# Local Einstein relation for fractals

L. Padilla, J. L. Iguain  Instituto de Investigaciones Físicas de Mar del
Plata (IFIMAR) and
Departamento de Física FCEyN, Universidad Nacional de Mar del Plata,
Deán Funes 3350, 7600 Mar del Plata, Argentina<EMAIL_ADDRESS>
###### Abstract

We study single random walks and the electrical resistance for fractals
obtained as the limit of a sequence of periodic structures. In the long-scale
regime, power laws describe both the mean-square displacement of a random walk
as a function of time and the electrical resistance as a function of length.
We show that the corresponding power-law exponents satisfy the Einstein
relation. For shorter scales, where these exponents depend on length, we find
how the Einstein relation can be generalized to hold locally. All these
findings were analytically derived and confirmed by numerical simulations.

Keywords: Fractals, Power-law behaviours, Einstein relation.

##  1 Introduction

Fractals are characterized by quantities that exhibit power-law behaviour in
space or time. More precisely, as scale invariance occurs for integer powers
of a characteristic length, pure power laws are modulated by logarithmic
periodic functions, that describe the departures from the main trend at
intermediate scales. These modulations have been the object of recent interest
and considerable effort has been devoted toward understanding the relation
between log-periodicity and discrete-scale invariance [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13].

For a given fractal and some related observables, which show (modulated)
power-law behaviours, a problem of interest is to determine whether or not the
exponents associated with these quantities are independent. Sometimes we can
expect a relation as a consequence of underlying physical laws. This is, for
example, the case of the mass $m$, the electric resistance $R$ and the mean-
square-displacement (MSD) $\Delta r^{2}$ for a single random walker. On a
fractal, the first two grow with length $l$ as $m(l)\sim l^{d_{f}}$ and
$R(l)\sim l^{\zeta}$, while the last one grows with time $t$ as $\Delta
r^{2}(t)\sim t^{2/d_{w}}$. The exponents $d_{f}$, $\zeta$ and $d_{w}$ are
known as the fractal, resistance and walk exponents, respectively, and these
power-law behaviours hold for scales large enough to ensure self-similarity.
In an $d$-dimensional euclidean space, the diffusion coefficient $D$ and
conductivity $\sigma$ are related by the Einstein equation [14]

$\sigma=\frac{e^{2}\rho}{k_{B}T}D.$ (1)

Here, $D=\lim_{t\to\infty}\Delta r^{2}(t)/2t$, $\rho$ and $e$ are the density
and charge of mobile particles, $T$ is the temperature and $k_{B}$ is the
Boltzmann constant. Equation (1) is one of the forms of the fluctuation-
dissipation theorem, and can be used together with simple scaling heuristic
arguments, to argue that the fractal, walk, and resistance exponents satisfy
the Einstein relation [14]

$d_{f}=d_{w}-\zeta,$ (2)

This property has been shown to hold asymptotically for some finitely ramified
fractals [15, 16]; which has been used to analyze the periodicity of the
oscillations in dynamic observables, in the first attempts to understand log-
periodic modulation[17]. Einstein relation was also investigated for random
walks on weighted graphs [18], and, more recently, for karst networks
structures [19].

A deterministic fractal can be obtained as the limit of a sequence of periodic
structures. In this procedure, the period increases at every step as $L^{n}$
($n=0,1,2,...$), where $L$ is a basic characteristic length scale. Self-
similarity is manifested in power-law behaviours, which occur for long enough
scales. However, this does not always hold for shorter lengths. Thus, the
local slopes of the observables as a function of time or length, in log-log
scales, are variable quantities, which approach constant values only
asymptotically.

In this work we argue that the local fractal, walk, and resistance exponents
are related through an equation that generalizes (2). This generalization is
obtained analytically, following the steady-state method for the calculation
of the effective diffusion coefficients for periodic substrates [20]. To
further strengthen our findings we perform numerical simulations for two
models of fractals; which confirm the theoretical predictions.

The paper is organized as follows. In Sec. 2 we relate the diffusion
coefficient and the unit cell resistance for a periodic structure. In Sec. 3
we derive the Einstein relation for self-similar systems. In Sec. 4 we
generalize this relation for scale-dependent exponents. In Sec. 5 we confirm
the generalized relation by numerical simulations performed on models of
asymptotic self-similar substrates. Finally, we give our conclusions in Sec.
6.

##  2 Periodic systems

In this section we address the problem of the diffusion coefficient for a
periodic substrate. We follows the steady-state method developed in reference
[20]. We start by introducing the periodic substrate with unit cell of linear
dimension $l$, schematized in figure 1, where the points represent sites, and
the arrows represent hopping rates. On this structure, a mobile particle can
jump between connected sites according to the hopping rates $k^{\prime}s$ (for
the sake of clarity only a few sites and arrows were highlighted). We focus on
a steady-state of non-interacting particles flowing with a constant current
density $j$.

Figure 1: Two nearest-neighbor cells $f$ and $f+1$, for a periodic substrate
with linear size period $l$. The points represent sites, which can be occupied
by mobile particles. The arrows represent hopping rates between pairs of
sites. For clarity, only a few sites and hopping rates were highlighted.
$n_{r}^{(f)}$ corresponds to the number of particles in the internal site $r$
of cell $f$

As shown in [20], this steady-state consists of a set of microscopic currents
distributed with the same periodicity as the substrate. In figure 1 two
nearest-neighbor (NN) unit cells are depicted schematically where, for
example, $n_{s}^{(f)}$ represents the number of particles in site $r$
(internal index) of cell $f$. Because of the mentioned periodicity, we get
that for given pair of connected sites with internal indices $s$ and $t$,

$i_{rs}^{(f)}=i_{rs}^{(f+1)},$ (3)

where $i_{rs}^{(f)}$ is the current from site $s$ to site $r$ in cell $f$. In
addition, as hopping rates do not depend on the cell either but only on the
internal indices, the last equation can be rewritten as

$k_{sr}(n_{s}^{(f)}-n_{r}^{(f)})=k_{sr}(n_{s}^{(f+1)}-n_{r}^{(f+1)}),$ (4)

or

$n_{s}^{(f+1)}-n_{s}^{(f)}=n_{r}^{(f+1)}-n_{r}^{(f)}.$ (5)

Therefore, in the steady-state, the difference in the occupation number for a
given site and the equivalent site in a NN cell is the same for all sites.

The relation of the steady-state problem with the diffusion coefficient $D$ is
provided by Fick’s law

$j=-D\frac{\Delta n}{l^{2}},$ (6)

which is valid for distances larger than $l$. Here $\Delta n$ corresponds to
the particle number difference for NN cells. Note that $D$ also determines the
mean-square displacement $\Delta^{2}x$ of a single random walker on the same
structure, which behaves as

$\Delta^{2}x(t)=2Dt;$ (7)

for time long enough for $\Delta x\gg l$.

Figure 2: Schematics of the equivalence between Fick’s law (left) and Ohm’s
law (right). In the mapping particles have unitary charge, while the other
quantities are related as $V=n$, and $R=1/k$.

Transforming the steady-state problem into an equivalent electrical problem is
straightforward. Indeed, for particles of unitary electric charge, a mapping
between Fick’s law and Ohm’s law results by identifying particle number with
electrostatic potential ($V_{a}=n_{a}$) and hopping rate with conductance
($k=1/R$). In figure 2 we represent this mapping for every pair of connected
sites. Following this analogy, we see that in the electric problem, the
potential difference for a pair of equivalent sites in NN cells takes the
constant value

$\Delta V=n_{r}^{(i+1)}-n_{r}^{(i)},$ (8)

and that the difference between particle populations

$\Delta n=\sum_{r=1}^{M}(n_{r}^{(i+1)}-n_{r}^{(i)})=M\Delta V,$ (9)

is proportional to the potential difference $\Delta V$, where the constant of
proportionality $M$ corresponds to the number of sites per unit cell.

Thus, according to equation (6), we can conclude that, given a periodic
substrate with unit cell of linear dimension $l$ and $M$ sites, the diffusion
coefficient and the potential difference between two equivalent sites in NN
cells, are connected through the relation

$D=-j\frac{l^{2}}{M\Delta V},$ (10)

where $j$ is the steady-state current density.

##  3 Self-similar substrates

Deterministic fractals are usually built by a recursive procedure, that
results in a sequence of structures called generations. A generation consists
of a periodic array of sites connected by bonds. The process begins with a
basic periodic structure (zeroth generation). At every step the unit cell is
scaled by a factor $L$ and the building rules ensure that self-similarity is
obtained after a large number of iterations.

Following equation (10), the diffusion coefficient $D_{p}$ for the generation
$p$ and the potential difference $\Delta V_{p}$ between two equivalent points
in NN unit cells are related as

$D_{p}=-j\frac{L^{2p}}{M_{p}\Delta V_{p}},$ (11)

where $M_{p}$ is the number of sites in the unit cell, and $L^{p}$ is its
linear dimension. Then, for two consecutive generations $p$ and $p+1$, through
which the same steady-state current flows, we obtain

$\frac{D_{p}}{D_{p+1}}=L^{-2}\frac{M_{p+1}}{M_{p}}\frac{\Delta V_{p+1}}{\Delta
V_{p}}.$ (12)

Now, since for a fractal the number of sites in a box with linear dimension
$l$ behaves as $m(l)\sim l^{d_{f}}$ (i. e., $d_{f}$ is the fractal dimension
defined through box-counting),
$M_{p+1}/M_{p}=(L^{(p+1)}/L^{p})^{d_{f}}=L^{d_{f}}$, and the last equation can
be rewritten as

$\frac{D_{p}}{D_{p+1}}=L^{d_{f}-2}\frac{\Delta V_{p+1}}{\Delta V_{p}},$ (13)

As previously shown [7, 8], a perfect diffusive self-similar structure
corresponds to a ratio $D_{p}/D_{p+1}$ which does not depend on $p$, i. e.,

$\frac{D_{p}}{D_{p+1}}=1+\lambda,$ (14)

with $\lambda$ a positive constant. In this model, the mean-square
displacement for a single random walker behaves as

$\Delta^{2}x(t)=f(t)t^{2\nu}.$ (15)

The modulation $f(t)$ is a log-periodic function, $f(t\tau)=f(t)$, and both
$\nu$ and $\tau$ can be analytically calculated in terms of $L$ and $\lambda$:

$\nu=\frac{\displaystyle 1}{\displaystyle
2+\frac{\displaystyle\log(1+\lambda)}{\displaystyle\log(L)}}$ (16)
$\tau=L^{1/\nu}$ (17)

The important partial conclusion in the context of this work is that,
according to above discussion, a perfect diffusive self-similar structure
implies a power-law behaviour for the resistance as a function of length.
Indeed, equations (13) and (14) leads to

$\frac{\Delta V_{p+1}}{\Delta V_{p}}=L^{1/\nu-d_{f}},$ (18)

where we have used $1+\lambda=L^{1/\nu-2}$, from equation (16). Thus, for a
perfect diffusive self-similar fractal the potential difference, which
corresponds to steady-state current, scales with length $l$ as

$\Delta V\sim l^{\zeta},$ (19)

where the exponent $\zeta$ is given by

$\zeta=1/\nu-d_{f};$ (20)

which is the Einstein relation (2), with $d_{w}=1/\nu$.

##  4 Local exponents

We consider now a generic substrate for which diffusive self-similarity is
reached only asymptotically. Let us assume a ratio between consecutive
diffusion coefficients, that depends on the generation $p$, as

$\frac{D_{p}}{D_{p+1}}=1+\lambda_{p}.$ (21)

where, $\\{\lambda_{p}:\,p=1,2,...\\}$ is a sequence of non-negative real
numbers, with $\displaystyle\lim_{p\to\infty}\lambda_{p}=\lambda$.

Because of this limit, at long enough times a single random walk on this
substrate will show a MSD behaviour as in equation (15), and, as pointed out
before, for large enough lengths the potential difference will behave as in
equation (19); with $\nu$ and $\zeta$ given by equations (16) and (20).

In this section we focus on local exponents, which correspond to the slopes in
log-log scales for finite length or time. As shown for example in [8], on a
substrate on which diffusion coefficients for generations $p$ and $p+1$
satisfy equation (21), the MSD for a single random walker behaves as

$\Delta^{2}x(t)\sim t^{2\nu_{p}},\;\;\;\mbox{for}\;\;\;L^{p}\lesssim\Delta
x\lesssim L^{p+1},$ (22)

with the local exponent $\nu_{p}$ given by

$\nu_{p}=\frac{\displaystyle 1}{\displaystyle
2+\frac{\displaystyle\log(1+\lambda_{p})}{\displaystyle\log(L)}}\;\;\cdot$
(23)

Then, after rearranging this equation as $1+\lambda_{p}=L^{1/\nu_{p}-2}$,
which corresponds to the left hand side of equation (13), we obtain

$\frac{\Delta V_{p+1}}{\Delta V_{p}}=L^{1/\nu_{p}-d_{f}}.$ (24)

Thus, we expect that the potential difference scales with length $l$ as

$\Delta V(l)\sim l^{\zeta_{p}},\;\;\;\mbox{for}\;\;\;L^{p}\lesssim l\lesssim
L^{p+1},$ (25)

and that the local exponents satisfy the relation

$\zeta_{p}=1/\nu_{p}-d_{f}.$ (26)

Therefore, local slopes in log-log scales for the resistance as a function of
length and for MSD of a single random walker as a function of time are related
for all scales through equation (26); which generalizes the Einstein relation.

##  5 Numerical simulations

We study numerically the steady-state that corresponds to a unitary current on
two models, for which diffusive self-similarity appears asymptotically. At
finite lengths, the local random-walk exponent $\nu_{p}$ is not constant.
Thus, we expect an also variable resistance exponent $\zeta_{p}$, related to
the former through equation (26).

The first model is a substrate built on a square lattice. A random walk
consists in a particle hopping among NN sites. If sites are connected by a
bond, the hopping rate is $k=1/4$. If the sites are not connected, the hopping
rate is $k=0$. A fractal is obtained by deleting some bonds. The
characteristic scale factor is $L=3$, and the unit cells for the first, the
second and the third generations are depicted schematically in figure 3. For
every generation the unit cell can be separated from the rest by cutting four
bonds. As shown in a previous work, the mass on this structure shows a power-
law behaviour with $d_{f}=2$. However, the random walk exponent $\nu_{p}$
grows with time and approaches a value $\nu<1/2$ when $t\to\infty$ [8].

We have run numerical simulations on the unit cell of the sixth generation, to
reach the steady-state in which a unitary current flows between the left and
right extremes. In figure 4 we plot with symbols the potential differences for
lengths $x=3^{i}$ ($i=0,1,...,6$), which are the unit cell linear sizes for
the generations zero to six. In the same figure, we plot a line using the
relation (26) and the numerical values for $\nu_{p}$, which are the outcomes
of random walk simulations reported in reference [8]. Notice that both data
set fall on the same curve, which confirms the relation (26).

Figure 3: Substrate in two dimensions, which results in scale-dependent walk
and resistance exponents. The schematics correspond to the unit cells for the
first, second and third generations. The segments represent bonds between
sites. Figure 4: Potential difference as a function of length for a unitary
current flowing trough the unit cell of the sixth generation substrate in
figure 3. The symbols correspond to simulations of the steady-state. The line
was plotted with the exponents $\zeta_{p}$ from equation (26) and the values
of $\nu_{p}$ which result from random-walk numerical simulations.  Figure 5:
Schematics of the one-dimensional random-walk model. We begin with a
homogeneous lattice, and a hopping rate $k_{0}$ between nearest-neighbor
sites. Then, hopping rates are reset to $k_{j}$ for transitions between sites
$j$ and $j+1$ for every $j$ such that $mod(j,L^{n})=0$, and for $n=1,2,...$.
In this example, $L=5$.

The second model is a generalization of the one-dimensional self-similar model
introduced in [7]. We start with a single random walk on a one-dimensional
lattice, with a hopping rate $k_{0}$ between any pair of NN sites. This
homogeneous case corresponds to generation zero. We introduce a natural number
$L$ to build the other generations.

In the first generation, we reset to $k_{1}<k_{0}$ the hopping rate for every
pair of sites $j$ and $j+1$, with $mod(j,L)=0$. The other hopping rates
remains as in zeroth generation.

In the second generation, we reset to $k_{2}<k_{1}$ the hopping rate for every
pair of sites $j$ and $j+1$, with $mod(j,L^{2})=0$. The other hopping rates
remains as in first generation.

This recursion follows indefinitely, in such a way that generation $n$ is
obtained from generation $n-1$ after resetting to $k_{n}<k_{n-1}$ the hopping
rate for every pair of sites $j$ and $j+1$, with $mod(j,L^{n})=0$. In figure 5
we show an schematics for $L=5$.

If we ask for perfect self-similarity for diffusion, i. e. equation (14), the
hopping rates are found iteratively as in reference [7]. For the more general
case of equation (21), the sequence of hopping rates is given by

$\frac{1}{k_{i}}=\frac{1}{k_{i-1}}+\frac{L^{i}\lambda_{i-1}}{k_{0}}\prod_{j=0}^{i-2}(1+\lambda_{j}),\;\;\;\;\;\mbox{for}\;\;i=1,2,3...$
(27)

We test the validity of the relation (26) among the local exponents for a
family of substrates given by

$\lambda_{p}=\lambda\,(1-2^{-p/5.}).$ (28)

At short enough lengths these substrates are nearly homogeneous
($\lambda_{p}\approx 0$ for $p\ll 5$), while, on the other extreme, self-
similarity for diffusion is reached for lengths much larger than $L^{5}$. The
local random walk exponent (23) decreases with length and approaches
asymptotically $\nu$ in equation (16). Thus, the variation of $\nu_{p}$ in
space increases with $\lambda$ and, because of equation (26), the same should
occur with the variation of $\zeta_{p}$. This is an interesting model, because
the variation of the exponents with length can be adjusted through the
parameter $\lambda$.

Figure 6: Potential difference as a function of length for unitary current on
the one-dimensional model with $\lambda_{p}=\lambda\,(1-2^{-p/5.})$, and
$L=2$. (Main) Symbols correspond to data obtained with numerical simulations
on a tenth-generation substrate. Lines were drawn using the values of
theoretical exponents. From bottom to top, $\lambda=1$ (red), $\lambda=2$
(green), $\lambda=4$ (violet), $\lambda=5$ (blue). (Inset) More detailed
structure for $\lambda=2$.

We have run numerical simulations for the steady-state that corresponds to a
unitary current flowing on this model, with $L=2$ and $\lambda=1,2,4,5$. All
substrates were built until generation $10$. In figure 6-main we plot with
symbols the potential difference as a function of the length $x$, for
$x=2^{j}$ ($j=0,1,...,9$). The lines correspond to the exponents $\zeta_{p}$
obtained from equations (26) and (23). Note the excellent agreement between
theory and simulations. The inset in the same figure shows substructure of
$\Delta V$ for $\lambda=2$.

##  6 Conclusions

We have studied first the connection between single random walks and steady-
state potential difference for substrates with spatial periodicity. Then, by
considering a sequence of periodic systems, a common procedure for
deterministic fractal construction, we find that the length dependent fractal,
walk and resistance exponents, for the substrate obtained in the infinite
limit of this sequence, satisfy, at every length scale, the relation (26).
This can be considered as a local version of the Einstein relation (2). We
have tested our predictions numerically for two models. The first model is a
fractal in two dimensions, while the the second is a fractal in one dimension.
Both models lead to length-dependent exponents at intermediate scales. The
excellent agreement between the outcomes of these simulations and the
theoretical predictions supports the validity of the mentioned relation among
exponents, not only in the asymptotic self-similar limit but also locally, for
all length scales.

## Acknowledgments

We are grateful to H. O. Mártin for useful discussions. This research was
supported by the Universidad Nacional de Mar del Plata, 15/E1040, and the
Consejo Nacional de Investigaciones Científicas y Técnicas, PIP1748/21.

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