional partial differential equation:

$\frac{{\partial }^{\beta }p\left(x,t\right)}{\partial t^{\beta }}=\frac{{\partial }^{\alpha }p\left(x,t\right)}{\partial \left|x\right|{}^{\alpha }}+\frac{t^{-\beta }}{\mathrm{\Gamma }\left(1-\beta \right)}\delta \left(x\right)\text{.}$

Empirical tests of the above formalism in different markets can be found in Scalas et al. (2004), Scalas (2006), Politi and Scalas (2007), Politi and Scalas (2008), and Sazuka et al. (2009).

The formalism of CTRW assumes that the waiting times are sequences of independent and identically distributed random variables. This assumption might be too restrictive. Engle and Russell (1998) introduced the Autoregressive Conditional Duration (ACD) model. The ACD model has been followed by many similar models where conditional duration or some function of duration is modeled as an autoregressive process. However, it has been observed empirically that the decay of the autocorrelation function of the duration process can be very slow. To model this behavior, Jasiak (1999) introduced the Fractionally Integrated Autoregressive Conditional Duration (FIACD) model. The formalism is similar to the FIGARCH described above.

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REACTIONS IN DISORDERED MEDIA MODELLED BY FRACTALS

A. BLUMEN , ... G. ZUMOFEN , in Fractals in Physics, 1986

2 FRACTALS IN TIME AND SPACE

The basic feature of fractals is their self-similarity, ¹⁴ i.e. their invariance (either individually, or in their ensemble) under the group of dilatation operations. Many examples for geometrical (i.e. spatial) fractals are presented in these proceedings, so that we find it advisable to start our considerations with the temporal ones.

As stressed recently, the continuous time-random walk (CTRW) treatment of charge transport in amorphous materials, (such as used by Scher and Montroll ⁴ ) is based on a fractal set of event times. ^{15, 16} Let ψ(t) be the probability density that an event occurs at time t after the previous event has taken place. A simple example is the Poisson process:

(4) ${\psi }^{\text{P}}(\text{t})=\text{b}\text{exp}(-\text{bt})$

One constructs now readily a dilatationalIy symmetric distribution, by taking in the following manner account of events occuring on all time scales: ¹⁶

(5) $\psi (\text{t})=\frac{1-\text{N}}{\text{N}}\underset{\text{j}=1}{\overset{\infty }{\Sigma }}{\text{N}}^{\text{j}}{\text{b}}^{\text{j}}\text{exp}(-{\text{tb}}^{\text{j}})$

with N<1. As is evident, the distribution (5) is a normalized sum of Poisson-terms and

(6) $\psi (\text{bt})=\psi (\text{t})/(\text{Nb})-(1-\text{N})\text{exp}(-\text{tb})/\text{N}$

For later applications we need b<N, so that b<1 and thus at longer times ψ(bt) ≃ ψ(t)/Nb. The last expression is equivalent to

(7) $\psi (\text{t})\sim \frac{1}{{\text{t}}^{1+\gamma }}$

with γ = ℓnN/ℓnb. Equation (7) shows directly the temporal scaling of ψ(t), i.e. its fractal nature in time.

To show that Eq. (5) arises very naturally let us consider the distribution of carrier release times from low-lying traps to the conduction band. ¹⁷ This distribution is fundamental in the multiple trapping (MT)-formalism. For activated processes the rates depend exponentially on the energy, so that an equidistant level spacing E_j = jΔE leads to rates proportional to exp (-E_j/kT) = b^j, with b = exp (-ΔE/kT). Furthermore the density of states in the energy tail is often itself exponential in energy, exp (-E_j/kT_o) (where we introduced an effective temperature T_o) so that the density of states follows N^j with N = exp (-ΔE/kT_o). Thus, in this example γ = ℓnN/ℓnb = T/T_o for T<T_o, and one has dispersive transport below T_o.

Scaling also carries over to quantities related to ψ(t). Let χ_n(t) denote the probability that exactly n events occured in time t. This basic quantity of the CTRW-formalism is simply related to ψ(t) via its Laplace-transform:

(8) $L[{\text{x}}_{\text{n}}(\text{t})]\equiv {\text{x}}_{\text{n}}(\text{u})={[\psi (\text{u})]}^{\text{n}}[1-\psi (\text{u})]/\text{u}$

where ψ(u) = ℒ[ψ(t)].

In Fig. 1 we present the set of χ_n(t) for the Poisson process. Eq. (4). The curves show pronounced maxima which, with increasing n shift to longer times. From the log-log plot no scaling is evident.

FIGURE 1. Probabilities χ_n(t) that exactly n steps occurred during t for a Poisson waiting time distribution, ψ^P(t), Eq. (4), and ψ^P(τ_e) = 1/e.

To display scaling we have to study ψ(t)-forms which behave algebraically at long times. A typical example is the function ψ₂(t), wich belongs to the family of functions defined through: ^{4, 18}

(9) ${\psi }_{\text{n}}(\text{t})=C_{\text{n}}{\text{a}}^2[\text{exp}({\text{a}}^2\text{t})]{\text{i}}^{\text{n}}\text{erfc}({\text{at}}^{1/2})$

where the iⁿ erfc(z) are repeated integrals of the error function and the c_n are normalization constants. The function ψ₂(t) has no first moment, ψ₂(u) = (1+u^1/2/a)⁻², i.e. γ = 0.5 in Eq. (7). In Fig. 2 we show the corresponding χ_n(t). As is evident by inspection, the curves scale very well at long times, and their slope is also given by γ = 0.5. Indeed, for qualitative arguments one may well approximate the χ_n(t) through χ₀(t) in the long-time regime, and we will use this property further in the text.

FIGURE 2. Same as in Fig. 1, where the waiting-time distribution ψ₂(t) is given by Eq. (9), and ψ₂(τ_e) = 1/e.

We now turn our attention to the more common geometrical fractals. Many stochastically disordered structures, like percolation clusters at criticality, aggregates constructed by diffusion-limited growth, linear and branched polymers, epoxy-resins and various porous materials have been characterized as being self-similar under length scaling. ¹⁹ As we know, however, fractals need not be stochastic. Well-known examples are the deterministicalIy built Sierpinski-gaskets, whose generators in d-dimensional Euclidean spaces are hypertetrahedrons consisting of d+1 hypertetrahedra of sidelength downscaled by a factor of two. ¹⁴ Extensions obtain by changing the generator forms, both by choosing other scaling factors than 2 and by varying the coverage of the original tetrahedron. ^{20, 21} This procedure allows to obtain pseudo-Sierpinski structures with prespecified fractal dimensions. Additonally, one may also obtain deterministic fractals by a direct set-product; thus a Sierpinski gasket multiplied by a linear chain gives rise to a Tobleronestructure. ^{21, 22}

After specifying the fractals in time and in space we are now ready to also connect the two aspects. As already mentioned, spatial disorder is often accompanied by temporal disorder, and both facets appear in the study of dynamical processes, such as transient flow through porous rocks and migration over percolative systems, in which the site energies or the interactions (barriers) are randomly distributed. In previous work we have extended the CTRW to fractal lattices and have studied for several classes of waiting-time distributions ψ(t) the mean squared displacement <r ² (t)> and the decay of the population of the walkers due to trapping. We have shown that the interplay of the two stochastic (spatial and temporal) aspects leads to interesting new behaviors. ^{13, 23}

As an example consider the mean squared displacement of a walker on a geometrical fractal, when the waiting-times between steps are fixed. One has ²⁴

(10) $<{\text{r}}_{\text{n}}^2>\sim {\text{n}}^{\tilde{\text{d}}/\overline{\text{d}}}$

where n is the number of steps and $\tilde{\text{d}}$ and $\tilde{\text{d}}$ are the fractal and the spectral (fracton) dimension, respectively. The CTRW analog of Eq. (10) is

(11) $<{\text{r}}^2(\text{t})>=\underset{\text{n}=0}{\overset{\infty }{\Sigma }}<{\text{r}}_{\text{n}}^2>{\text{x}}_{\text{n}}(\text{t})$

where the χ_n(t) is the probability of having performed exactly n steps during the time t, which probability is given in the Laplace-domain by Eq. (8). In general, Eq. (11) leads to complex expressions. However, for fractals the pattern followed by Eq. (11) is readily obtainable. For χ_n(t) which scale at long times (see Fig. 2) one has for a fixed t value:

(12) $x_{\text{n}}(\text{t})\simeq \{\begin{array}{cc}{\text{x}}_0(\text{t})& \text{for}\text{n}<{\text{n}}_{\text{max}}(\text{t})\\ 0& \text{otherwise}\end{array}$

Using the normalization relation for the χ_n(t) and Eq. (12) one retrieves the time-dependence of n_max:

(13) $1=\underset{\text{n}=0}{\overset{\infty }{\Sigma }}{\text{x}}_{\text{n}}(\text{t})\simeq \underset{n=0}{\overset{{\text{n}}_{\text{max}}}{\Sigma }}{\text{x}}_0(\text{t})={\text{x}}_0(\text{t}){\text{n}}_{\text{max}}(\text{t})$

and hence, for χ₀(t) ∼ t^−γ one has n_max ∼ t^γ.

The same argument applied to Eq. (11) gives:

(14) $<{\text{r}}^2(\text{t})>\sim \underset{\text{n}=0}{\overset{{\text{n}}_{\text{max}}}{\Sigma }}{\text{n}}^{\tilde{\text{d}}/\overline{\text{d}}}{\text{x}}_0(\text{t})\sim {\text{x}}_0(\text{t}){\text{n}}_{\text{max}}^{1+\tilde{\text{d}}/\overline{\text{d}}}$

which, with help of Eq. (13), is:

(15) $<{\text{r}}^2(\text{t})>/\sim {\text{n}}_{\text{max}}^{\tilde{\text{d}}/\overline{\text{d}}}\sim t^{\gamma (\tilde{\text{d}}/\overline{\text{d}}}$

thus establishing the subordination ¹³ (i.e. the multiplicative behavior of the exponents) for the two kinds of disorder.

A similar argument applies also to S(t), the mean number of sites visited by the walker in time t. For walks with fixed waiting-times one has ${\text{S}}_{\text{n}}\sim {\text{n}}^{\tilde{\text{d}}/2}$ for $\tilde{\text{d}}<2$ and S_n ∼n otherwise. Hence

(16) $\text{S}(\text{t})\sim \{\begin{array}{cc}{\text{t}}^{\gamma \tilde{d}/2}& \text{for}\tilde{\text{d}}<2\\ {\text{t}}^{\gamma }& \text{for}\tilde{\text{d}}>\text{2}\end{array}$

i.e. another example for subordination. ¹³

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FRACTAL-LIKE EXCITON DYNAMICS: GEOMETRICAL AND ENERGETICAL DISORDER

Raoul KOPELMAN , in Fractals in Physics, 1986

3 RESULTS AND DISCUSSION

We see (Table 1) that all samples exhibit a fractal-like (heterogeneous) behavior (h > 0) at low temperatures. There is a reduction in h at higher temperatures. This is consistent with the subordination theorem: ¹⁴ d′_s = βd_s where β is the parameter characterizing the anomalous hopping time distribution (e.g. for continuous time random walk). We also see that some samples have h → 0 for higher temperatures. This implies that the "fractal-like" effects are entirely due to energy disorder, so that at higher temperatures the energy disorder parameter W is small compared to T. For other samples, the effective geometry may indeed be fractal-like at room temperature. Obviously our d′_s provide only lower limits for the "real" (geometric) d_s. We note that for vycor porous glass (at a low temperature) d′_s ≈ 1.1, which is consistent with the literature value ¹⁵ of d_f $\underset{˜}{<}$ 2 (remembering that for fractals d_s ≤ d_f ≤ d where d is the embedding Euclidean dimension). We emphasize that in all these samples, and especially in the thin films, most of the bulk might be quite crystalline (i.e., Euclidean), but our slow experimental time scales assure us that only the sluggish kinetics in the fractal-like regions do contribute to our observations.

For the larger pore membranes, at higher temperatures, we observe the same behavior as in deposited films, i.e., no delayed fluorescence on the millisecond time-scale. In short, this approach might characterize crystalline domain sizes. In addition, we have found excellent correlation between the fractal-like kinetics and a number of spectroscopic features: 1) The spectral bandwidths (W); 2) Observation of super-trap (betamethylnaphthalene) emissions; 3) Typical photophysical product (excimer) emissions; 4) Typical photochemical product (radical) emissions.

We have certainly established a correlation between fractal-like exciton annihilation kinetics and the geometrical constrains and/or energetic disorder of a number of samples of pure naphthalene. We have shown an analogous behavior to that of isotopic mixed naphthalene crystals (percolation clusters), in contrast to the behavior of perfect, pure naphthalene crystals. Further work is in progress with the aim to unscramble the contributions of fractal-like geometry from those of energy disorder (fractal-like behavior on the potential energy surfaces or a "fractal" waiting-time distribution of the excitation hopping).

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Mathematical Statistical Physics

Gérard Ben Arous , Jiří Černý , in Les Houches, 2006

4.2 Scaling limit

Observe that the Bouchaud model (for a = 0) can be expressed as a time change of the simple random walk. The time-change process is crucial for us:

Definition 4.3.

Let S(0) = 0 and let S(k), k ∈ ℕ, be the time of the k ^th jump of X. For s ∈ ℝ we define S(s) = S(⌊s⌋). We call S(s) the clock process. Obviously, X(t) = Y(k) for all S(k) ≤ t < S(k + 1).

The following result shows that the limit of the d-dimensional Bouchaud model and its clock process on ℤ ^d (d ≥ 2) is trivial, in the sense that it is identical with the scaling limit of the much simpler ("completely annealed") dynamics of the CTRW.

Theorem 4.4 (Scaling limit of BTM on ℤ ^d ).

Let

(4.7) $f\left(n\right)=\{\begin{array}{ll}{n}^{\alpha /2}{\left(logn\right)}^{\left(1-\alpha \right)/2},\hfill & ifd=2,\hfill \\ n^{\alpha /2},\hfill & ifd\ge 3.\hfill \end{array}$

Then for all d ≥ 2 and for a.e. τ,

(4.8) $\frac{C_d\left(\alpha \right)X\left(nt\right)}{f\left(n\right)}\stackrel{n\to \infty }{⟶}{\Psi }_d\left(t\right)and\frac{S\left(C_d{\left(\alpha \right)}^{-2}f{\left(n\right)}^{2}s\right)}{n}\stackrel{n\to \infty }{⟶}V\left(s\right).$

weakly in the Skorokhod topology on D([0, T], ℝ ^d ) (the space of cadlag functions from [0, T] to ℝ ^d ). If G_d (0) denotes Green's function of the d-dimensional random walk at 0, then

(4.9) $C_d\left(\alpha \right)=\{\begin{array}{ll}\sqrt{2{\pi }^{1-\alpha }\Gamma \left(1-\alpha \right)\Gamma \left(1+\alpha \right)},\hfill & ifd=2,\hfill \\ \sqrt{d{G}_d{\left(0\right)}^{\alpha }\Gamma \left(1-\alpha \right)\Gamma \left(1+\alpha \right)},\hfill & ifd\ge 3.\hfill \end{array}$

The main ideas of the proof of this theorem will be explained in Section 4.4. At this place, let us only compare the fractional-kinetics process Ψ _d with the F.I.N. diffusion Z. Both these processes are defined as a time change of the Brownian motion B_d (t). The clock processes however differ considerably. For d = 1, the clock equals ϕ(t) = ∫ ℓ(t, x)ρ(dx), where ρ is the random speed measure obtained as the scaling limit of the environment. Moreover, since ℓ is the local time of the Brownian motion B ₁, the processes B ₁ and ϕ are dependent.

For d ≥ 2, the Brownian motion B_d and the clock process, i.e. the stable subordinator V, are independent. The asymptotic independence of the clock process S and the location Y of the BTM is a very remarkable feature distinguishing d ≥ 2 and d = 1. It explains the "triviality" of the scaling limit in dimension d ≥ 2, but is, by no means, trivial matter to prove. We will come back to an intuitive explanation of the independence in Section 5.3. Note also that nothing like a scaling limit of the random environment appears in the definition of Ψ _d , moreover, the convergence holds τ-a.s. The absence of the scaling limit of the environment in the definition of Ψ _d transforms into the non-Markovianity of Ψ _d . Note however that it is considerably easier to control the behaviour of Ψ _d than of Z even if Ψ _d is not Markov: many quantities related to Ψ _d can be computed explicitly, as can be seen from Proposition 4.2.

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Neutron Scattering - Applications in Biology, Chemistry, and Materials Science

Tilo Seydel , in Experimental Methods in the Physical Sciences, 2017

2.3.4 Fractional Generalization of the Diffusion Equation

In biological macromolecules such as proteins, internal molecular motions are rarely of a simple Brownian type. Complex relaxation modes rather manifest themselves in decay functions that deviate from the simple Debye-relaxation, equation (19). Possible generalizations and extensions are given by the Kohlrausch-Williams-Watts stretched exponential function [13, 63] (cf. Equation (5) and Volume 44, Section 6.8.1, of this book series [11])

(30) $\begin{array}{l}\hfill G(t)\propto exp(-t^{\beta })\end{array}$

with 0 ≤ β ≤ 1, and by the inverse power law

(31) $\begin{array}{l}\hfill G(t)\propto \frac{1}{{(1+t)}^{\alpha }}.\end{array}$

Beyond these rather empirical models, an extension of the models describing diffusion has been pursued. Brownian diffusion models are based on Markovian random walk models, which assume that the diffusive process at any moment in time contains no memory of its preceding states. Later, a systematic generalization of the models describing the "random walks" of the diffusing particles in space and time has been achieved. This has led to so-called continuous time random walk models [64]. To this effect, the diffusion equation (eq. (8)) has been generalized to contain noninteger derivatives in time [65],

(32) $\begin{array}{l}\hfill {}^C{D}^{\alpha }G(r,t)=K_{\alpha }{\nabla }^{2}G(r,t),\end{array}$

where G(r, t) is the van Hove autocorrelation function and

(33) $\begin{array}{l}\hfill {}^C{D}^{\alpha }\left[\mathrm{\Phi }\right](t)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t\frac{{\mathrm{\Phi }}^{\prime }(\xi )}{{(t-\xi )}^{\alpha }}d\xi \end{array}$

(where ^C denotes "Caputo"-type) is the operator defining the generalized fractional derivative of the function Φ = Φ(t). Therein, Φ′ denotes the first-order integer derivative of Φ and Γ(x) is the gamma function (not to be confused with the Lorentzian spectral linewidth also being assigned the symbol Γ in different contexts).

(34) $\begin{array}{l}\hfill K_{\alpha }=\frac{{\sigma }_c^2}{{\tau }_c^{\alpha }}\end{array}$

is the generalized fractional diffusion coefficient. Therein, σ _c and τ _c are the characteristic spatial and time scales, respectively, and the scalar 0 < α ≤ 1 is a real number interpreted as the memory parameter of the system. For α = 1, the diffusion would be of the classical Fickian type (see equation (8)). In an analog way, the diffusion equation may be further generalized to contain noninteger derivatives in space, but we will not discuss this possibility here. The model of fractional diffusion has been successfully applied for instance to describe internal molecular relaxations in proteins [66, 67] as well as to describe the intermediate-time diffusion of proteins in polymeric solutions [68].

The spatial Fourier transform of equation (32) is solved by the Mittag-Leffler function and thus gives the intermediate scattering function due to fractional diffusion as

(35) $\begin{array}{l}\hfill I(q,t)\propto E_{\alpha ,1}\left(-{\left|\frac{t}{\tau }\right|}^{\alpha }\right)\text{with}\tau ={\left(K_{\alpha }{q}^2\right)}^{-1/\alpha }\end{array}$

with the fractional relaxation time τ and the Mittag-Leffler function

(36) $\begin{array}{l}\hfill E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }\frac{z^n}{\mathrm{\Gamma }(\alpha n+\beta )}.\end{array}$

The Mittag-Leffler function (cf. figure 6) constitutes the generalization of the exponential function $exp(z)=E_{1,1}(z)$ , as it can be easily seen that equation (36) for α = 1, β = 1 is the Taylor series of the exponential function. Remarkably, the relation $\tau ={\left(K_{\alpha }{q}^2\right)}^{-1/\alpha }$ between the generalized diffusion coefficient and the fractional relaxation time fully describes the dependence of the measured spectra on the scattering vector and therefore links the spatial and temporal components of the model. It can be seen as a generalization of the simple Brownian diffusion τ = (Dq ²)⁻¹ and has been verified for the first time in neutron spectra from silk fibers [30]. In equation (5), the Kohlrausch-Williams-Watts model is consequently replaced by the Fourier transform of the Mittag-Leffler function to establish the fractional generalization of the scattering function. The approach of describing diffusion using the fractional calculus is mathematically elegant. Moreover, this approach can describe the dynamics in suitable complex systems by just two characteristic quantities, namely K _α and α. It is the dependence of these quantities on system parameters such as the tensile stress on a biological fiber that then provides very useful insights. Moreover, recent molecular dynamics simulations of several different prototypical proteins support the view that the internal molecular motions of individual proteins can quite generally be interpreted in terms of models of fractional diffusion over a very wide range of time scales [69]. However, as a general criticism to the fractional dynamics approach it may be noted that it does not convey direct specific information on the confinement geometry of the molecular diffusive processes, as opposed to the approach described in section 2.3.3. Concluding this paragraph, we remark that the terms "fractional" and "fractal" are not to be confused. Nevertheless, while "fractional" essentially refers to the derivatives of noninteger order given by equation (32), it is often an underlying "fractal" geometry that causes "fractional" diffusion.

Figure 6. The Mittag-Leffler function E _α,β (t) (solid line for α = 0.7, β = 1), equation (36), interpolates between the (stretched) exponential decay (equation (30), dashed line) at small times and the inverse power law (equation (31), dotted line) at large times. (Figure rendered in MATHEMATICA [9].)

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Numerical Methods

L. Zheng , X. Zhang , in Modeling and Analysis of Modern Fluid Problems, 2017

8.6.1 Fractional Anomalous Diffusion

The research of anomalous diffusion of particles with new constitutive relations has attracted much attention in recent years. A large number of mathematical models have been proposed, among them, the comb structure model (Arkhincheev, 2007; Iomin, 2013) shown in Fig. 8.44 is one of the most significant models that can be used to simulate the transport process of particles well. The model displays two special characteristics: one is that the migration (particles transport) and proliferation (the increase or decrease of particles number) are independent of each other, namely the migration-proliferation dichotomy (Fedotov and Iomin, 2007) is considered; the other is that the diffusion and convection velocity in the x-direction only happen along the x-axis and the diffusion in the y-direction is perpendicular to the x-axis. The comb structure is a widely used model to simulate various pertinent situations, such as cell transport (Iomin, 2005), spiny dendrites (Iomin and Méndez, 2013), percolation clusters (Arkhincheev and Baskin, 1991), and so on.

Figure 8.44. The sketch framework of comb model.

The anomalous transport of particles in comb structure can be seen as a special case of continuous time random walk and the 1-D diffusion in comb model is described by the time fractional Fokker–Planck equation (Iomin, 2006) with the time fractional derivative of order α—the classical one corresponds to the time fractional derivative of order 1/2. Iomin (2006) discussed the fractional transport of cancer cells due to self-entrapping. His work indicated that the distribution function of the fractional transport depends on the scaled proliferation rate and the order of the fractional derivative. Baskin and Iomin (2004) studied specific properties of particles transported by comb model and the diffusive transport of particles leads to subdiffusion, which corresponds 0   < β  <   1 in the relationship between the mean square displacement and parameter t: $〈x^2\left(t\right)〉\sim t^{\beta }$ . Moreover, the log-normal distribution with exponentially fast spreading was obtained for the cases that transport exponent μ in $<x^2\left(t\right)>\sim t^{\mu }$ approaches infinity. Lenzi et al. (2013) investigated a diffusive process in the comb model by considering the effects of drift terms, which represents an external force acting on the system; their results showed that an anomalous spreading may present different diffusive regimes connected to anomalous diffusion and stationary states. For more papers about the comb model, see Refs. (Arkhincheev, 2010; Arkhincheev et al., 2011; Iomin and Baskin, 2005).

The 1-D constitutive relationship (Méndeza and Iomin, 2013) to describe the comb model is derived by the Fick's first law of diffusion:

(8.152) $j=-D\frac{\partial P\left(x,t\right)}{\partial x}\text{,}$

the corresponding time fractional continuity equation is:

(8.153) $\frac{{\partial }^{\alpha }P\left(x,t\right)}{\partial t^{\alpha }}=-divj\text{,}$

where j refers to the diffusion flux, D denotes the diffusion coefficient, and α  =   1/2 corresponds to the classical comb model. P(x,t) is the distribution function defined by $P\left(x,t\right)={\int }_{-\infty }^{\infty }{P}_1\left(x,y,t\right)dy$ where P ₁(x,y,t) refers to the particles distribution at the special location (x,y) and time t. The symbol $\frac{{\partial }^{\alpha }}{\partial t^{\alpha }}$ stands for the time fractional derivative based on the Caputo's definition (Podlubny, 1999), given by:

(8.154) $\frac{{\partial }^{\alpha }j}{\partial t^{\alpha }}=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{1}{{\left(t-\tau \right)}^{\alpha }}\frac{\partial j\left(x,\tau \right)}{\partial \tau }d\tau \text{,}$

The classical Fick's first law of diffusion has been the most successful law for studying diffusion problem in various pertinent situations. However, an unphysical property is that it issues an infinite velocity of propagation due to the fact that there is a finite amount of particles at larger distances from the origin even for very small times.

Cattaneo model (Cattaneo, 1948; Gómez et al., 2007) overcomes the shortcoming of the Fick's first law of diffusion well by introducing the relaxation time term. The modified Fick's first law of diffusion is as follows:

(8.155) $j+\xi \frac{\partial j}{\partial t}=-D\frac{\partial P\left(x,t\right)}{\partial x}\text{,}$

where the propagation velocity (Compte and Metzler, 1997) v  =   (D/ξ)^1/2, ξ is a nonnegative constant and refers particularly to the relaxation time of diffusion. The limit ξ  →   0 corresponds to the classical Fick's first law of diffusion with an infinite velocity of propagation.

Compte and Metzler (1997) proposed three possible generalized Cattaneo equations, each one supported by a different scheme, and the properties of these generalizations are studied in both the long-time and the short-time regimes. One generalized Cattaneo constitutive relationship is as follows:

(8.156) $j+\xi \frac{{\partial }^{\alpha }j}{\partial t^{\alpha }}=-D\frac{\partial P\left(x,t\right)}{\partial x}\text{.}$

Qi and Jiang (2011) presented exact solution for the space fractional Cattaneo diffusion equation:

(8.157) $j+\xi \frac{{\partial }^{\alpha }j}{\partial t^{\alpha }}=-D\frac{{\partial }^{\mu -1}P\left(x,t\right)}{\partial {\left|x\right|}^{\mu -1}},1<\mu \le 2.$

Here, the symbol $\frac{{\partial }^{\mu -1}}{\partial {\left|x\right|}^{\mu -1}}$ stands for the space Riesz fractional operator and the corresponding definition is given in Ref. (Qi and Jiang, 2011). For more recent studies about the Cattaneo models, see Refs. (Atanackovic et al., 2007; Liu et al., 2013b; Xu et al., 2013; Qi and Guo, 2014).

However, the model only involves the partial time derivative and is considered only as a "place holder" for a more complete formulation. The frame-indifferent generalization of Cattaneo's law was developed by Christov (2009) by the implementation of the Oldroyd upper-convective derivative. Christov (2007) proposed a generalized Fourier's law model by incorporating spatial memory into the constitutive relation; the integral and differential versions of the memory terms in the constitutive relation are discussed. The model also considers the material invariance and has been applied well in analyzing the flow and heat transfer of various fluids (Han et al., 2014; Mustafa, 2015).

Motivated by the previously mentioned works, we study fractional anomalous diffusion in comb structure; the higher spatial gradient is introduced in the constitutive relation between the flux and particles distribution. In addition, the effects of Cattaneo–Christov flux diffusion and convection are also taken into consideration. The new constitutive relation is described by:

(8.158) $j+\xi \left[\frac{\partial j}{\partial t}+u·\nabla j-j·\nabla u+\left(\nabla ·u\right)j\right]=-D\frac{\partial P\left(x,t\right)}{\partial x}$

The numerical discretization method with the L1-and L2-approxiamations for fractional derivative is used to obtain the numerical solution. The effects of the involved parameters on the particles distribution are analyzed and discussed in detail.

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Random Walks in Random Environments

L.V. Bogachev , in Encyclopedia of Mathematical Physics, 2006

Continuous-Time RWRE

As in the discrete-time case, a random walk on $\mathbb{Z}$ with continuous time is a homogeneous Markov chain X _t ,t∈[0,∞), with state space $\mathbb{Z}$ and nearest-neighbor (or at least bounded) jumps. The term "Markov" as usual refers to the "lack of memory" property, which amounts to saying that from the entire history of the process development up to a given time, only the current position of the walk is important for the future evolution while all other information is irrelevant.

Since there is no smallest time unit as in the discrete-time case, it is convenient to describe transitions of X _t in terms of transition rates characterizing the likelihood of various jumps during a very short time. More precisely, if p _xy (t):=P{X _t =y∣X ₀=x} are the transition probabilities over time t, then for h→0

[44] $\begin{array}{l}{p}_{\text{<mi mathvariant="italic" is="true">xy</mi>}}(h)=c_{\text{<mi mathvariant="italic" is="true">xy</mi>}}h+o(h)(x\ne y)\\ p_{\text{<mi mathvariant="italic" is="true">xx</mi>}}(h)=1-h{\displaystyle \sum _{y\ne x}}{c}_{\text{<mi mathvariant="italic" is="true">xy</mi>}}+o(h)\end{array}$

Equations for the functions p _xy (t) can then be derived by adapting the method of decomposition commonly used for discrete-time Markov chains (cf. the section "Transience and recurrence"). Here it is more convenient to decompose with respect to the "last" step, that is, by considering all possible transitions during a small increment of time at the end of the time interval [0,t+h]. Using Markov property and eqn [44] we can write

$\begin{array}{l}{p}_{0x}(t+h)=h{\displaystyle \sum _{y\ne x}}{p}_{0y}(t)c_{\text{<mi mathvariant="italic" is="true">yx</mi>}}\\ +p_{0x}(t)\left(1-h{\displaystyle \sum _{y\ne x}}{c}_{\text{<mi mathvariant="italic" is="true">xy</mi>}}\right)+o(h)\end{array}$

which in the limit h→0 yields the master equation (or Chapman–Kolmogorov's forward equation)

[45] $\begin{array}{l}\frac{\text{d}}{\text{d}t}{p}_{0x}(t)={\displaystyle \sum _{y\ne x}}\{c_{\text{<mi mathvariant="italic" is="true">yx</mi>}}{p}_{0y}(t)-c_{\text{<mi mathvariant="italic" is="true">xy</mi>}}{p}_{0x}(t)\}\\ p_{0x}(0)={\delta }_0(x)\end{array}$

where δ₀(x) is the Kronecker symbol.

Continuous-time RWRE are therefore naturally described via the randomized master equation, that is, with random transition rates. The canonical example, originally motivated by Dyson's study of the chain of harmonic oscillators with random couplings, is a symmetric nearest-neighbor RWRE, where the random transition rates c _xy are nonzero only for y=x±1 and satisfy the condition c _x,x+1=c _x+1,x , otherwise being i.i.d. (see Alexander et al. (1981)). In this case, the problem [45] can be formally solved using the Laplace transform, leading to the equations

[46] $s+G_0^{+}+G_0^{-}={[{{\displaystyle \hat{p}}}_0(s)]}^{-1}$

[47] $s+G_x^{-}+G_x^{+}=0(x\ne 0)$

where G _x ⁻,G _x ⁺ are defined as

[48] $G_x^{\pm }:=c_{x,x\pm 1}\frac{{{\displaystyle \hat{p}}}_{0x}(s)-{{\displaystyle \hat{p}}}_{0,x\pm 1}(s)}{{{\displaystyle \hat{p}}}_{0x}(s)}$

and ${{\displaystyle \hat{p}}}_{0x}(s):={\int }_0^{\infty }{p}_{0_x}(t){\text{e}}^{-\text{<mi mathvariant="italic" is="true">st</mi>}}\text{d}t$ . From eqns [47] and [48] one obtains the recursion

[49] $\begin{array}{l}{G}_x^{\pm }:={\left(\frac{1}{c_{x,x\pm 1}}+\frac{1}{s+G_{x\pm 1}^{+}}\right)}^{-1}\\ x=0,\pm 1,\pm 2,\dots \end{array}$

The quantities G ₀ ^± are therefore expressed as infinite continued fractions depending on s and the random variables c _x,x±1,c _x,x±2,…. The function ${{\displaystyle \hat{p}}}_{00}(s)$ can then be found from eqn [46].

In its generality, the problem is far too hard, and we shall only comment on how one can evaluate the annealed mean

$\mathbb{E}{{\displaystyle \hat{p}}}_{00}(s)=\mathbb{E}{(s+G_0^{+}+G_0^{-})}^{-1}$

According to eqn [49], the random variables G ₀ ⁺,G ₀ ⁻ are determined by the same algebraic formula, but involve the rate coefficients from different sides of site x, and hence are i.i.d. Furthermore, eqn [49] implies that the random variables G ₀ ⁺,G ₁ ⁺ have the same distribution and, moreover, G ₁ ⁺ and c ₀₁ are independent. Therefore, eqn [49] may be used as an integral equation for the unknown density function of G ₀ ⁺. It can be proved that the suitable solution exists and is unique, and although an explicit solution is not available, one can obtain the asymptotics of small values of s, thereby rendering information about the behavior of p ₀₀(t) for large t. More specifically, one can show that if $c_{*}:={(\mathbb{E}{c}_{01}^{-1})}^{-1}$ , then

$\mathbb{E}{{\displaystyle \hat{p}}}_{00}(s)\sim {(4c_{*}s)}^{-1/2},s\to 0$

and so by a Tauberian theorem

[50] $\mathbb{E}{p}_{00}(t)\sim {(4\pi c_{*}t)}^{-1/2},t\to \infty$

Note that asymptotics [50] appears to be the same as for an ordinary symmetric random walk with constant transition rates c _x,x+1=c _x+1,x =c _*, suggesting that the latter provides an EMA for the RWRE considered above.

This is further confirmed by the asymptotic calculation of the annealed mean square displacement, ${\mathit{E}}_0{X}_t^2\sim 2c_{*}t$ as t→∞ (Alexander et al. 1981). Moreover, Kawazu and Kesten (1984) proved that X _t is asymptotically normal:

[51] ${\displaystyle \underset{t\to \infty }{\text{lim}}}{\mathit{P}}_0\left\{\frac{X_t}{\sqrt{2c_{*}t}}\le x\right\}=\text{Φ}(x)$

Therefore, if c _*>0, then the RWRE has the same diffusive behavior as the corresponding ordered system, with a well-defined diffusion constant D=c _*.

In the case where c _*=0 (i.e., $\mathbb{E}{c}_{01}^{-1}=\infty$ ), one may expect that the RWRE exhibits subdiffusive behavior. For example, if the density function of the transition rates is modeled by

$f_a(u)=(1-\alpha )u^{-\alpha }{\mathbf{1}}_{\{0<u<1\}}(0<\alpha <1)$

then, as shown by Alexander et al. (1981),

$\begin{array}{l}\mathbb{E}{p}_{00}(t)\sim C_{\alpha }{t}^{-(1-\alpha )/(2-\alpha )}\\ {\mathit{E}}_0{X}_t^2\sim C_{\alpha }^{\prime }{t}^{2(1-\alpha )/(2-\alpha )}\end{array}$

In fact, Kawazu and Kesten (1984) proved that in this case t ^−α/(1+α) X _t has a (non-Gaussian) limit distribution as t→∞.

To conclude the discussion of the continuous-time case, let us point out that some useful information about recurrence of X _t can be obtained by considering an imbedded (discrete-time) random walk ${{\displaystyle \stackrel{\sim }{X}}}_n$ , defined as the position of X _t after n jumps. Note that continuous-time Markov chains admit an alternative description of their evolution in terms of sojourn times and the distribution of transitions at a jump. Namely, if the environment ω is fixed, then the random sojourn time of X _t in each state x is exponentially distributed with mean 1/c _x , where c _x :=∑ _y≠x c _xy , while the distribution of transitions from x is given by the probabilities p _xy =c _xy /c _x .

For the symmetric nearest-neighbor RWRE considered above, the transition probabilities of the imbedded random walk are given by

$\begin{array}{l}{p}_x:=p_{x,x+1}=\frac{c_{x,x+1}}{c_{x-1,x}+c_{x,x+1}}\\ q_x:=p_{x,x-1}=1-p_x\end{array}$

and we recognize here the transition law of a random walk in the random bonds environment considered in the previous subsection (cf. eqn [41]). Recurrence and zero asymptotic velocity established there are consistent with the results discussed in the present section (e.g., note that the CLT for both X _n , eqn [43], and X _t , eqn [51], does not involve any centering). Let us point out, however, that a "naive" discretization of time using the mean sojourn time appears to be incorrect, as this would lead to the scaling t=nδ₁ with ${\delta }_1:=\mathbb{E}{(c_{-1,0}+c_{01})}^{-1}$ , while from comparing the limit theorems in these two cases, one can conclude that the true value of the effective discretization step is given by ${\delta }_{*}:={(2c_{*})}^{-1}=(1/2)\mathbb{E}{c}_{01}^{-1}$ . In fact, by the arithmetic–harmonic mean inequality we have δ_*>δ₁, which is a manifestation of the RWRE's diffusive slowdown.

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