Finite-Velocity Diffusion in Random Media

A Correction to this article was published on 30 August 2020

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We investigated a diffusion-like equation with a bounded speed of signal propagation (the so called telegrapher’s equation) in a random media. We discuss some properties of the mean-value solution in a well-defined perturbation theory. The frequency-dependent effective-velocity of propagation is studied in the long and short time regime. We show that due to the wave-like character of telegrapher’s equation the effective-velocity is a complex dispersive function in time. Exact results and asymptotic perturbative long-time behaviors (for an exponential space-correlated binary disorder) are presented, showing their agreement and corroborating the goodness of the effective medium approximation in continuous system.

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  • 30 August 2020

    Erratum: Finite-Velocity Diffusion in Random Media. [J Stat Phys 179, 729���747 (2020)]


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M.O.C. thanks funding provided by CONICET (Grant No. PIP 112-201501-00216, CO).

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Correspondence to Manuel O. Cáceres.

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Communicated by Gregory Schehr.


Appendix A: Telegrapher’s Equation on Random Media

The simplest kind of absorptive (electromagnetic) wave model is telegrapher’s equation (1), there \(T^{-1}\) is a measure of the wave attenuation and v is a characteristic velocity, both parameters given in terms of Maxwell’s equations and Ohm’s law, that is:

$$\begin{aligned} T=\varepsilon /4\pi \sigma \qquad \text {and }\qquad v^{2}=c^{2}/\mu \varepsilon , \end{aligned}$$

here \(\varepsilon \) is the dielectric and \(\mu \) the permeability constants, c the light velocity, while \(\sigma \) is the conductivity (responsible of Joule effects in a waveguides).

In the context of an electronic circuit with inductance L, resistance R, capacity C and leakage conductance per unit length G, the telegrapher parameters T and v can be identified as [25]:

$$\begin{aligned} T=\left( G/C-R/L\right) ^{-1}\qquad \text {and }\qquad v=\left( LC\right) ^{-1/2}. \end{aligned}$$

In the context of a diffusion-like process we noted that telegrapher’s equation (1) can alternative be transformed to an integral equation with a time-memory kernel:

$$\begin{aligned} \partial _{t}P\left( x,t\right) =v^{2}\int _{0}^{t}e^{-(t-\tau )/T}\ \partial _{x}^{2}P\left( x,\tau \right) d\tau \qquad \text {with }\qquad \left. \partial _{t}P\left( x,t\right) \right| _{t=0}=0, \end{aligned}$$

and conditions for positive and normalization of solutions. In this form T and v can also be related with parameters of a persistent lattice random walk [8, 11, 12]. The last non-local in time equation (A1) can also be considered a generalized master equation [26]. Therefore, the continuous-time random-walk (CTRW) theory, with internal states [27,28,29], allows successfully to be used to consider further generalizations of telegrapher’s equation with fractional calculus in the context of fractal-time and/or fractal-space persistent random-walk dynamics [2, 22].

Interestingly, telegrapher’s equation (1) can also be introduced form a phenological point of view considering a non-local in time Fick’s law [8]:

$$\begin{aligned} J\left( x,t\right) =-v^{2}\int _{0}^{t}e^{-(t-\tau )/T}\partial _{x}P\left( x,\tau \right) d\tau . \end{aligned}$$

Combining this equation with the conservation law: \(\partial _{t}P\left( x,t\right) +\partial _{x}J\left( x,t\right) =0\), the telegrapher equation (1) is recovered.

In addition, if in the non-local Fick’s law (A2) we consider a random velocity field (as possible effects from the heterogeneous media) we can replace \(v^{2}\rightarrow v^{2}+\xi \left( x\right) \) in Eq. (A2). Then, noting that:

$$\begin{aligned} \partial _{t}J\left( x,t\right) =-\left( v^{2}+\xi \left( x\right) \right) \partial _{x}P\left( x,t\right) -J\left( x,t\right) /T\ \end{aligned}$$

and using the conservation law, the telegrapher equation (2) (in a random media) is obtained.

Appendix B: Projector Operator Techniques Revisited

Projector operator techniques has a long history in physics, for a comprehensive review see [23]. Consider the Laplace transform of Eq. (2) and initial conditions: \(P\left( x,t=0\right) =\delta \left( x\right) \), \(\left. \partial _{t}P\left( x,t\right) \right| _{t=0}=0\). After some algebra we get:

$$\begin{aligned} \left( s^{2}+\frac{s}{T}-v^{2}\partial _{x}^{2}\right) P\left( x,s\right) =\left( s+\frac{1}{T}\right) \delta \left( x\right) +\partial _{x}\xi \left( x\right) \partial _{x}P\left( x,s\right) . \end{aligned}$$

If we apply the projectors operator \({\mathcal {P}}\) and \({\mathcal {Q}}=\left( {\mathbf {1}}-{\mathcal {P}}\right) \) to Eq. (B1) we get:

$$\begin{aligned} {\varvec{\Lambda }}{\mathcal {P}}P\left( x,s\right)= & {} \left( s+\frac{1}{T} \right) \delta \left( x\right) +{\mathcal {P}}{\varvec{\Theta }}_{\xi }\left( {\mathcal {P}}+{\mathcal {Q}}\right) P\left( x,s\right) \end{aligned}$$
$$\begin{aligned} {\varvec{\Lambda }}{\mathcal {Q}}P\left( x,s\right)= & {} {\mathcal {Q}}{\varvec{\Theta }} _{\xi }\left( {\mathcal {P}}+{\mathcal {Q}}\right) P\left( x,s\right) , \end{aligned}$$

where we have defined the differential operator

$$\begin{aligned} {\varvec{\Lambda }}=\left( s^{2}+\frac{s}{T}-v^{2}\partial _{x}^{2}\right) , \end{aligned}$$


$$\begin{aligned} {\varvec{\Theta }}_{\xi }=\partial _{x}\xi \left( x\right) \partial _{x}. \end{aligned}$$

Solving \({\mathcal {Q}}P\left( x,t\right) \) in terms of \({\mathcal {P}}P\left( x,t\right) \) and replacing this expression in (B2) we finally get a formal result in terms of the integral convolutional operator \({\varvec{\Lambda }}^{-1}\) with kernel:

$$\begin{aligned} g\left( x,s\right) = \exp \left( -\left| \frac{x}{v}\right| \sqrt{s\left( s+1/T\right) }\right) \big / 2\sqrt{v^{2}s\left( s+1/T\right) }. \end{aligned}$$

In fact, the Green function of operator (B4) is \(g\left( k,s\right) =\left( s^{2}+\frac{s}{T}+v^{2}k^{2}\right) ^{-1}\), thus in real space we get:

$$\begin{aligned} g\left( x,s\right) =\frac{1}{2\pi }\int _{-\infty }^{+\infty }e^{ikx}\frac{dk }{\left( s^{2}+\frac{s}{T}+v^{2}k^{2}\right) }, \end{aligned}$$

this integral can be done by residues and the result is (B5). Using (B5) and the formal expression for \({\mathcal {P}}P\left( x,t\right) \), gives, after some algebra, the exact equation:

$$\begin{aligned} {\varvec{\Lambda }}{\mathcal {P}}P\left( x,s\right) =\left( s+\frac{1}{T}\right) \delta \left( x\right) +{\mathcal {P}}{\varvec{\Theta }}_{\xi }\sum _{n=0}^{\infty }\left[ {\varvec{\Lambda }}^{-1}{\mathcal {Q}}{\varvec{\Theta }}_{\xi }\right] ^{n} {\mathcal {P}}P\left( x,s\right) , \end{aligned}$$

Sum terms in Eq. (B6) take care of the average over the disorder and it is represented in its integral form as:

$$\begin{aligned}&{\mathcal {P}}\partial _{x}\sum _{n=0}^{\infty }\int _{-\infty }^{+\infty }dx_{1}\ldots \int _{-\infty }^{+\infty }dx_{n}\ \left\langle \xi \left( x\right) \xi \left( x_{1}\right) \ldots \xi \left( x_{n}\right) \right\rangle _{T} \\&\quad \times \left[ \partial _{x_{1}}^{2}g\left( x-x_{1},s\right) \right] \cdots \left[ \partial _{x_{n}}^{2}g\left( x_{n-1}-x_{n},s\right) \right] \partial _{x_{n}}\left\langle P\left( x_{n},s\right) \right\rangle , \nonumber \end{aligned}$$

where the object \(\left\langle \xi \left( x\right) \xi \left( x_{1}\right) \ldots \xi \left( x_{n}\right) \right\rangle _{T}\) is a Terwiel cumulant of order \(n+1\), that is:

$$\begin{aligned} \left\langle \xi \left( x\right) \xi \left( x_{1}\right) \ldots \xi \left( x_{n}\right) \right\rangle _{T}\equiv {\mathcal {P}}\xi \left( x\right) {\mathcal {Q}}\xi \left( x_{1}\right) {\mathcal {Q}}\ldots {\mathcal {Q}}\xi \left( x_{n}\right) . \end{aligned}$$

Using the space-convolution structure of (B6) we can straightforwardly take the Fourier transform, from which we get the explicit result:

$$\begin{aligned} {\mathcal {P}}P\left( k,s\right) =\frac{\left( s+1/T\right) }{s\left( s+1/T\right) +\left( v^{2}+\gamma \left( k,s\right) \right) k^{2}},\ P\left( k,t=0\right) =1,\left. \partial _{t}P\left( k,t\right) \right| _{t=0}=0, \nonumber \\ \end{aligned}$$

where the function \(\gamma \left( k,s\right) \) is given by:

$$\begin{aligned} \gamma \left( k,s\right)= & {} \sum _{n=0}^{\infty }\int _{-\infty }^{+\infty }dy_{1}e^{iky_{1}}\ldots \int _{-\infty }^{+\infty }dy_{n}e^{iky_{n}}\ \left\langle \xi \left( 0\right) \xi \left( y_{1}\right) \ldots \xi \left( y_{n}\right) \right\rangle _{T} \\&\times \left[ \partial _{y_{1}}^{2}g\left( 0-y_{1},s\right) \right] \ldots \left[ \partial _{y_{n}}^{2}g\left( y_{n-1}-y_{n},s\right) \right] , \nonumber \end{aligned}$$

this is just (8). To prove (B10) note that the first term in the sum of (B6) explicitly read:

$$\begin{aligned} {\mathcal {P}}{\varvec{\Theta }}_{\xi }\left\langle P\left( x,s\right) \right\rangle =\partial _{x}\left\langle \xi \left( x\right) \right\rangle \partial _{x}\left\langle P\left( x,s\right) \right\rangle , \end{aligned}$$

which cancels if the mean-value of \(\xi \left( x\right) \) is zero. The second term read:

$$\begin{aligned} {\mathcal {P}}{\varvec{\Theta }}_{\xi }{\varvec{\Lambda }}^{-1}{\mathcal {Q}}{\varvec{\Theta }}_{\xi }\left\langle P\left( x,s\right) \right\rangle= & {} {\mathcal {P}} {\varvec{\Theta }}_{\xi _{x}}\int _{-\infty }^{+\infty }dx_{1}g\left( x-x_{1},s\right) {\mathcal {Q}}{\varvec{\Theta }}_{\xi _{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\= & {} {\mathcal {P}}{\varvec{\Theta }}_{\xi _{x}}\int _{-\infty }^{+\infty }dx_{1}g\left( x-x_{1},s\right) {\mathcal {Q}}\partial _{x_{1}}\xi \left( x_{1}\right) \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle , \nonumber \\ \end{aligned}$$

here the last integral can be integrated by parts to get:

$$\begin{aligned} {\mathcal {P}}{\varvec{\Theta }}_{\xi }{\varvec{\Lambda } }^{-1}{\mathcal {Q}}{\varvec{\Theta }}_{\xi }\left\langle P\left( x,s\right) \right\rangle= & {} -{\mathcal {P}} {\varvec{\Theta }}_{\xi _{x}}\int _{-\infty }^{+\infty }dx_{1}\left[ \partial _{x_{1}}g\left( x-x_{1},s\right) \right] {\mathcal {Q}}\xi \left( x_{1}\right) \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\= & {} -{\mathcal {P}}\partial _{x}\xi \left( x\right) \int _{-\infty }^{+\infty }dx_{1}{\mathcal {Q}}\xi \left( x_{1}\right) \left[ \partial _{x}\partial _{x_{1}}g\left( x-x_{1},s\right) \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\= & {} -\partial _{x}\int _{-\infty }^{+\infty }dx_{1}{\mathcal {P}}\xi \left( x\right) {\mathcal {Q}}\xi \left( x_{1}\right) \left[ \partial _{x}\partial _{x_{1}}g\left( x-x_{1},s\right) \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\= & {} \partial _{x}\int _{-\infty }^{+\infty }dx_{1}\left\langle \xi \left( x\right) \xi \left( x_{1}\right) \right\rangle _{T}\ \left[ \partial _{x_{1}}^{2}g\left( x-x_{1},s\right) \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle . \nonumber \\ \end{aligned}$$

If we now introduce the Fourier transform in (B13) and use the stationary behavior of Terwiel’s cumulants we get:

$$\begin{aligned}&{\mathcal {F}}_{k}\left[ \hbox {B13}\right] \nonumber \\&\quad \equiv \int _{-\infty }^{+\infty }dx\ e^{ikx}\partial _{x}\int _{-\infty }^{+\infty }dx_{1}\left\langle \xi \left( x\right) \xi \left( x_{1}\right) \right\rangle _{T}\left[ \partial _{x_{1}}^{2}g\left( x-x_{1},s\right) \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\&\quad =(-ik)\int _{-\infty }^{+\infty }dx\ e^{ik(x-x_{1}+x_{1})}\int _{-\infty }^{+\infty }dx_{1}\left\langle \xi \left( x\right) \xi \left( x_{1}\right) \right\rangle _{T}\left[ \partial _{x_{1}}^{2}g\left( x-x_{1},s\right) \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\&\quad =(-ik)\int _{-\infty }^{+\infty }dx_{1}e^{ikx_{1}}\left[ \int _{-\infty }^{+\infty }dx\ e^{ik(x-x_{1})}\left\langle \xi \left( x\right) \xi \left( x_{1}\right) \right\rangle _{T}\left[ \partial _{x_{1}}^{2}g\left( x-x_{1},s\right) \right] \right] \partial _{x_{1}}\left\langle P\left( x_{1},s\right) \right\rangle \nonumber \\&\quad =-k^{2}\left[ \int _{-\infty }^{+\infty }dy_{1}\ e^{iky_{1}}\left\langle \xi \left( 0\right) \xi \left( y_{1}\right) \right\rangle _{T}\left[ \partial _{y_{1}}^{2}g\left( 0-y_{1},s\right) \right] \right] \left\langle P\left( k,s\right) \right\rangle . \end{aligned}$$

Similar calculus can be done to any term appearing in (B6), therefore leading to (B9) and (B10).

Appendix C: Resuming Techniques on Telegrapher’s Equation

Terwiel’s cumulant contributions of neighbors sites in a discrete random systems, were presented in the pioneer work [16]. In this Appendix we briefly introduce these ideas for the (continuous) telegrapher equation.

Consider the infinite sum of integrals (B10); there, expressions like \(\left[ \partial _{x}^{2}g\left( x,s\right) \right] \) appear n-times. Using (9) we can write (8) in the form:

$$\begin{aligned}&\gamma \left( k,s\right) \nonumber \\&\quad ={\mathcal {P}}\sum _{p=0}^{\infty }\int _{-\infty }^{+\infty }dy_{1}\ldots \int _{-\infty }^{+\infty }dy_{p}e^{ik(y_{1}+\cdots y_{p})}\xi \left( 0\right) {\mathcal {Q}}F_{0,y_{1}}\xi \left( y_{1}\right) {\mathcal {Q}}\cdots \xi \left( y_{p-1}\right) {\mathcal {Q}}F_{y_{p-1},y_{p}}\xi \left( y_{p}\right) , \nonumber \\ \end{aligned}$$

where \(F_{x,y}\equiv \left[ s\left( s+1/T\right) g\left( x-y,s\right) -\delta \left( x-y\right) \right] /v^{2}\).

One important difference with the resumation techniques in continuous models is that the “diagonal” part of \(F_{x,y}\) has two contributions because \( g\left( 0,s\right) \ne 0\). So following previous experience we introduce a first possible resumation consisting in integrating all \(\delta \left( x\right) \) contributions coming from \(F_{x,y}\), and they are of the form:

$$\begin{aligned}&{\mathcal {P}}\left( \xi \left( 0\right) -\xi \left( 0\right) {\mathcal {Q}}\frac{1 }{v^{2}}\xi \left( 0\right) +\xi \left( 0\right) {\mathcal {Q}}\frac{1}{v^{2}} \xi \left( 0\right) {\mathcal {Q}}\frac{1}{v^{2}}\xi \left( 0\right) -\cdots \right) \\&\quad ={\mathcal {P}}\sum _{p=0}^{\infty }\left( -\xi \left( 0\right) \mathcal { Q}\frac{1}{v^{2}}\right) ^{p}\xi \left( 0\right) \equiv {\mathcal {P}}\psi \left( 0\right) , \end{aligned}$$

here the random operator \(\psi \left( x\right) \) at site \(x=0\), defined in (11), has been used. Integral (C1) can therefore be reordered leading to:

$$\begin{aligned} \gamma \left( k,s\right)= & {} {\mathcal {P}}\sum _{n=0}^{\infty }\left( \frac{ s\left( s+1/T\right) }{v^{2}}\right) ^{n}\int _{-\infty }^{+\infty }dy_{1}\nonumber \\&\ldots \int _{-\infty }^{+\infty }dy_{n}e^{-iky_{n}}\psi \left( 0\right) {\mathcal {Q}}g\left( 0-y_{1},s\right) \psi \left( y_{1}\right) {\mathcal {Q}}g\left( y_{1}-y_{2},s\right) \cdots \nonumber \\&\times \cdots \psi \left( y_{n-2}\right) {\mathcal {Q}}g\left( y_{n-2}-y_{n-1},s\right) \psi \left( y_{n-1}\right) {\mathcal {Q}}g\left( y_{n-1}-y_{n},s\right) \psi \left( y_{n}\right) , \nonumber \\ \end{aligned}$$

which is (10).

In terms of the continuous field operator \(\psi \left( x\right) \), a difference in this integral-resumation is noted with respect to the one coming from a discrete model [16]. There the same discrete-resumation is restricted to terms in which a given \(\psi _{n_{j}}\) at lattice site \(n_{j}\) does not coincide with its nearest neighbors \(\psi _{n_{j\pm 1}}\) at sites \(n_{j\pm 1}\). As we mention before this effect comes form the non-diagonal behavior of the continuous function \(F_{x,y}\). In addition to this, in discrete models a random field \(\xi _{n_{j}}\) is assumed to be statistical independent from any other \(\xi _{n_{i}}\) if \( n_{i}\ne n_{j}\). As mentioned this is not the case in continuous space-correlated models.

1. On the (Continuous) Random Operator \(\psi \left( x\right) \)

In order to study the small s behavior of \(\gamma \left( k,s\right) \) it is convenient to rewrite the random operator \(\psi \left( x\right) \) into a more friendly way to be used with any statistics of the random field \(\xi \left( x\right) \). From (11) we can write:

$$\begin{aligned} \left[ 1+\frac{1}{v^{2}}\xi \left( x\right) \left( {\mathbf {1}}-{\mathcal {P}} \right) \right] \psi \left( x\right) \left[ \bullet \right] =\xi \left( x\right) \left[ \bullet \right] , \end{aligned}$$

working out the LHS of (C3) we get:

$$\begin{aligned} \left( 1+\frac{1}{v^{2}}\xi \left( x\right) \right) \psi \left( x\right) \left[ \bullet \right] =\xi \left( x\right) \left[ \bullet \right] +\frac{1}{ v^{2}}\xi \left( x\right) {\mathcal {P}}\psi \left( x\right) \left[ \bullet \right] , \end{aligned}$$

dividing by \(\left( 1+\frac{1}{v^{2}}\xi \left( x\right) \right) \) we get:

$$\begin{aligned} \psi \left( x\right) \left[ \bullet \right]= & {} \left( 1+\frac{1}{v^{2}}\xi \left( x\right) \right) ^{-1}\left\{ \xi \left( x\right) +\frac{1}{v^{2}}\xi \left( x\right) {\mathcal {P}}\psi \left( x\right) \right\} \left[ \bullet \right] \nonumber \\\equiv & {} \left\{ M\left( x\right) +\frac{M\left( x\right) }{v^{2}}{\mathcal {P}} \psi \left( x\right) \right\} \left[ \bullet \right] , \end{aligned}$$

where \(M\left( x\right) \) is a random number written in terms of the field \( \xi \left( x\right) \):

$$\begin{aligned} M\left( x\right) =\left( \frac{\xi \left( x\right) }{1+\xi \left( x\right) /v^{2}}\right) . \end{aligned}$$

Now, we can solve \({\mathcal {P}}\psi \left( x\right) \) by applying \({\mathcal {P}} \) to (C5)

$$\begin{aligned} {\mathcal {P}}\psi \left( x\right) \left[ \bullet \right]= & {} {\mathcal {P}}\left\{ M\left( x\right) +\frac{M\left( x\right) }{v^{2}}{\mathcal {P}}\psi \left( x\right) \right\} \left[ \bullet \right] \nonumber \\= & {} \left\{ {\mathcal {P}}M\left( x\right) +{\mathcal {P}}\left( \frac{M\left( x\right) }{v^{2}}\right) {\mathcal {P}}\psi \left( x\right) \right\} \left[ \bullet \right] , \end{aligned}$$

then we get:

$$\begin{aligned} \left\{ 1-\frac{\left\langle M\left( x\right) \right\rangle }{v^{2}}\right\} {\mathcal {P}}\psi \left( x\right) \left[ \bullet \right] ={\mathcal {P}}M\left( x\right) \left[ \bullet \right] , \end{aligned}$$

from which,

$$\begin{aligned} {\mathcal {P}}\psi \left( x\right) \left[ \bullet \right] =\left\{ 1-\left\langle M\left( x\right) \right\rangle /v^{2}\right\} ^{-1}{\mathcal {P}} M\left( x\right) \left[ \bullet \right] . \end{aligned}$$

Introducing (C9) into (C5) we arrive to a simpler expression for the random operator \(\psi \left( x\right) \):

$$\begin{aligned} \psi \left( x\right) \left[ \bullet \right] =M\left( x\right) \left\{ 1+ \frac{1/v^{2}}{\left\{ 1-\left\langle M\left( x\right) \right\rangle /v^{2}\right\} }{\mathcal {P}}M\left( x\right) \right\} \left[ \bullet \right] . \end{aligned}$$

Long time behavior of \(\gamma \left( k,s\right) \) could be studied from Terwiel’s cumulants contributions in the low-frequency regime, that is, in the small Laplace parameter s from \(\gamma \left( k,s\right) =\sum _{n=1}^{\infty }\gamma _{n}\left( k,s\right) \), with:

$$\begin{aligned} \gamma _{n}\left( k,s\right)= & {} \left( \frac{s\left( s+1/T\right) }{v^{2}} \right) ^{n}\int _{-\infty }^{+\infty }dy_{1}\ldots \int _{-\infty }^{+\infty }dy_{n}e^{-iky_{n}}\ \left\langle \psi \left( 0\right) \psi \left( y_{1}\right) \ldots \psi \left( y_{n}\right) \right\rangle _{T} \nonumber \\ \\&\times g\left( 0-y_{1},s\right) \ldots g\left( y_{n-1}-y_{n},s\right) . \nonumber \end{aligned}$$

Nevertheless, a crucial point in this formula is the fact that \(\left\langle \psi \left( x\right) \right\rangle \ne 0\). Thus to any order \({\mathcal {O}} \left( s^{\nu }\right) \) there are infinite diagrammatic contributions, see [16]. Thus a better arrangement must be done to consider a suitable perturbation to calculate \(\gamma _{n}\left( k,s\sim 0\right) \). In fact the new reordering, around some EMA leads to a well-defined perturbation theory in the small s.

2. The EMA for Telegrapher’s Equation on Random Media

As we mention before the EMA consists in defining a new random operator \( {\tilde{\psi }}\left( x\right) \). In this way introducing the changes \( (v^{2}+\Gamma )\equiv {\tilde{v}}^{2}\), \(\left( \xi \left( x\right) -\Gamma \right) \equiv {\tilde{\xi }}\left( x\right) \) and \(g\left( x,s\right) \rightarrow {\tilde{g}}\left( x,s\right) \) we can write \({\tilde{\psi }}\), from (C10), in the form:

$$\begin{aligned} {\tilde{\psi }}\left( x\right) \left[ \bullet \right] ={\tilde{M}}\left( x\right) \left\{ 1+\frac{1/{\tilde{v}}^{2}}{\left\{ 1-\left\langle {\tilde{M}}\left( x\right) \right\rangle /{\tilde{v}}^{2}\right\} }{\mathcal {P}}{\tilde{M}}\left( x\right) \right\} \left[ \bullet \right] , \end{aligned}$$


$$\begin{aligned} {\tilde{M}}\left( x\right) =\left( \frac{\left( \xi \left( x\right) -\Gamma \right) }{1+\left( \xi \left( x\right) -\Gamma \right) /(v^{2}+\Gamma )} \right) . \end{aligned}$$

Note that from (C12) we can immediately calculate \(\left\langle \psi \left( x\right) \right\rangle \) and therefore the self-consistent equation \( {\mathcal {P}}{\tilde{\psi }}\left( x\right) \left[ 1\right] =0\) implies \( \left\langle {\tilde{M}}\left( x\right) \right\rangle =\)\(\left\langle \left( \xi \left( x\right) -\Gamma \right) (v^{2}+\Gamma )/(v^{2}+\xi \left( x\right) )\right\rangle =0\), which gives (20).

Then expression for the effective-velocity \({\tilde{\gamma }}\left( k,s\right) =\sum _{n=1}^{\infty }{\tilde{\gamma }}_{n}\left( k,s\right) \) can be read from (C11) as:

$$\begin{aligned} {\tilde{\gamma }}_{n}\left( k,s\right)= & {} \left( \frac{s\left( s+1/T\right) }{ {\tilde{v}}^{2}}\right) ^{n}\int _{-\infty }^{+\infty }dy_{1}\ldots \int _{-\infty }^{+\infty }dy_{n}e^{-iky_{n}}\left\langle {\tilde{\psi }}\left( 0\right) {\tilde{\psi }}\left( y_{1}\right) \cdots {\tilde{\psi }}\left( y_{n}\right) \right\rangle _{T} \nonumber \\&\times {\tilde{g}}\left( 0-y_{1},s\right) \ldots {\tilde{g}}\left( y_{n-1}-y_{n},s\right) , \end{aligned}$$

which is (14). From this formula higher orders terms of \({\mathcal {O}} (s^{\nu })\) can be calculated considering contributions from \(n\ge 1\).

Appendix D: Space-Correlated Binary Disorder

Equation (C14) is a general expression for the calculation of the dispersive function \({\tilde{\gamma }}\left( k,s\right) \) in orders of \(s^{\nu } \), and for any statistics of the random field \(\xi \left( x\right) \). An important model of disorder is the binary one, which allows for considering different options for modeling its n-points correlation functions. In general we have to deal with the random operator \({\tilde{\psi }}\left( x\right) \) given in (C12). Nevertheless, in the particular case of an equiprobable binary disorder this random operator adopts a simpler structure. Using the stationary probability of the symmetric binary disorder: \(P\left( \xi \right) =\left( \delta _{\xi ,\Delta }+\delta _{\xi ,-\Delta }\right) /2\)), the expression of \({\tilde{\psi }}\left( x\right) \left[ \bullet \right] \) can be simplified as follows. Introducing \({\tilde{M}}\left( x\right) \) from (C13) (noting that \(\left\langle {\tilde{M}}\left( x\right) \right\rangle =0\) and \(\Gamma =-\Delta ^{2}/v^{2}\)) in (C12) we get:

$$\begin{aligned} {\tilde{\psi }}\left( x\right) \left[ \bullet \right]= & {} {\tilde{M}}\left( x\right) \left\{ 1+\frac{1}{(v^{2}+\Gamma )}{\mathcal {P}}{\tilde{M}}\left( x\right) \right\} \left[ \bullet \right] \nonumber \\= & {} \frac{(v^{4}-\Delta ^{2})}{v^{2}}\left( \frac{v^{2}\xi \left( x\right) +\Delta ^{2}}{v^{2}+\xi \left( x\right) v^{2}}\right) \left( 1+{\mathcal {P}} \left( \frac{v^{2}\xi \left( x\right) +\Delta ^{2}}{v^{2}+\xi \left( x\right) v^{2}}\right) \right) \left[ \bullet \right] . \end{aligned}$$

Using binary disorder we also get that \({\mathcal {P}}\left( \frac{v^{2}\xi \left( x\right) +\Delta ^{2}}{v^{2}+\xi \left( x\right) v^{2}}\right) \left[ f\right] =\frac{1}{v^{2}}{\mathcal {P}}\xi \left( x\right) \left[ f\right] \), then introducing this expression in (D1) and taking the mean value, we obtain after some algebra:

$$\begin{aligned} {\mathcal {P}}{\tilde{\psi }}\left( x\right) \left[ \bullet \right] =\frac{ (v^{4}-\Delta ^{2})}{v^{2}}\left( \frac{1}{v^{2}}\right) {\mathcal {P}}\left( \xi \left( x\right) +\frac{1}{v^{2}}\xi \left( x\right) {\mathcal {P}}\xi \left( x\right) \right) \left[ \bullet \right] , \end{aligned}$$

from which we can write a simpler operational expression:

$$\begin{aligned} {\tilde{\psi }}\left( x\right) \left[ \bullet \right] =\frac{(v^{4}-\Delta ^{2}) }{v^{4}}\left[ \xi \left( x\right) +\frac{1}{v^{2}}\xi \left( x\right) {\mathcal {P}}\xi \left( x\right) \right] \left[ \bullet \right] . \end{aligned}$$

With this expression for \({\tilde{\psi }}\left( x\right) \left[ \bullet \right] \), any n-points Terwiel’s cumulant associated with the symmetric binary disorder can straightforwardly be calculated, so we can proceed to study the effective-velocity \(\gamma \left( k,s\right) \) order-by-order in s from (C14).

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Cáceres, M.O. Finite-Velocity Diffusion in Random Media. J Stat Phys 179, 729–747 (2020).

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  • Telegrapher’s equation
  • Random media
  • Effective Medium Approximation
  • Exact solutions

Mathematics Subject Classification

  • 82C44
  • 82D30
  • 60G60
  • 60H25
  • 58J65