# Stochastic dynamics with multiplicative dichotomic noise: heterogeneous
telegrapher’s equation, anomalous crossovers and resetting

Trifce Sandev<EMAIL_ADDRESS>Research Center for Computer Science
and Information Technologies, Macedonian Academy of Sciences and Arts, Bul.
Krste Misirkov 2, 1000 Skopje, Macedonia  Institute of Physics & Astronomy,
University of Potsdam, D-14776 Potsdam-Golm, Germany  Institute of Physics,
Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius
University, Arhimedova 3, 1000 Skopje, Macedonia  Ljupco Kocarev
<EMAIL_ADDRESS>Research Center for Computer Science and Information
Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2,
1000 Skopje, Macedonia  Faculty of Computer Science and Engineering, Ss. Cyril
and Methodius University,
PO Box 393, 1000 Skopje, Macedonia  Ralf Metzler<EMAIL_ADDRESS>Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm,
Germany  Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic
of Korea  Aleksei Chechkin<EMAIL_ADDRESS>Institute of Physics &
Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany  Faculty of
Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of
Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland  Akhiezer
Institute for Theoretical Physics National Science Center “Kharkiv Institute
of Physics and Technology”, Akademichna 1, Kharkiv 61108, Ukraine

###### Abstract

We analyze diffusion processes with finite propagation speed in a non-
homogeneous medium in terms of the heterogeneous telegrapher’s equation. In
the diffusion limit of infinite-velocity propagation we recover the results
for the heterogeneous diffusion process. The heterogeneous telegrapher’s
process exhibits a rich variety of diffusion regimes including hyperdiffusion,
ballistic motion, superdiffusion, normal diffusion and subdiffusion, and
different crossover dynamics characteristic for complex systems in which
anomalous diffusion is observed. The anomalous diffusion exponent in the short
time limit is twice the exponent in the long time limit, in accordance to the
crossover dynamics from ballistic diffusion to normal diffusion in the
standard telegrapher’s process. We also analyze the finite-velocity
heterogeneous diffusion process in presence of stochastic Poissonian
resetting. We show that the system reaches a non-equilibrium stationary state.
The transition to this non-equilibrium steady state is analysed in terms of
the large deviation function.

##  I Introduction

It is well known that the Green’s function of the classical diffusion
equation, the Gaussian distribution, has non-zero values for any $x$ at $t>0$,
which means that some of the particles move with an arbitrarily chosen large
velocity. To avoid this unphysical property, a finite-velocity diffusion
process governed by the so-called telegrapher’s (or Cattaneo) equation was
introduced, and a corresponding persistent random walk model was proposed.
Historically, the telegrapher’s equation has been derived by Heaviside for a
voltage $u$ along a lossy transmission line in electrodynamic theory heaviside
,

$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}u(x,t)+\frac{\partial}{\partial t}u(x,t)=D\frac{\partial^{2}}{\partial
x^{2}}u(x,t).$ (1)

Here $\tau$ is a time parameter, and $D$ is the diffusion coefficient, which
relates to a finite propagation velocity $v=\sqrt{D/\tau}$. Contrary to the
diffusion equation which is parabolic, the telegrapher’s equation is a
hyperbolic partial differential equation. A simple interpretation of this
process is that the probability flux is delayed over time by the interval
$\tau$ with respect to the probability gradient,
$J(x,t+\tau)=-D\frac{\partial}{\partial x}u(x,t)$. Assuming $\tau\ll t$, then

$\displaystyle J(x,t)+\tau\frac{\partial}{\partial
t}J(x,t)=-D\frac{\partial}{\partial x}u(x,t).$ (2)

This equation was proposed by Cattaneo in 1948 cattaneo1  (see also cattaneo2
; new ) to extend the standard constitutive relation. Combining this equation
with continuity equation

$\displaystyle\frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial
x}J(x,t),$ (3)

one obtains the telegrapher’s equation (1) that is often alternatively
referred to as Cattaneo equation. The persistent random walk was suggested
first by Fürth furth  and Taylor taylor , who considered it as a suitable
model for transport in turbulent diffusion, while Goldstein gave solutions of
various forms of the telegrapher’s equation goldstein  (see also Jagher ). The
telegrapher’s equation can be considered as a particular case of a spatio-
temporally coupled Lévy walk model with exponential waiting time probability
density zumofen ; klafter ; levywalkreview . Extended Poisson-Kac theory
provides a unifying framework for stochastic processes with finite propagation
velocity and was developed recently klages . The telegrapher’s equation was
also used to study finite-velocity diffusion on a comb epl  and in random
media jsp , as well as the telegraph processes with random velocities jstor .
Fractional generalisations of the telegrapher’s equation were considered in
compte ; prw1 ; TK ; polito ; awad ; gorska , while non-Markovian discrete-
time versions of the telegraph process were studied in polito2 . For more
details on the Cattaneo equation, as well as derivation and application of the
telegrapher’s equation, we refer to the literature, see, e.g., weiss2002 ;
joseph ; spigler ; masoliverentropy .

In the telegrapher’s equation (1) it is assumed that the diffusion coefficient
$D$ and the time interval $\tau$ are constants. In present paper we consider
the case of space-dependent diffusion coefficient. In pure diffusion models, a
space-dependent diffusivity is introduced to describe heterogeneous diffusion
process (HDP), i.e., relative diffusion of passive tracers in the atmosphere
richardson ; yaglom , transport processes in heterogeneous media denisov1 ;
Haggerty ; Dentz ; 8 ; jstat ; diss ; chchme ; ch1 ; ch3 ; ch2 ; me2020 ;
edgarpre , and on random fractals Procaccia , including comb structures 6 ;
csf . The mean first passage time and related search problems 9 ; santos ,
ergodicity breaking chchme ; ch1 ; ch3  and infinite ergodic theory for HDPs
barkaiarxiv , as well as Lévy flights in inhomogeneous media 10  have been
investigated, as well.

In section 2 we introduce the telegrapher’s equation for a finite-velocity
HDP. We derive a general solution of the problem in section 3. Exact results
for the probability density function (PDF) are obtained, and the asymptotic
behaviours are analysed. In section 4 we present general result for the MSD,
for which we observe different crossovers between diffusion regimes in the
system. Several special cases are recovered, as well. We then introduce
exponential resetting in the heterogeneous telegrapher’s equation in section 5
and report exact results for the PDF and MSD. It is shown that in the long
time limit the system approaches a non-equilibrium stationary state (NESS).
The transition to the NESS is analysed in terms of the large deviation
function. In section 6 we summarise our findings.

##  II From the Langevin equation with dichotomic noise to the telegrapher’s
equation

The master equation for a persistent random walk leading to a Langevin
equation with dichotomic noise was considered in kac ; balakrishnan . The
Langevin equation takes the form

$\displaystyle\dot{x}(t)=v\,\zeta(t),$ (4)

where $v$ is a positive constant with physical dimension of a speed, and
$\zeta(t)$ is a stationary dichotomic Markov process that jumps between two
states $\pm 1$ with the mean rate $\nu$, i.e., the inverse mean sojourn time
for each state. The corresponding equation for the PDF $P(x,t)$ of such a
process is the telegrapher’s equation (1), where $\tau=\frac{1}{2\nu}$ and
$D=v^{2}\tau$. For an elegant and simple derivation of equation (1) starting
from equation (4) we refer the reader to Ref. balakrishnan . In the limit
$\nu\rightarrow\infty$, $v\rightarrow\infty$ such that $D$ is a finite
constant the dichotomic noise reduces to a Gaussian white noise, and the
diffusion equation is obtained.

There exist different generalisations of the standard telegrapher’s equation
for inhomogeneous cases weiss2002 ; masoliver1994 ; masoliverweiss ; brissaud
; kitahara ; sancho0 ; ratanov99 . In this paper, we consider the form
originating from the general nonlinear Langevin equation with multiplicative
dichotomic noise

$\displaystyle\dot{x}(t)=v(x)\zeta(t),$ (5)

where $v(x)>0$ is a position-dependent speed, and $\zeta(t)$ is the same
dichotomic process as in equation (4). The result is

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}P(x,t)+\frac{1}{\tau}\frac{\partial}{\partial
t}P(x,t)=\frac{\partial}{\partial x}\left\\{v(x)\frac{\partial}{\partial
x}\left[v(x)P(x,t)\right]\right\\},$ (6)

where $v(x)=\sqrt{D(x)/\tau}$. A detailed derivation of the heterogeneous
telegrapher’s equation from the Langevin equation (5) is given in Ref.
ratanov99 , see Theorem 3.1. Other forms of the heterogeneous telegrapher’s
equation can be derived for a voltage and current in a inhomogeneous lossy
transmission line, from the generalised Fick’s law with position-dependent
diffusivity, as well as from the persistent random walk in inhomogeneous
medium. For details please see Appendix A. The motivation to consider
heterogeneous models comes from the application of the telegrapher’s equation
in the description of turbulent diffusion turb ; turb1 ; turb2 , as well as in
cosmic-ray transport cosmic . Moreover, the heterogeneous telegrapher’s
equation may also be important in the description of turbulent relative
dispersion of particle pairs turbulent1 ; turbulent2  and represents a
generalization of the Richardson model richardson , since it takes into
consideration the long-time correlation of the Lagrangian relative velocity of
a particle pair, which exists in turbulent flows sawford ; sokolov .

##  III Solution of the heterogeneous telegrapher’s equation

###  III.1 Solution for $x_{0}\neq 0$

In what follows we consider equation of form (6) for power-law position
dependent speed $v(x)=v_{\alpha}|x|^{\frac{\alpha}{2}}$, where $v_{\alpha}>0$
is with physical dimension $[v_{\alpha}]=\text{m}^{1-\alpha/2}s^{-1}$. Thus,
we write the heterogeneous telegrapher’s equation (6) in the form

$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}P(x,t)+\frac{\partial}{\partial
t}P(x,t)={\mathcal{D}_{\alpha}}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}P(x,t)\right]\right\\},$ (7)

where $\mathcal{D}_{\alpha}=v^{2}_{\alpha}\tau$ is a diffusion coefficient
with physical dimension $[\mathcal{D}_{\alpha}]=\text{m}^{2-\alpha}s^{-1}$.
For $\tau\rightarrow 0$ and $v_{\alpha}\rightarrow\infty$ such that
$\mathcal{D}_{\alpha}=const$, equation (7) becomes the heterogeneous
(infinite-velocity) diffusion equation chchme ,111Here we note that we use
$p(x,t)$ for the PDF of the HDP, while $P(x,t)$ is the PDF of the
heterogeneous telegrapher’s process.

$\displaystyle\frac{\partial}{\partial
t}p(x,t)=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}p(x,t)\right]\right\\},$ (8)

which is derived from the Langevin equation in the Stratonovich interpretation
chchme

$\displaystyle\dot{x}(t)=\sqrt{2\,\mathcal{D}_{\alpha}|x|^{\alpha}}\,\eta(t),$
(9)

with position dependent diffusion coefficient, where $\eta(t)$ is a white
Gaussian noise of zero mean. Here we use $\alpha<2$ to ensure the growth
condition for existence and uniqueness of the solution of a Markovian
stochastic differential equation, see Ref. chchme . The case with $\alpha=2$
requires a separate consideration and is related to the problem of geometric
Brownian motion in the Stratonovich interpretation pre2020 .

To solve equation (7), we consider the initial conditions222See the discussion
of initial conditions in Refs. weiss2002 ; ejp ; ratanovbook .

$\displaystyle P(x,t=0)=\delta(x-x_{0}),\quad\frac{\partial}{\partial
t}P(x,t=0)=0,$ (10)

and the boundary conditions are set to zero at infinity, i.e.,

$\displaystyle P(\pm\infty,t)=0,\quad\frac{\partial}{\partial
x}P(\pm\infty,t)=0.$

The Laplace transform333The Laplace transform is defined by
$\hat{f}(s)=\mathcal{L}\left[f(t)\right]=\int_{0}^{\infty}f(t)\,e^{-st}dt$,
while the inverse Laplace transform by
$f(t)=\mathcal{L}^{-1}\left[\hat{f}(s)\right]=\frac{1}{2\pi\imath}\int_{c-\imath\infty}^{c+\imath\infty}\hat{f}(s)\,e^{st}ds$.
of equation (7) yields

$\displaystyle s(1+\tau s)\hat{P}(x,s)-(1+\tau
s)\delta(x-x_{0})=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{P}(x,s)\right]\right\\},$ (11)

which can be rewritten in the form

$\displaystyle
s\hat{P}(x,s)-\delta(x-x_{0})=\mathcal{D}_{\alpha}\frac{1}{1+\tau
s}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{P}(x,s)\right]\right\\}.$ (12)

We note that by inverse Laplace transform, we obtain an equivalent formulation
for equation (7),

$\displaystyle\frac{\partial}{\partial
t}P(x,t)=\mathcal{D}_{\alpha}\int_{0}^{t}K(t-t^{\prime},\tau)\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}P(x,t^{\prime})\right]\right\\}dt^{\prime},$
(13)

where

$K(t,\tau)=\frac{1}{\tau}e^{-t/\tau}\quad\rightarrow\quad\hat{K}(s,\tau)=\frac{1}{1+s\,\tau}.$

Therefore, the heterogeneous telegrapher’s equation (7) can be considered as a
heterogeneous diffusion equation with an non-local memory kernel. Such an
equation with exponential memory kernel was analysed for $\alpha=0$ in
sokolov2002 ; chechkinpre2021 .

For $\tau=0$, we obtain the HDP equation in Laplace space,

$\displaystyle
s\hat{p}(x,s)-\delta(x-x_{0})=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{p}(x,s)\right]\right\\}.$ (14)

Using $s\rightarrow s(1+\tau s)$ in equation (14) we obtain

$\displaystyle s(1+\tau s)\hat{p}(x,s(1+\tau
s))-\delta(x-x_{0})=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{p}(x,s(1+\tau s))\right]\right\\}.$ (15)

Now, one can see that by substituting

$\displaystyle\hat{P}(x,s)=(1+s\,\tau)\,\hat{p}(x,s(1+s\,\tau)),$ (16)

from equation (15) we arrive at equation (12). Thus, we can directly obtain
the solution $P(x,t)$ from the solution $p(x,t)$. Relation (16) is not
affected by the inhomogeneity in space and was derived in sokolov2002  for the
standard telegrapher’s equation, Eq. (13) with $\alpha=0$, by using the
subordination approach. The solution of the diffusion equation for the HDP (8)
is given by (see equation (18) in hdpjpa2022 )

$\displaystyle
p(x,t)=\frac{|x|^{1/p-1}}{\sqrt{4\pi\mathcal{D}_{p}t}}\times\exp\left(-\frac{p^{2}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|^{2}}{4\mathcal{D}_{p}t}\right),$
(17)

where $\mathcal{D}_{\alpha}\rightarrow\mathcal{D}_{p}$ and
$p=\frac{2}{2-\alpha}$, and which in Laplace space reads

$\displaystyle\hat{p}(x,s)=\frac{|x|^{1/p-1}}{2\sqrt{\mathcal{D}_{p}}}s^{-1/2}\times\exp\left(-\frac{p}{\sqrt{\mathcal{D}_{p}}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|s^{1/2}\right).$
(18)

By integration of (18) it is shown that
$\int_{-\infty}^{\infty}\hat{p}(x,s)\,dx=\frac{1}{s}$ (see Appendix A in Ref.
hdpjpa2022 ), which means that $p(x,t)$ is normalized. Therefore, from Eq.
(16), we have

$\displaystyle\int_{-\infty}^{\infty}\hat{P}(x,s)\,dx$
$\displaystyle=(1+s\,\tau)\int_{-\infty}^{\infty}\hat{p}(x,s(1+s\,\tau))\,dx=(1+s\,\tau)\frac{1}{s(1+s\,\tau)}=\frac{1}{s},$
(19)

which means that the PDF $P(x,t)$ is normalized, as well. We also note that in
the limit $\alpha\rightarrow 2$ and $x,x_{0}>0$, the PDF (18) turns into the
log-normal distribution for geometric Brownian motion pre2020 . Thus, for the
PDF $P(x,t)$, we obtain in Laplace space

$\displaystyle\hat{P}(x,s)=$
$\displaystyle\frac{|x|^{1/p-1}}{2v_{p}}(\tau^{-1}+s)(s+\tau^{-1})^{-1/2}s^{-1/2}$
$\displaystyle\times\exp\left(-\frac{p}{v_{p}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|(s+\tau^{-1})^{1/2}s^{1/2}\right),$
(20)

where $v_{p}=\sqrt{\mathcal{D}_{p}/\tau}$. This result can be rewritten in the
form

$\displaystyle\hat{P}(x,s)=\frac{|x|^{1/p-1}}{2v_{p}}(\tau^{-1}+s)\hat{L}(x,s),$
(21)

with

$\displaystyle\hat{L}(x,s)=$ $\displaystyle(s+\tau^{-1})^{-1/2}s^{-1/2}$
$\displaystyle\times\exp\left(-\frac{p}{v_{p}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|(s+\tau^{-1})^{1/2}s^{1/2}\right).$
(22)

The PDF then becomes

$\displaystyle
P(x,t)=\frac{|x|^{1/p-1}}{2v_{p}}\left[\tau^{-1}L(x,t)+\frac{\partial}{\partial
t}L(x,t)\right],$ (23)

where

$\displaystyle L(x,t)=$
$\displaystyle\theta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)e^{-\frac{t}{2\tau}}$
$\displaystyle\times
I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}\left[\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right).$
(24)

Finally, we obtain the form

$\displaystyle P(x,t)=$
$\displaystyle\frac{|x|^{1/p-1}}{2}e^{-\frac{t}{2\tau}}\delta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)$
$\displaystyle+\frac{|x|^{1/p-1}}{4v_{p}\tau}\theta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)e^{-\frac{t}{2\tau}}\left[I_{0}(\xi)+\frac{t}{2\tau}\frac{I_{1}(\xi)}{\xi}\right],$
(25)

where

$\displaystyle\xi=\frac{\sqrt{t^{2}-\frac{p^{2}}{v_{p}^{2}}\left[\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2\tau}.$
(26)

Here we note that $I_{\nu}(z)$ is the modified Bessel function of the first
kind with asymptotic behavior $I_{\nu}(z)\sim\frac{e^{z}}{\sqrt{2\pi z}}$.
Thus, in the long time limit, from (III.1) we arrive at the PDF (17) for the
heterogeneous diffusion equation, while in short time limit we have

$\displaystyle
P(x,t)=\frac{|x|^{1/p-1}}{2}\delta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right).$
(27)

For $\alpha=0$, i.e., $p=1$, we obtain the solution for the standard
telegrapher’s equation goldstein  (see also equation (16) in Ref.
masporraweiss , equation (2.5.3) in Ref. ratanovbook  or equation (6) in
masoliverreset )

$\displaystyle P(x,t)=$
$\displaystyle\frac{e^{-\frac{t}{2\tau}}\delta\left(vt-|x-x_{0}|\right)}{2}$
$\displaystyle+\frac{e^{-\frac{t}{2\tau}}}{4v\tau}\theta\left(vt-|x-x_{0}|\right)\left[I_{0}\left(\frac{\sqrt{v^{2}t^{2}-|x-x_{0}|^{2}}}{2v\tau}\right)+vt\frac{I_{1}\left(\frac{\sqrt{v^{2}t^{2}-|x-x_{0}|^{2}}}{2v\tau}\right)}{\sqrt{v^{2}t^{2}-|x-x_{0}|^{2}}}\right].$
(28)

For $x_{0}=0$, the PDF (III.1) reduces to

$\displaystyle
P(x,t)=\frac{|x|^{1/p-1}}{2}e^{-\frac{t}{2\tau}}\delta\left(v_{p}t-p|x|^{1/p}\right)+$
$\displaystyle\frac{|x|^{1/p-1}}{4v_{p}\tau}e^{-\frac{t}{2\tau}}\theta\left(v_{p}t-p|x|^{1/p}\right)$
$\displaystyle\times\left[I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}|x|^{2/p}}}{2v_{p}\tau}\right)+v_{p}t\frac{I_{1}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}|x|^{2/p}}}{2v_{p}\tau}\right)}{\sqrt{v_{p}^{2}t^{2}-p^{2}|x|^{2/p}}}\right].$
(29)

A graphical representation of the PDF (III.1) for different parameter values
is shown in figure 1. We observe the distinct finite-velocity propagation
where the PDF drops to zero beyond the front
$v_{p}t=p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|$.
We compare the PDFs for the telegrapher’s equation and the diffusion equation
in figure 2. Due to the finite-velocity propagation, we see the drop to zero
of the PDF for the telegrapher’s process (red solid lines), while
instantaneous propagation is observed for the HDP (blue dashed lines). We also
note that the PDF is unimodal for $\alpha>0$ and bimodal for $\alpha<0$. Here
we note that the contribution of the delta functions which ensure
normalization of the PDF at the endpoints $x$, that satisfy
$v_{p}t=p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|$,
are not shown in the figures. We have already proven in Eq. (19) that the PDF
satisfies the normalization condition $\int_{-\infty}^{\infty}P(x,t)\,dx=1$,
however, for the readers’ convenience, in Appendix B we provide additional
proof of the normalization directly by integration of the PDF (23).

(a) (b)
(c) (d)

Figure 1: PDF (III.1) for $x_{0}=5$, $\tau=1$, $v_{\alpha}=1$. (a) $\alpha=0$,
$t=2$ (blue dashed line), $t=5$ (red solid line), $t=10$ (black dotted line).
(b) $\alpha=0.5$, $t=2$ (blue dashed line), $t=5$ (red solid line), $t=10$
(black dotted line). (c) $\alpha=-0.5$, $t=2$ (blue dashed line), $t=5$ (red
solid line), $t=10$ (black dotted line). (d) Comparison between PDFs for
$t=7$, and $\alpha=0.5$ (blue dashed line), $\alpha=0$ (red solid line),
$\alpha=-0.5$ (black dotted line). The delta functions at the endpoints, which
ensure normalization of the PDFs are not shown in the figure.

(a)
(b) (c)

Figure 2: PDF (III.1) for $\tau=1$, $v_{\alpha}=1$ (blue dashed line) versus
PDF (17) (red solid line) for $\mathcal{D}_{\alpha}=1$, at $t=3.5$. (a)
$\alpha=0$ – standard telegrapher’s process versus standard diffusion; (b)
$\alpha=0.5$ (long-time superdiffusion); (c) $\alpha=-0.5$ (long-time
subdiffusion). The contribution of the delta function at the endpoints in the
solution of the telegrapher’s equation (blue dashed lines), which ensure
normalization of the PDF, are not shown in the figure.

###  III.2 Alternative solution for $x_{0}=0$

For the specific initial condition at the origin, $P(x,0)=\delta(x)$, we find
the solution of equation (7) in Laplace space yielding

$\displaystyle s(1+\tau s)\hat{P}(x,s)-(1+\tau
s)\delta(x)=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{P}(x,s)\right]\right\\}.$ (30)

From here, by differentiation with respect to $x$, we find

$\displaystyle s(1+\tau s)\hat{P}(x,s)-(1+\tau
s)\delta(x)=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{(2\theta(x)-1)\frac{\alpha}{2}|x|^{\alpha-1}\hat{P}(x,s)+|x|^{\alpha}\frac{\partial}{\partial
x}\hat{P}(x,s)\right\\},$ (31)

where $\theta(x)$ is the Heaviside step function. By further differentiation
with respect to $x$, we obtain

$\displaystyle s(1+\tau s)\hat{P}(x,s)-(1+\tau s)\delta(x)=$
$\displaystyle\mathcal{D}_{\alpha}\left\\{\frac{\alpha(\alpha-1)}{2}|x|^{\alpha-2}\hat{P}(x,s)+\alpha|x|^{\alpha-1}\delta(x)\hat{P}(x,s)\right.$
$\displaystyle\left.+\frac{3\alpha}{2}\left[2\theta(x)-1\right]|x|^{\alpha-1}\frac{\partial}{\partial
x}\hat{P}(x,s)+|x|^{\alpha}\frac{\partial^{2}}{\partial
x^{2}}\hat{P}(x,s)\right\\}.$ (32)

We see that equation (III.2) is invariant with respect to inversion
$x\rightarrow-x$, and we use $y=|x|$. Equation (III.2), then becomes

$\displaystyle s(1+\tau s)\hat{P}(y,s)-(1+\tau s)\delta(x)=$
$\displaystyle\mathcal{D}_{\alpha}\frac{(\alpha-1)\alpha}{2}y^{\alpha-2}\hat{P}(y,s)+\mathcal{D})_{\alpha}\alpha
y^{\alpha-1}\hat{P}(y,s)\delta(x)$
$\displaystyle+\mathcal{D}_{\alpha}\frac{3\alpha}{2}y^{\alpha-1}\frac{\partial}{\partial
y}\hat{P}(y,s)+2\mathcal{D}_{\alpha}y^{\alpha}\frac{\partial}{\partial
y}\hat{P}(y,s)\delta(x)+\mathcal{D}_{\alpha}y^{\alpha}\frac{\partial^{2}}{\partial
y^{2}}\hat{P}(y,s),$ (33)

where
$\hat{P}(|x|,s)=\mathcal{C}(s)\hat{f}(|x|,s)=\mathcal{C}(s)\hat{f}(y,s)$, and
$\mathcal{C}(s)$ is a function of $s$. From equation (III.2), we obtain a
system of two equations

$\displaystyle\frac{\partial^{2}}{\partial
y^{2}}\hat{f}(y,s)+\frac{3\alpha/2}{y}\frac{\partial}{\partial
y}\hat{f}(y,s)+\left[-\frac{s(1+s\tau)}{\mathcal{D}_{\alpha}}y^{-\alpha}+\frac{(\alpha-1)\alpha}{2}\frac{1}{y^{2}}\right]\hat{f}(y,s)=0,$
(34)
$\displaystyle-(1+s\tau)=\mathcal{C}(s)\,\mathcal{D}_{\alpha}\left.\left[\alpha
y^{\alpha-1}\hat{f}(y,s)+2y^{\alpha}\frac{\partial}{\partial
y}\hat{f}(y,s)\right]\right|_{y=0}.$ (35)

Equation (34) is the Lommel-type equation

$\displaystyle
z^{\prime\prime}(y)+\frac{1-2\beta^{\prime}}{y}z^{\prime}(y)+\left[\left(a\alpha^{\prime}y^{\alpha^{\prime}-1}\right)^{2}+\frac{\beta^{\prime
2}-\nu^{2}\alpha^{\prime 2}}{y^{2}}\right]z(y)=0,$ (36)

which solution is given by

$z(y)=y^{\beta^{\prime}}Z_{\nu}\left(ay^{\alpha^{\prime}}\right),$

where $Z_{\nu}(y)=C_{1}J_{\nu}(y)+C_{2}N_{\nu}(y)$ and $J_{\nu}(y)$ and
$N_{\nu}(y)$ are the Bessel functions of first and second kind, respectivelly.
Therefore, we have

$\displaystyle\hat{f}(y,s)=y^{\frac{2-3\alpha}{4}}\,Z_{\frac{1}{2}}\left(\imath\frac{2}{2-\alpha}\sqrt{\frac{s(1+s\tau)}{\mathcal{D}_{\alpha}}}y^{\frac{2-\alpha}{2}}\right)=y^{\frac{2-3\alpha}{4}}\,K_{\frac{1}{2}}\left(\frac{2}{2-\alpha}\sqrt{\frac{s(1+s\tau)}{\mathcal{D}_{\alpha}}}y^{\frac{2-\alpha}{2}}\right),$
(37)

where $Z_{\nu}(\imath z)=C_{1}I_{\nu}(z)+C_{2}K_{\nu}(z)$, and $I_{\nu}(z)$
and $K_{\nu}(y)$ are the modified Bessel functions bookGR , where $K_{\nu}(z)$
satisfies the zero boundary conditions at infinity. The PDF then reads

$\displaystyle\hat{P}(x,s)=\mathcal{C}(s)\,|x|^{\frac{2-3\alpha}{4}}\,K_{\frac{1}{2}}\left(\frac{2}{2-\alpha}\sqrt{\frac{s(1+s\tau)}{\mathcal{D}_{\alpha}}}|x|^{\frac{2-\alpha}{2}}\right).$
(38)

From equation (35), by using the series representation of $K_{\nu}(y)$

$\displaystyle
K_{\nu}(z)\sim\frac{\Gamma(\nu)}{2}\left(\frac{z}{2}\right)^{-\nu}\left[1+\frac{z^{2}}{4(1-\nu)}+\dots\right]+\frac{\Gamma(-\nu)}{2}\left(\frac{z}{2}\right)^{\nu}\left[1+\frac{z^{2}}{4(\nu+1)}+\dots\right],$
(39)

for $z\rightarrow 0$ and $\nu\notin Z$, for $\mathcal{C}(s)$ we obtain

$\displaystyle\mathcal{C}(s)=\mathcal{D}_{\alpha}^{-3/4}\,\frac{s^{-1/4}(1+s\tau)^{3/4}}{\sqrt{(2-\alpha)\pi}}.$
(40)

The PDF in Laplace space then reads

$\displaystyle\hat{P}(x,s)$
$\displaystyle=\mathcal{D}_{\alpha}^{-3/4}\,\frac{s^{-1/4}(1+s\tau)^{3/4}}{\sqrt{(2-\alpha)\pi}}|x|^{\frac{2-3\alpha}{4}}K_{\frac{1}{2}}\left(\frac{2}{2-\alpha}\sqrt{\frac{s(1+s\tau)}{\mathcal{D}_{\alpha}}}|x|^{\frac{2-\alpha}{2}}\right)$
$\displaystyle=\frac{|x|^{-\alpha/2}}{2v_{\alpha}}\left(s+\tau^{-1}\right)^{1/2}s^{-1/2}\exp\left(-\frac{2}{2-\alpha}\frac{s^{1/2}\left(s+\tau^{-1}\right)^{1/2}}{v_{\alpha}}|x|^{(2-\alpha)/2}\right),$
(41)

where we use $K_{\frac{1}{2}}(x)=\sqrt{\pi/(2x)}\,e^{-x}$. This is the same
result as (III.1) for $x_{0}=0$, as it should be.

##  IV Calculation of the mean squared displacement

From the PDF (16), we find the MSD
$\left\langle\hat{x}^{2}(s)\right\rangle_{\tau}=\int_{-\infty}^{\infty}x^{2}\,\hat{P}(x,s)\,dx$,

$\displaystyle\left\langle\hat{x}^{2}(s)\right\rangle_{\tau}$
$\displaystyle=(1+s\,\tau)\int_{-\infty}^{\infty}x^{2}\,\hat{p}(x,s(1+s\,\tau))\,dx=(1+s\,\tau)\left\langle\hat{x}^{2}(s(1+s\tau))\right\rangle_{0},$
(42)

where $\left\langle
x^{2}(u)\right\rangle_{0}=\frac{\Gamma\left(1+2p\right)}{p^{2p}}\frac{(\mathcal{D}_{p}u)^{p}}{\Gamma\left(1+p\right)}\,{{}_{1}}F_{1}\left(-p,\frac{1}{2},-p^{2}\frac{|x_{0}|^{2/p}}{4\mathcal{D}_{p}u}\right)$
is the MSD for $\tau=0$, see equation (20) in Ref. hdpjpa2022 . Here
${}_{1}F_{1}(a,b,z)$ is the confluent hypergeometric function of the first
kind.

For $x_{0}=0$, the MSD is $\left\langle
x^{2}(u)\right\rangle_{0}=\frac{\Gamma\left(1+2p\right)}{p^{2p}}\frac{(\mathcal{D}_{p}u)^{p}}{\Gamma\left(1+p\right)}$,
i.e.,
$\left\langle\hat{x}^{2}(s)\right\rangle_{0}=\frac{\Gamma\left(1+2p\right)}{p^{2p}}\mathcal{D}_{p}u^{p}s^{-p-1}$,
and thus, from equation (42) we find

$\displaystyle\left\langle x^{2}(t)\right\rangle_{\tau}$
$\displaystyle=\frac{\Gamma\left(1+2p\right)}{p^{2p}}\left(\frac{\mathcal{D}_{p}}{\tau}\right)^{p}\mathcal{L}^{-1}\left[\frac{s^{-p-1}}{(s+\tau^{-1})^{p}}\right]=\frac{\Gamma\left(1+2p\right)\left(\mathcal{D}_{p}\tau\right)^{p}}{p^{2p}}\left(\frac{t}{\tau}\right)^{2p}E_{1,2p+1}^{p}\left(-\frac{t}{\tau}\right),$
(43)

where

$\displaystyle
E_{\rho,\beta}^{\delta}(z)=\sum_{n=0}^{\infty}\frac{(\delta)_{n}}{\Gamma(\rho
n+\beta)}\frac{z^{n}}{n!}$ (44)

is the three-parameter Mittag-Leffler function prabhakar , and
$(\delta)_{n}=\Gamma(\delta+n)/\Gamma(\delta)$ is the Pochhammer symbol. To
perform the inverse Laplace transform in Eq. (43) we use the formula, see Eq.
(5.1.33) in Ref. mainardi book ,

$\displaystyle\mathcal{L}^{-1}\left[\frac{s^{\rho\delta-\beta}}{(s^{\rho}+\lambda)^{\delta}}\right]=t^{\beta-1}E_{\rho,\beta}^{\delta}\left(-\lambda{t}^{\rho}\right),$
(45)

with $|\lambda/s^{\rho}|<1$ (note that in Eq. (43) we have $\rho\rightarrow
1$, $\delta\rightarrow p$, and $\rho\delta-\beta\rightarrow-p-1$, which means
$\beta\rightarrow 2p+1$). From the definition of the Mittag-Leffler function
and the known formula garrappa

$\displaystyle
E_{\rho,\beta}^{\delta}(-z)=\frac{z^{-\delta}}{\Gamma(\delta)}\sum_{n=0}^{\infty}\frac{\Gamma(\delta+n)}{\Gamma(\beta-\rho(\delta+n))}\frac{(-z)^{-n}}{n!},$
(46)

with $z>1$, and $0<\rho<2$, we find the asymptotic behavior of the MSD in the
short and long time limits,

$\displaystyle\left\langle
x^{2}(t)\right\rangle_{\tau}\sim\frac{\Gamma\left(1+2p\right)}{p^{2p}}\left(\mathcal{D}_{p}\tau\right)^{p}\left\\{\begin{array}[]{l
l l}\frac{\left(t/\tau\right)^{2p}}{\Gamma\left(1+2p\right)},\quad t/\tau\ll
1,\\\ \\\ \frac{(t/\tau)^{p}}{\Gamma\left(1+p\right)},\quad t/\tau\gg
1.\end{array}\right.$ (50)

From these results we take the following conclusions: (i) for $0<\alpha<2$ we
have a crossover from hyperdiffusion $\left\langle x^{2}(t)\right\rangle\simeq
t^{\mu_{1}}$, $\mu_{1}=\frac{4}{2-\alpha}$, $\mu_{1}>2$ to $\left\langle
x^{2}(t)\right\rangle\simeq t^{\mu_{2}}$, $\mu_{2}=\frac{2}{2-\alpha}$, which
means: (a) either superdiffusion for $0<\alpha<1$, (b) ballistic motion for
$\alpha=1$, (c) or hyperdiffusion for $1<\alpha<2$; (ii) for $\alpha=0$ we
observe a crossover from ballistic motion $\left\langle
x^{2}(t)\right\rangle\simeq t^{2}$, to normal diffusion $\left\langle
x^{2}(t)\right\rangle\simeq t$; (iii) for $-2<\alpha<0$ we have a crossover
from superdiffusion $\left\langle x^{2}(t)\right\rangle\simeq t^{\mu_{1}}$,
$\mu_{1}=\frac{4}{2-\alpha}$, $1<\mu_{1}<2$ to subdiffusion with $\left\langle
x^{2}(t)\right\rangle\simeq t^{\mu_{2}}$, $\mu_{2}=\frac{2}{2-\alpha}$,
$1/2<\mu_{2}<1$; (iv) for $\alpha=-2$ we have a crossover from normal
diffusion $\left\langle x^{2}(t)\right\rangle\simeq t^{\mu_{1}}$,
$\mu_{1}=\frac{4}{2-\alpha}=1$ to subdiffusion with $\left\langle
x^{2}(t)\right\rangle\simeq t^{\mu_{2}}$,
$\mu_{2}=\frac{2}{2-\alpha}=\frac{1}{2}$; (iv) for $\alpha<-2$ we obtain a
crossover from subdiffusion $\left\langle x^{2}(t)\right\rangle\simeq
t^{\mu_{1}}$, $\mu_{1}=\frac{4}{2-\alpha}$, $0<\mu_{1}<1$ to subdiffusion with
$\left\langle x^{2}(t)\right\rangle\simeq t^{\mu_{2}}$,
$\mu_{2}=\frac{2}{2-\alpha}$, $0<\mu_{2}<1/2$. Therefore, various diffusive
crossovers are observed, rendering the considered model a suitable basis for
the description of anomalous dynamics in complex systems. The obtained results
are summarized in Table 1. In figure 3, we show the graphical representation
of the MSD (43) where we observe the characteristic crossover dynamics from
$\langle x^{2}(t)\rangle\sim t^{2p}$ to $\langle x^{2}(t)\rangle\sim t^{p}$.

|  MSD – short time behavior |  MSD – long time behavior
---|---|---
|  $\langle x^{2}(t)\rangle\sim t^{\mu_{1}}$, $\mu_{1}=4/(2-\alpha)$ |  $\langle x^{2}(t)\rangle\sim t^{\mu_{2}}$, $\mu_{2}=\mu_{1}/2$
$1<\alpha<2$ |  $\mu_{1}>4$ – hyperdiffusion |  $\mu_{2}>2$ – hyperdiffusion
$\alpha=1$ |  $\mu_{1}=4$ – hyperdiffusion |  $\mu_{2}=2$ – ballistic motion
$0<\alpha<1$ |  $2<\mu_{1}<4$ – hyperdiffusion |  $1<\mu_{2}<2$ – superdiffusion
$\alpha=0$ |  $\mu_{1}=2$ – ballistic motion |  $\mu_{2}=1$ – normal diffusion
$-2<\alpha<0$ |  $1<\mu_{1}<2$ – superdiffusion |  $1/2<\mu_{2}<1$ – subdiffusion
$\alpha=-2$ |  $\mu_{1}=1$ – normal diffusion |  $\mu_{2}=1/2$ – subdiffusion
$\alpha<-2$ |  $0<\mu_{1}<1$ – subdiffusion |  $0<\mu_{2}<1/2$ – subdiffusion
Table 1: Characteristic crossover regimes in finite-velocity HDPs Figure 3:
MSD (43) for $\tau=1$, $\mathcal{D}_{\alpha}=1$, and $\alpha=0.5$ (blue dashed
line), $\alpha=0$ (red solid line), $\alpha=-0.5$ (black dotted line).

##  V Finite-velocity HDP with stochastic resetting

###  V.1 Probability density function and non-equilibrium stationary state

We now turn to the analysis of the effect of stochastic resetting on the
finite-velocity HDP. We consider a Poissonian stochastic resetting mechanism
evans1 ; evans2  with instantaneous resetting events, whereas we leave the
case of non-instantaneous resetting for future research. Thus from the simple
renewal equation approach we deduce evans2014 ; mendez2019 ; bodrova1 ;
bodrova2

$\displaystyle
P_{r}(x,t)=e^{-rt}P(x,t)+\int_{0}^{t}r\,e^{-rt^{\prime}}P(x,t^{\prime})\,dt^{\prime},$
(51)

where $P(x,t)$ is the PDF (III.1) in absence of resetting, which means that
each resetting event to the initial position $x_{0}$ renews the process at a
rate $r$. In this equation, the first term on the right-hand side corresponds
to the fraction that there is no resetting event up to time $t$, while the
second term describes multiple resetting events up to time $t$. The standard
telegrapher’s equation, which is a special case of our model for $p=1$
($\alpha=0$), in presence of stochastic resetting was analysed in
masoliverreset ; rt1 . Numerous examples of space-dependent diffusion in soft
matter systems and recent experimental advances exper  motivated new studies
on inhomogeneous diffusion processes with resetting. Thus, the particular
cases of heterogeneous diffusion processes with $\alpha=1$ and stochastic
Poissonian resetting were considered ray1 ; ray2 . In this context, our study
is a natural generalization of these very recent advances. Moreover, in view
of the importance of resetting phenomena in the context of search problems, we
can speculate that the heterogeneous telegrapher’s equation with resetting
represents a toy model of random search in a turbulent environment. We also
note that run-and-tumble particle motion under stochastic resetting rt1 ; rt2
; rt3 ; rt4  and more generalized models of Lévy walks under resetting xu ;
xu2  are of current interest.

By Laplace transform of equation (51), it follows that

$\displaystyle\hat{P}_{r}(x,s)=\frac{s+r}{s}\hat{P}(x,s+r).$ (52)

Using this relation, and in combination with equation (11), we arrive at the
relation

$\displaystyle\tau\left[s^{2}\hat{P}_{r}(x,s)-s\delta(x-x_{0})\right]$
$\displaystyle+(2r\tau+1)\left[s\hat{P}_{r}(x,s)-\delta(x-x_{0})\right]$
$\displaystyle=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}\hat{P}_{r}(x,s)\right]\right\\}-r(r\tau+1)\left[\hat{P}_{r}(x,s)-\frac{1}{s}\delta(x-x_{0})\right].$
(53)

By inverse Laplace transform we derive the corresponding telegrapher’s
equation with position-dependent diffusion coefficient in the presence of
stochastic resetting,

$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}P_{r}(x,t)+(2r\tau+1)\frac{\partial}{\partial
t}P_{r}(x,t)=\mathcal{D}_{\alpha}\frac{\partial}{\partial
x}\left\\{|x|^{\frac{\alpha}{2}}\frac{\partial}{\partial
x}\left[|x|^{\frac{\alpha}{2}}P_{r}(x,t)\right]\right\\}-r(r\tau+1)\left[P_{r}(x,t)-\delta(x-x_{0})\right].$
(54)

Next, we will show that in the long time limit the system reaches a non-
equilibrium stationary state (NESS). From Eqs. (III.1) and (52), we obtain

$\displaystyle\hat{P}_{r}(x,s)=$
$\displaystyle\frac{|x|^{1/p-1}}{2v_{p}}\frac{(s+r+\tau^{-1})^{1/2}(s+r)^{1/2}}{s}$
$\displaystyle\times\exp\left(-\frac{p}{v_{p}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|(s+r+\tau^{-1})^{1/2}(s+r)^{1/2}\right).$
(55)

From here, from the final value theorem
$\lim_{t\rightarrow\infty}f(t)=\lim_{s\rightarrow 0}s\hat{f}(s)$ schiff , we
find the NESS

$\displaystyle P_{r}^{st}(x)$
$\displaystyle=\lim_{t\rightarrow\infty}P_{r}(x,t)=\lim_{s\rightarrow
0}s\hat{P}_{r}(x,s)$
$\displaystyle=\frac{|x|^{1/p-1}}{2v_{p}}\sqrt{r(r+\tau^{-1})}\times\exp\left(-\sqrt{r(r+\tau^{-1})}\frac{p}{v_{p}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right).$
(56)

For $\tau\rightarrow 0$ (note that
$v_{p}=\sqrt{\mathcal{D}_{p}/\tau}\rightarrow\infty$), we recover the result
for HDPs with stochastic resetting, see equation (26) in Ref. hdpjpa2022 ,

$\displaystyle
p_{r}^{st}(x)=\frac{|x|^{1/p-1}}{2\sqrt{\mathcal{D}_{p}/r}}\times\exp\left(-\frac{p}{\sqrt{\mathcal{D}_{p}/r}}\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right).$
(57)

A graphical representation of the NESS is given in figure 4. For $\alpha=0$
($p=1$) this is in fact a Laplace distribution masoliverreset ; rt1

$\displaystyle
P_{r}^{st}(x)=\frac{\sqrt{r(r+\tau^{-1})}}{2v}\times\exp\left(-\frac{\sqrt{r(r+\tau^{-1})}}{v}\left|x-x_{0}\right|\right).$
(58)

In figure 4(a) we observe that the PDF has a cusp at $x_{0}>0$ since the
resetting mechanism introduces a source of probability at $x_{0}$. For
$\alpha>0$ we observe another cusp at $x=0$ since for small $x$ the intensity
of the multiplicative noise in the Langevin equation becomes very small such
that the particle spends more time around the origin before it is reset to the
initial position $x_{0}$. For $\alpha<0$ the PDF shows an anti-cusp at $x=0$
since for small $x$ the intensity of the multiplicative noise becomes very
large, and the particle does not spend much time near the origin. For
$x_{0}=0$, see figure 4(b), the PDFs show a cusp only at $x=0$.

(a) (b)

Figure 4: NESS (V.1) for $\tau=1$, $v_{\alpha}=1$, $r=0.01$ and $\alpha=0.5$
(blue dashed line), $\alpha=0$ (red solid line), $\alpha=-0.5$ (black dotted
line). (a) $x_{0}=5$, (b) $x_{0}=0$.

###  V.2 Transition to the non-equilibrium stationary state

In order to find the relaxation dynamics to the NESS, we consider the renewal
equation (51). We see that in the long time limit the dominant term is the
integral term, which will be estimated by the Laplace approximation for large
$t$. For the integral, we obtain (see Appendix C)

$\displaystyle\int_{0}^{t}r\,e^{-rt^{\prime}}P(x,t^{\prime})\,dt^{\prime}\sim
r\frac{|x|^{1/p-1}}{4v_{p}\sqrt{\tau\pi}}\sqrt{t}\int_{0}^{1}\left[1+\frac{\tau_{0}}{\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{p}^{2}}}}\right]\frac{e^{-t\Phi(\tau_{0},w)}}{\left(\tau_{0}^{2}-\frac{w^{2}}{v_{p}^{2}}\right)^{1/4}}d\tau_{0},$
(59)

where

$\displaystyle\Phi(\tau_{0},w)=\left(r+\frac{1}{2\tau}\right)\tau_{0}-\frac{1}{2\tau}\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{p}^{2}}},$
(60)

and

$w=\frac{p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|}{t}.$

Note that the integral in (59) is always convergent in spite of the
singularity arising from the denominator in the integrand. From the Laplace
approximation arfken

$\displaystyle\mathcal{I}(t)\approx
e^{-t\,f(z_{0})}g(z_{0})\sqrt{\frac{2\pi}{t|f^{\prime\prime}(z_{0})|}},$ (61)

of the integral

$\displaystyle\mathcal{I}(t)=\int_{0}^{1}e^{-tf(z)}g(z)\,dz$ (62)

for large $t$, which requires the evaluation of the minimum of the function
$f(z)$, i.e., $f^{\prime}(z_{0})=0$, if $0<z_{0}<1$ (if the extremum point
$z_{0}$ is outside the integration limits, $z_{0}>1$, then the approximation
result is calculated at $z_{0}=1$), we find the integral (59). The extremum
point can be calculated from
$\left.\frac{\partial}{\partial\tau_{0}}\Phi(\tau_{0},w)\right|_{\tau_{0}=\tau_{0}^{\ast}}=0$,
which gives

$\displaystyle\tau_{0}^{\ast}=\frac{2r\tau+1}{\sqrt{(2r\tau+1)^{2}-1}}\frac{w}{v_{p}}.$
(63)

From here, we find that the PDF behaves as

$\displaystyle P_{r}(x,t)\sim e^{-t\,I(w)},$ (64)

where the large deviation function (LDF) reads

$\displaystyle
I(w)=\left\\{\begin{array}[]{ll}\sqrt{r(r+\tau^{-1})}\frac{w}{v_{p}},&\quad\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\leq\frac{2\sqrt{r\tau(r\tau+1)}}{p(2r\tau+1)}v_{p}t,\\\
\\\
\left(r+\frac{1}{2\tau}\right)-\frac{1}{2\tau}\sqrt{1-\frac{w^{2}}{v_{p}^{2}}},&\quad\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\geq\frac{2\sqrt{r\tau(r\tau+1)}}{p(2r\tau+1)}v_{p}t.\end{array}\right.$
(68)

From here we conclude that the length scale grows like
$\xi(t)\sim\left(v_{p}t\right)^{p}$.

For $x_{0}=0$, we have ($w=p\frac{|x|^{1/p}}{t}$)

$\displaystyle
I(|x|/\xi(t))=\left\\{\begin{array}[]{ll}p\sqrt{r(r+\tau^{-1})}\frac{(|x|/t^{p})^{1/p}}{v_{p}},&\quad|x|\leq\left(\frac{2\sqrt{r\tau(r\tau+1)}}{p(2r\tau+1)}\right)^{p}v_{p}^{p}t^{p},\\\
\\\
\left(r+\frac{1}{2\tau}\right)-\frac{1}{2\tau}\sqrt{1-p^{2}\frac{(|x|/t^{p})^{2/p}}{v_{p}^{2}}},&\quad|x|\geq\left(\frac{2\sqrt{r\tau(r\tau+1)}}{p(2r\tau+1)}\right)^{p}v_{p}^{p}t^{p},\end{array}\right.$
(72)

and the length scale is $\xi(t)\sim\left(v_{p}t\right)^{p}$. The trajectories
corresponding to the first line of equation (72) are relaxed to the NESS (note
that the LDF corresponds to the one of the PDF (V.1), as it should), while
those satisfying the second line of equation (72) are not relaxed and are
still in transient regime. The boundary between the NESS region and the
transient region moves with a non-constant velocity $v(t)\sim
v_{p}^{p}t^{p-1}$ (see figure 5). For $\tau\rightarrow 0$ the LDF reduces to
the one for the HDP hdpjpa2022 . For the standard telegrapher’s equation
($p=1$, i.e., $\alpha=0$), the LDF becomes

$\displaystyle
I(|x|/t)=\left\\{\begin{array}[]{ll}\sqrt{r(r+\tau^{-1})}\frac{|x|/t}{v},&\quad|x|\leq\frac{2\sqrt{r\tau(r\tau+1)}}{2r\tau+1}vt,\\\
\\\
\left(r+\frac{1}{2\tau}\right)-\frac{1}{2\tau}\sqrt{1-\frac{x^{2}/t^{2}}{v^{2}}},&\quad|x|\geq\frac{2\sqrt{r\tau(r\tau+1)}}{2r\tau+1}vt.\end{array}\right.$
(76)

For $\tau\rightarrow 0$ (HDP), we arrive at the known LDF ldf_paper

$\displaystyle
I(|x|/t)=\left\\{\begin{array}[]{ll}\sqrt{r/D}\,\frac{|x|}{t},&|x|\leq\sqrt{4Dr}\,t,\\\
\\\
r+\frac{1}{4D}\left(\frac{|x|}{t}\right)^{2},&|x|\geq\sqrt{4Dr}\,t.\end{array}\right.$
(80)

In figure 5, we show the boundaries between the regions in which the particles
have already relaxed to the NESS and the region in which the particles are in
a transient regime. It is evident that the length scale depends on the
parameter $\alpha$.

(a)
(b) (c)

Figure 5: Boundary between the region where the NESS is achieved and the
transient region for $\tau=1$, $v_{\alpha}=1$, $r=0.01$ and (a) $\alpha=0$ –
standard telegrapher’s process, (b) $\alpha=0.5$, (c) $\alpha=-0.5$.

###  V.3 Mean squared displacement

The MSD can be calculated from equation (52), yielding

$\displaystyle\langle
x^{2}(s)\rangle_{r}=\frac{s+r}{s}\langle\hat{x}^{2}(x,s+r)\rangle_{\tau},$
(81)

where $\langle\hat{x}^{2}(x,s)\rangle_{\tau}$ is given by equation (42). From
here we conclude that in the short time limit ($s\rightarrow\infty$) the MSD
in presence of resetting behaves analogously to the MSD in absence of
resetting $\langle x^{2}(t)\rangle_{r}=\langle\hat{x}^{2}(x,t)\rangle_{\tau}$,
while in the long time limit ($s\rightarrow 0$) it saturates to $\langle
x^{2}(t)\rangle_{r}=r\langle\hat{x}^{2}(x,r)\rangle_{\tau}$.

For $x_{0}=0$, the MSD becomes

$\displaystyle\left\langle x^{2}(t)\right\rangle_{r}$
$\displaystyle=\frac{\Gamma\left(1+2p\right)\left(\mathcal{D}_{\alpha}/\tau\right)^{p}}{p^{2p}}\mathcal{L}^{-1}\left[\frac{s^{-1}(s+r)^{-p}}{(s+r+\tau^{-1})^{p}}\right]$
$\displaystyle=\frac{\Gamma\left(1+2p\right)\left(\mathcal{D}_{\alpha}\tau\right)^{p}}{p^{2p}\tau}\int_{0}^{t}e^{-rt^{\prime}}\left(\frac{t^{\prime}}{\tau}\right)^{2p-1}E_{1,2p}^{p}\left(-\frac{t^{\prime}}{\tau}\right)\,dt^{\prime}.$
(82)

In the short time limit it behaves as the MSD in absence of resetting,
equation (43),

$\displaystyle\left\langle x^{2}(t)\right\rangle_{r}$
$\displaystyle\sim\frac{\Gamma\left(1+2p\right)\left(\mathcal{D}_{\alpha}\tau\right)^{p}}{p^{2p}\tau}\int_{0}^{t}\left(\frac{t^{\prime}}{\tau}\right)^{2p-1}E_{1,2p}^{p}\left(-\frac{t^{\prime}}{\tau}\right)\,dt^{\prime}$
$\displaystyle=\frac{\Gamma\left(1+2p\right)\left(\mathcal{D}_{\alpha}\tau\right)^{p}}{p^{2p}}\left(\frac{t}{\tau}\right)^{2p}E_{1,2p+1}^{p}\left(-\frac{t}{\tau}\right),$
(83)

while in the long time limit the MSD saturates to $\langle
x^{2}(t)\rangle_{r}\sim\frac{1}{r^{p}(r+\tau^{-1})^{p}}$, due to the resetting
mechanism. A graphical representation of the MSD (V.3) is shown in figure 6,
where the its saturation due to the stochastic resetting is clearly observed.

Figure 6: MSD (V.3) for $\tau=1$, $\mathcal{D}_{\alpha}=1$, $r=0.01$, and
$\alpha=0.5$ (blue dashed line), $\alpha=0$ (red solid line), $\alpha=-0.5$
(black dotted line).

##  VI Summary

We reported exact results for the heterogeneous telegrapher’s equation. A rich
range of different diffusion regimes were observed, such as a crossover from
hyperdiffusion to either superdiffusion, ballistic motion, or hyperdiffusion
with different exponent, from ballistic motion to normal diffusion, from
superdiffusion to subdiffusion, from normal diffusion to subdiffusion, or from
subdiffusion with larger exponent to subdiffusion with lower exponent.
Therefore, the considered model is suitable to describe anomalous diffusion in
complex systems exhibiting characteristic crossover dynamics, including
finite-velocity diffusion in random media. We also analysed the finite-
velocity HDP with stochastic resetting and we showed that the system reaches a
NESS. The transition to the NESS was analysed in terms of the large deviation
function. We also found the boundaries between the region in which the system
relaxed to the NESS and the transient region as a function of $\alpha$. Exact
results for the MSD under resetting were obtained, as well. The anomalous
diffusive regime saturates in the long time limit due to the resetting
mechanism.

Future research could be related to the investigation of ergodic properties of
finite-velocity HDPs in absence and presence of resetting chchme ; ch1 ; ch3 ;
pregbm ; jpagbm ; wei , including also corresponding higher-dimensional
formulations ch2 ; me2020 . Infinite- and finite-velocity HDPs in presence of
time-dependent resetting pal_t , non-instantaneous bodrova2 ; radice1  and
space-time coupled returns pal_njp , HDPs in presence of resetting in an
interval pal_int ; roldan  and bounded in complex potential csf_reset , as
well as discrete space-time resetting models discrete models  for HDPs, are
other topics worth investigating.

## Acknowledgements

TS was supported by the Alexander von Humboldt Foundation. AC acknowledges
support of the Polish National Agency for Academic Exchange (NAWA). TS and LK
also acknowledge support from the bilateral Macedonian-Chinese research
project 20-6333, funded under the intergovernmental Macedonian-Chinese
agreement. The authors acknowledge financial support by the German Science
Foundation (DFG, Grant No. ME 1535/12-1).

##  Appendix A Different forms of heterogeneous telegrapher’s equation

Here we note that one can derive different form of the heterogeneous
telegrapher’s equations for a voltage and current in a lossy transmission
inhomogeneous line book , which have the form

$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}I(x,t)+\frac{\partial}{\partial t}I(x,t)=\frac{\partial}{\partial
x}\left[D(x)\frac{\partial}{\partial x}I(x,t)\right],$ (84)
$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}V(x,t)+\frac{\partial}{\partial
t}V(x,t)=D(x)\frac{\partial^{2}}{\partial x^{2}}V(x,t),$ (85)

where $\tau=L/R=\text{const}$, $R=\text{const}$ is the resistance,
$L=\text{const}$ is the inductance, $D(x)=\left[RC(x)\right]^{-1}$, and $C(x)$
is the capacitance.

Telegrapher’s equation of form (84) can also be derived from the continuity
equation

$\displaystyle\frac{\partial}{\partial t}n(x,t)+\frac{\partial}{\partial
x}J(x,t)=0,$ (86)

where $n(x,t)$ is the concentration and the flow of particles $J(x,t)$ obeys
the generalized Fick’s law with memory,

$\displaystyle
J(x,t)=\frac{1}{\tau}\int_{0}^{t}e^{(t-t^{\prime})/\tau}D(x)\frac{\partial}{\partial
x}n(x,t^{\prime})\,dt^{\prime},$ (87)

where $\tau$ is a time parameter. From equations (86) and (87), one arrives at
the heterogeneous telegrapher’s equation

$\displaystyle\tau\frac{\partial^{2}}{\partial
t^{2}}n(x,t)+\frac{\partial}{\partial t}n(x,t)=\frac{\partial}{\partial
x}\left[D(x)\frac{\partial}{\partial x}n(x,t)\right].$ (88)

Another form of the heterogeneous telegrapher’s equation can be derived from
the persistent random walk in inhomogeneous medium. It takes the form
weiss2002 ; masoliver1994 ; masoliverweiss

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}P(x,t)+\frac{1}{\tau}\frac{\partial}{\partial
t}P(x,t)=v(x)\frac{\partial}{\partial x}\left[v(x)\frac{\partial}{\partial
x}P(x,t)\right],$ (89)

where $v(x)$ is the position-dependent velocity.

##  Appendix B Normalization of the PDF

Here we provide detailed proof of the normalization condition
$\int_{-\infty}^{\infty}P(x,t)\,dx=1$. From Eq. (23), we have

$\displaystyle\int_{-\infty}^{\infty}P(x,t)\,dx=\mathcal{J}+\tau\,\frac{\partial}{\partial
t}\mathcal{J},$ (90)

with

$\displaystyle\mathcal{J}=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{-\infty}^{\infty}|x|^{1/p-1}$
$\displaystyle\theta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)$
$\displaystyle\times
I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}\left[\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right)dx=\mathcal{J}_{1}+\mathcal{J}_{2},$
(91)

where

$\displaystyle\mathcal{J}_{1}=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{0}^{\infty}|x|^{1/p-1}$
$\displaystyle\theta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)$
$\displaystyle\times
I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}\left[\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right)dx$
(92)

and

$\displaystyle\mathcal{J}_{2}=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{-\infty}^{0}|x|^{1/p-1}$
$\displaystyle\theta\left(v_{p}t-p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)$
$\displaystyle\times
I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-p^{2}\left[\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right)dx.$
(93)

For the first integral, by introducing $p\,x^{1/p}=y$, i.e., $x^{1/p-1}dx=dy$
and then $z=y-p\,\text{sgn}(x_{0})|x_{0}|^{1/p}$, i.e., $dy=dz$, we find

$\displaystyle\mathcal{J}_{1}$
$\displaystyle=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{0}^{\infty}\theta\left(v_{p}t-\left|y-p\,\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-\left[y-p\,\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right)dy$
$\displaystyle=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{0}^{\infty}\theta\left(v_{p}t-\left|z\right|\right)I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-z^{2}}}{2v_{p}\tau}\right)dz.$
(94)

Then we introduce new variable
$\frac{\sqrt{v_{p}^{2}t^{2}-z^{2}}}{2v_{p}\,t}=r$ to obtain

$\displaystyle\mathcal{J}_{1}=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\,\tau}\int_{0}^{\frac{t}{2\tau}}I_{0}(r)\frac{r\,dr}{\sqrt{\left(\frac{t}{2\tau}\right)^{2}-r^{2}}}=e^{-\frac{t}{2\tau}}\sinh\left(\frac{t}{2\tau}\right)$
(95)

For the second integral, we first introduce $z=-x$ and then $p\,z^{1/p}=y$ and
$k=y+p\,\text{sgn}(x_{0})|x_{0}|^{1/p}$, to find

$\displaystyle\mathcal{J}_{2}$
$\displaystyle=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{0}^{\infty}\theta\left(v_{p}t-\left|y+p\,\text{sgn}(x_{0})|x_{0}|^{1/p}\right|\right)I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-\left[y+p\,\text{sgn}(x_{0})|x_{0}|^{1/p}\right]^{2}}}{2v_{p}\tau}\right)dy$
$\displaystyle=\frac{e^{-\frac{t}{2\tau}}}{2v_{p}\tau}\int_{0}^{\infty}\theta\left(v_{p}t-\left|k\right|\right)I_{0}\left(\frac{\sqrt{v_{p}^{2}t^{2}-k^{2}}}{2v_{p}\tau}\right)dk=e^{-\frac{t}{2\tau}}\sinh\left(\frac{t}{2\tau}\right).$
(96)

Therefore,
$\mathcal{J}=2e^{-\frac{t}{2\tau}}\sinh\left(\frac{t}{2\tau}\right)=1-e^{-\frac{t}{\tau}}$.
Thus, we finaly obtain

$\displaystyle\int_{-\infty}^{\infty}P(x,t)\,dx=\left(1-e^{-\frac{t}{\tau}}\right)+\tau\,\frac{\partial}{\partial
t}\left(1-e^{-\frac{t}{\tau}}\right)=1-e^{-\frac{t}{\tau}}+e^{-\frac{t}{\tau}}=1,$
(97)

which completes the proof.

##  Appendix C Calculation of the integral in the renewal equation

Let us analyse the renewal equation

$\displaystyle
P_{r}(x,t)=e^{-rt}P(x,t)+\int_{0}^{t}r\,e^{-rt^{\prime}}P(x,t^{\prime})\,dt^{\prime}.$
(98)

For large time $t$ the integral term is dominant, and thus

$\displaystyle
P_{r}(x,t)\sim\int_{0}^{t}r\,e^{-rt^{\prime}}P(x,t^{\prime})\,dt^{\prime}.$
(99)

We introduce

$\displaystyle\xi^{\prime}=\frac{\sqrt{t^{\prime
2}-\frac{X^{2}}{v_{\alpha}^{2}}}}{2\tau},$ (100)

where

$X=p\left|\text{sgn}(x)|x|^{1/p}-\text{sgn}(x_{0})|x_{0}|^{1/p}\right|.$

We also use $t^{\prime}=t\tau_{0}$ ($dt^{\prime}=t\,d\tau_{0}$), from where it
follows

$\displaystyle\xi^{\prime}=\frac{t}{2\tau}\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{\alpha}^{2}}},$
(101)

where $w^{2}=X^{2}/t^{2}$. Since when $t$ is large then $\xi^{\prime}$ is also
large, and the Bessel function behaves as
$I_{\nu}(\xi^{\prime})\sim\frac{e^{\xi^{\prime}}}{\sqrt{2\pi\xi^{\prime}}}$.
Thus, for the integral, we obtain

$\displaystyle\int_{0}^{t}r\,e^{-rt^{\prime}}P(x,t^{\prime})\,dt^{\prime}$
$\displaystyle\sim\int_{0}^{t}r\,e^{-rt^{\prime}}\frac{|x|^{1/p-1}}{4v_{\alpha}\tau}e^{-\frac{t^{\prime}}{2\tau}}\left[I_{0}(\xi^{\prime})+\frac{t^{\prime}}{2\tau}\frac{I_{1}(\xi^{\prime})}{\xi^{\prime}}\right]dt^{\prime}$
$\displaystyle\sim
r\frac{|x|^{1/p-1}}{4v_{\alpha}\tau}\int_{0}^{t}e^{-\left(r+\frac{1}{2\tau}\right)t^{\prime}}\left[1+\frac{t^{\prime}}{2\tau}\frac{1}{\xi^{\prime}}\right]\frac{e^{\xi^{\prime}}}{\sqrt{2\pi\xi^{\prime}}}dt^{\prime}$
$\displaystyle=r\frac{|x|^{1/p-1}}{4v_{\alpha}\tau}\int_{0}^{t}\left[1+\frac{t^{\prime}}{2\tau}\frac{1}{\xi^{\prime}}\right]\frac{1}{\sqrt{2\pi\xi^{\prime}}}e^{-\left(r+\frac{1}{2\tau}\right)t^{\prime}+\xi^{\prime}}dt^{\prime}$
$\displaystyle=r\frac{|x|^{1/p-1}}{4v_{\alpha}\sqrt{\pi\tau}}\int_{0}^{t}\left[1+\frac{t^{\prime}}{\sqrt{t^{\prime
2}-\frac{X^{2}}{v_{\alpha}^{2}}}}\right]\frac{e^{-\left(r+\frac{1}{2\tau}\right)t^{\prime}+\frac{1}{2\tau}\sqrt{t^{\prime
2}-\frac{X^{2}}{v_{\alpha}^{2}}}}}{\left(t^{\prime
2}-\frac{X^{2}}{v_{\alpha}^{2}}\right)^{1/4}}dt^{\prime}$
$\displaystyle=r\frac{|x|^{1/p-1}}{4v_{\alpha}\sqrt{\tau\pi}}\int_{0}^{1}\left[1+\frac{t\tau_{0}}{\sqrt{t^{2}\tau_{0}^{2}-\frac{X^{2}}{v_{\alpha}^{2}}}}\right]\frac{e^{-\left(r+\frac{1}{2\tau}\right)t\tau_{0}+\frac{t}{2\tau}\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{\alpha}^{2}}}}}{\left(t^{2}\tau_{0}^{2}-\frac{X^{2}}{v_{\alpha}^{2}}\right)^{1/4}}td\tau_{0}$
$\displaystyle=r\frac{|x|^{1/p-1}}{4v_{\alpha}\sqrt{\tau\pi}}\sqrt{t}\int_{0}^{1}\left[1+\frac{\tau_{0}}{\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{\alpha}^{2}}}}\right]\frac{e^{-t\Phi(\tau_{0},w)}}{\left(\tau_{0}^{2}-\frac{w^{2}}{v_{\alpha}^{2}}\right)^{1/4}}d\tau_{0},$
(102)

where

$\displaystyle\Phi(\tau_{0},w)=\left(r+\frac{1}{2\tau}\right)\tau_{0}-\frac{1}{2\tau}\sqrt{\tau_{0}^{2}-\frac{w^{2}}{v_{\alpha}^{2}}}.$
(103)

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September 3, 2019

# Superconductivity in Nb5Ir3-xPtxO

Jiro Kitagawa and Shusuke Hamamoto Department of Electrical
EngineeringDepartment of Electrical Engineering Faculty of Engineering Faculty
of Engineering Fukuoka Institute of Technology Fukuoka Institute of Technology
3-30-1 Wajiro-higashi 3-30-1 Wajiro-higashi Higashi-ku Higashi-ku Fukuoka
811-0295 Fukuoka 811-0295 Japan Japan<EMAIL_ADDRESS>
###### Abstract

We have investigated the superconducting critical temperature $T_{\mathrm{c}}$
of solid solution Nb5Ir3-xPtxO with the Ti5Ga4-type structure. The both end-
members of Nb5Ir3O and Nb5Pt3O are reported to be superconductors with
$T_{\mathrm{c}}$ of 10.5 K and 3.8 K, respectively. Particularly Nb5Ir3O is
considered as a two-gap superconductor. The entire series of alloy hold the
Ti5Ga4-type structure and show the linear $x$ dependence of lattice
parameters. On the other hand, a nonlinear $T_{\mathrm{c}}$ vs $x$ plot was
obtained, suggesting that $T_{\mathrm{c}}$ is not determined by only lattice
parameters. The experimental result is discussed based on the Matthias rule
and compared with those of the other two-gap superconductors such as MgB2,
Mo8Ga41 and Nb3Sn.

superconductivity, Ti5Ga4-type, Nb5Ir3O

##  1 Introduction

In the Mn5Si3-type structure with the space group P63/mcm (No.193), the Mn
sites of 4d and 6g Wyckoff symmetries are occupied by early transition metals,
rare earth or alkaline earth elements, and the Si sites of 6g Wyckoff symmetry
by metalloid elements or post-transition metals. The Mn5Si3-type structure
possesses the interstitial 2b site, allowing the addition of light elements
such as oxygen, boron and carbon with no change of the space group. Such an
ordered variant of Mn5Si3-type structure is called as the Ti5Ga4 or
Hf5CuSn3-type structure. Figures 1(a) and 1(b) show the crystal structure of
Ti5Ga4-type Nb5Ir3O investigated in this study. The oxygen atom is surrounded
by Nb2 atoms in the octahedral site (6g site), which forms a face-sharing Nb26
chain along the $c$-axis. Another octahedral Ir atoms enclose the Nb1 atom (4d
site) also forming a one-dimensional atomic chain along the $c$-axis.
Therefore an expansion of lattice parameter $a$ would enhance the one-
dimensional nature of octahedral Nb26 and Nb1 atomic chains.

Figure 1: (a)(b) Crystal structure of Nb5Ir3O. (c) $c$ vs $a$ plot of
Ti5Ga4-type compounds. Except E-TM+5$d$-LTM group, the data of commercial
crystallographic database CRYSTMET are employed.

The anisotropy parameter of $c/a$ emphasizes the crystallographic feature of
Ti5Ga4-type compounds (see Fig. 1(c)). We have classified the compounds into
four groups AE, RE, AC and E-TM, where the Mn-sites are occupied by alkaline
earth, rare earth, actinide and early transition metal elements, respectively.
On going from AE, RE and E-TM (AC), $c/a$ seems to decrease, suggesting an
important role of atomic size of the Mn-site element for determining the
anisotropy. In the case of AE and RE, the rather large values of both lattice
parameters are probably due to a weak bonding between the parent Mn5Si3-type
compound and added interstitial elements, which is caused by the rather poor
electron counts in parent compound. We separate the E-TM group compounds in
which 5$d$ late transition metals like Ir and Pt occupy the Si sites from the
E-TM group, and denote E-TM+5$d$-LTM in Fig. 1(c). The E-TM+5$d$-LTM group
possesses $c/a$ smaller than that of E-TM group, which means the enhancement
of anisotropy. It should be noted that all compounds in the E-TM+5$d$-LTM
group are superconductors as mentioned below.

Although physical properties of many Ti5Ga4-type compounds are studied[1, 2,
3, 4], the superconductivity is reported only in several compounds[5, 6, 7, 8,
9] such as Nb5Ir3O, Nb5Pt3O, Nb5Ge3C0.3, Zr5Pt3Ox and Zr5Pt3Cx. For Nb5Ir3O,
the non oxygen-doped Nb5Ir3 is a superconductor[5] with the critical
temperature $T_{\mathrm{c}}=$ 9.3 K, and by increasing the oxygen
concentration, $T_{\mathrm{c}}$ progressively increases to 10.5 K. In this
case, the oxygen addition leads to the shrinkage of $c$ and the expansion of
$a$, which is regarded as the enhancement of one-dimensionality of Nb26 and
Nb1 atomic chains. The temperature dependence of specific heat of Nb5Ir3O
cannot be reproduced by a single exponential temperature dependence, but
explained by a two-gap model[5]. The bulk superconductivity with
$T_{\mathrm{c}}$ of 3.8 K has been confirmed for Nb5Pt3O by measuring the
specific heat, which seems to follow a single exponential temperature
dependence just below $T_{\mathrm{c}}$ but shows the deviation at low
temperature[6]. In this study, Pt-substitution effect on $T_{\mathrm{c}}$ is
investigated in the solid solution system Nb5Ir3-xPtxO to further elucidate
the superconductivity of Ti5Ga4-type compounds.

##  2 Materials and Methods

Polycrystalline samples were prepared using Nb powder (99.9%), Nb2O5 powder
(99.9%), Ir powder (99.99%) and Pt wire (99.9%). Nb5Ir3-xO was initially
prepared by arc melting a pellet made by pressing a homogenized mixture of Nb,
Nb2O5 and Ir powders. Then Nb5Ir3-xO was remelted with added Pt wire to form
the stoichiometric composition. The samples were remelted several times to
ensure the homogeneity of the samples. The weight loss during the arc melting
was negligible. Each as-cast sample was annealed in an evacuated quartz tube
at 800 ∘C for 4 days. A powder X-ray diffractometer (Shimadzu, XRD-7000L) with
Cu-K$\alpha$ radiation was used to measure the X-ray diffraction (XRD)
patterns of prepared samples. The temperature dependences of ac magnetic
susceptibility $\chi_{ac}$ (T) and electrical resistivity $\rho$ (T), between
2.8 K and 300 K, were measured using a closed-cycle He gas cryostat. The
mutual inductance method with an alternating filed of 5 Oe and 800 Hz was
employed to measure $\chi_{ac}$ (T).

##  3 Results and Discussion

Figure 2(a) shows the XRD patterns of representative samples, which can be
indexed by the Ti5Ga4-type structure except the minor impurity peaks indicated
by the asterisks. The impurity phase is assigned as Nb3Ir2-Nb3Pt2 solid
solution. The amount of Nb3Ir2, which is a superconductor[10] with
$T_{\mathrm{c}}\sim$ 2.3 K, in $x=$ 0 sample is rather large, but the
$T_{\mathrm{c}}$ is below the our lowest measurement temperature, and the
impurity phase would have no influence on our experimental results. With
increasing $x$, the peak positions with Miller indices ($h$ $k$ 0) and (0 0
$l$) shift to lower and higher 2$\theta$ angles, respectively. This implies
that the lattice parameter $a$ ($c$) increases (decreases) by substituting Pt
into Ir. The lattice parameters are obtained by the least square method[11]
and the $x$ dependence of these parameters are shown in Fig. 2(b). By
increasing $x$, $a$ ($c$) linearly expands (shrinks), which means an enhanced
one-dimensional nature of octahedral Nb26 and Nb1 atomic chains.

Figure 2: (a) XRD patterns of Nb5Ir3-xPtxO. The origin of each pattern is
shifted by an integer value for clarity. (b) $x$ dependence of lattice
parameters of Nb5Ir3-xPtxO.

$\chi_{ac}$ (T) of Nb5Ir3-xPtxO with 0.0$\leq x\leq$ 1.6 and those with
1.8$\leq x\leq$ 3.0 are shown in Figs. 3(a) and 3(b), respectively. Each
sample exhibits diamagnetic signal and the $T_{\mathrm{c}}$ was determined as
being the intercept of the linearly extrapolated diamagnetic slope with the
normal state signal (see the broken lines in the Fig. 3(a)).
$T_{\mathrm{c}}$’s of Nb5Ir3O and Nb5Pt3O are approximately 10.1 K and 4.3 K,
respectively, which are in agreement with the literature values[5, 6]. As $x$
is increased, $T_{\mathrm{c}}$ does not shift for the sample with $x\leq$ 0.6
and starts to decrease at $x\geq$ 0.6.

Figure 3: (a) Temperature dependence of $\chi_{ac}$ for Nb5Ir3-xPtxO with
$x$=0.0, 0.6, 0.8, 1.0 and 1.6. (b) Temperature dependence of $\chi_{ac}$ for
Nb5Ir3-xPtxO with $x$=1.8, 2.0, 2.2, 2.6, 2.8 and 3.0.

Figure 4 shows $\rho$ (T) of representative Nb5Ir3-xPtxO samples, which are
normalized by the room-temperature values listed in Table I. Except Nb5Pt3O,
each sample shows the zero resistivity below $T_{\mathrm{c}}$. For Nb5Pt3O
with low $T_{\mathrm{c}}$, zero resistivity could not be observed at the
lowest achievable temperature. All $\rho$ (T) curves deviate from the
linearity above $T_{\mathrm{c}}$ (see Fig. 4(b)), which is also observed[12,
13] in A15 superconductors like Nb3Sn or a pyrochloa superconductor of KOs2O6.
An additional scattering source[12, 13] is responsible for the deviation, and
Woodward and Cody[12] have presented a well-known empirical formula as
follows:

$\rho=\rho_{0}+AT+\rho_{1}exp(-\frac{T_{0}}{T})$ (1)

, where the first term means a residual resistivity, the second one phonon
part of $\rho$ and the third one describes anomalous temperature dependence.
We have also fitted $\rho$ (T) in Fig. 4(b) using eq. (1), and the well
reproducibilities as depicted by the solid curves are obtained. As can be seen
from Table I summarizing the fitting parameters, $\rho_{1}$ roughly
corresponds to $\rho$(RT)$-\rho_{0}$. For the almost samples, the third term
in eq.(1) dominates over the second one, and $T_{0}$ does not largely change
by the Pt substitution. We note that the deviation from the linearity is
commonly observed in the Ti5Ga4-type superconductors[8, 14] such as Zr5Pt3Ox
and Nb5Ge3C0.3.

Figure 4: Temperature dependence of $\rho$ for Nb5Ir3-xPtxO with $x$=0.0, 0.6, 1.6, 2.2, 2.6 and 3.0 at (a) low temperature and (b) between 2.8 and 300 K. The solid curves in (b) are $\rho$ (T) calculated by eq.(1). Table 1: Room-temperature $\rho$ and parameters obtained by the fitting using eq.(1) for Nb5Ir3-xPtxO with $x$=0.0, 0.6, 1.6, 2.2, 2.6 and 3.0. $x$ |  $\rho$ (RT) ($\mu\Omega$ cm) |  $\rho_{0}$ ($\mu\Omega$ cm) |  $A$ ($\mu\Omega$ cm/K) |  $\rho_{1}$ ($\mu\Omega$ cm) |  $T_{0}$ (K)
---|---|---|---|---|---
0.0 | 271 | 153 | 0.13 | 115 | 109
0.6 | 186 | 137 | 0.067 | 44 | 117
1.6 | 162 | 113 | 0.024 | 64 | 121
2.2 | 337 | 193 | 0.074 | 189 | 129
2.6 | 150 | 95 | 0.046 | 68 | 144
3.0 | 503 | 219 | 0.32 | 265 | 101

Figure 5(a) shows the $x$ dependence of $T_{\mathrm{c}}$, which is contrasted
with the linear $x$ dependence of lattice parameters. Although the latter
behavior may support the enhancement of one dimensional nature of Nb26 and Nb1
atomic chains with increasing $x$, leading to an increase of $T_{\mathrm{c}}$
as observed in Nb5Ir3Ox (see the Introduction), Nb5Ir3-xPtxO exhibits the
opposite behavior. The contrasted results between Nb5Ir3-xPtxO and Nb5Ir3Ox
indicates that the superconductivity of Nb5Ir3O is robust against the
substitution of Pt into Ir and is not largely affected by only the crystal
structure parameters. The electronic specific heat coefficient $\gamma$ of the
normal state reflects the electronic density of states at the Fermi level,
which is responsible for the magnitude of $T_{\mathrm{c}}$. The
$\gamma$-values of Nb5Ir3O and Nb5Pt3O are extracted to be 41 and 30 mJ/molK2,
respectively[6]. Therefore, the electronic density of states at the Fermi
level would partially determine $T_{\mathrm{c}}$. The valence electron
concentration (VEC) per atom is frequently useful tool for investigating the
effect of electronic density of states at the Fermi level. The
$T_{\mathrm{c}}$ vs VEC plot well describes the tendency of $T_{\mathrm{c}}$
for body-centered-cubic and A15 superconductors, and generally shows a broad
maximum at the specific VEC value (Matthias rule)[15]. Employing
$T_{\mathrm{c}}$ data of the Mn5Si3 or Ti5Ga4-type superconductors, we made
the $T_{\mathrm{c}}$ vs VEC plot as shown in Fig. 5(b). Assigned VEC of Nb,
Zr, Ir, Pt, O and C atoms are 5, 4, 9, 10, 6 and 4, respectively. The broad
maximum seems to exist at Nb5Ir3O, and the $x$ dependence of $T_{\mathrm{c}}$
in this study may follow the Matthias rule. Although the reliability of
Matthias rule for two-gap superconductors is not established and further study
is needed, we note that Nb3Sn which is well known familiar A15 superconductor
but recently regarded as a two-gap superconductor[16] follows the Matthias
rule.

Figure 5: (a) $x$ dependence of $T_{\mathrm{c}}$ determined by $\chi_{ac}$
measurements for Nb5Ir3-xPtxO. (b) $T_{\mathrm{c}}$ vs VEC plot for several
Mn5Si3 or Ti5Ga4-type superconductors.

Focusing on two-gap superconductors, the effects of atomic substitution on
$T_{\mathrm{c}}$ have been reported for several compounds. Mg1-xAlxB2 or
MgB2-xCx shows a linear suppression of $T_{\mathrm{c}}$ by increasing the
amount of atomic substitution[17]. Especially C-substitution leads to a more
rapid decrease of $T_{\mathrm{c}}$, which means the superconducting properties
of MgB2 are mainly responsible for boron. The similar result is confirmed in
V-substituted Mo8-xVxGa41, where the end members of Mo8Ga41 and V8Ga41 are 9.7
K two-gap superconductor and non superconductor, respectively[18]. On the
other hand, Nb3Sn provides the non monotonous substitution effect of
$T_{\mathrm{c}}$, especially when the counterpart of end member is also a
superconductor. For example, Nb3Sn1-xInx, in which Nb3In is 5 K
superconductor, shows the $x$ independent $T_{\mathrm{c}}$ up to
$x\sim$0.12[19]. So our study might indicate that two-gap superconductivity
ascribed to the Nb atoms is rather robust against the atomic replacement other
than Nb atom, if the counter part of end Nb-compound is also a superconductor.

##  4 Summary

We have determined $T_{\mathrm{c}}$ of the Ti5Ga4-type solid solution
Nb5Ir3-xPtxO, in which both end-members are superconductors. Especially
Nb5Ir3O is known as a two-gap superconductor with $T_{\mathrm{c}}=$ 10.5 K.
Although the lattice parameters linearly depend on $x$, $T_{\mathrm{c}}$
hardly changes up to $x\sim$ 0.6, suggesting that $T_{\mathrm{c}}$ is not
determined by only lattice parameters. Combining $T_{\mathrm{c}}$ data of the
other Mn5Si3 or Ti5Ga4-type superconductors, the $T_{\mathrm{c}}$ vs VEC plot
is constructed and may be explained by the Matthias rule. The atomic
substitution effect on $T_{\mathrm{c}}$ is compared among two-gap
superconductors such as MgB2, Mo8Ga41 and Nb3Sn. Our result is similar to that
of Nb3Sn1-xInx.

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