[p. 398l, §1]

The upsurge of fractal models in the study of transport properties
of disordered materials poses the question whether the Einstein relation
whose derivation depends on spatial homogeneity is also valid for fractals
or other inhomogeneous structures [1, 2].
Recently a general probabilistic analogue of the Einstein relation
as a connection between mean first-passage times and passage probabilities
in general Markov chains was identified [3, 4].
Here we present a more compact derivation.

[p. 398l, §2]

We will start with a simple probabilistic argument for the basic relation.
We then interpret the result and apply it to finitely ramified fractals.
Form these considerations we obtain the so-called fractal Einstein relation [5].
Finally, we use mean-first-passage times to give the star-triangle transformation
for Markov chains and calculate exactly the fracton (spectral) dimension of fractal trees.
We emphasize the relevance of our results for simulation and experiment.

[p. 398l, §3]

Let us begin by presenting a simple connection between first-passage times
and first-passage probabilities in a finite-dimensional Markov chain.
A finite-dimensional Markov chain can be visualized as a walker (or particle)
moving randomly between a finite number of states (sites).
The transitions of the walker from site to site
are governed by a transition matrix whose elements
give the single-step transition probabilities.
For simplicity we assume the chain to be ergodic, i. e.,
for every pair of sites there is a minimal integer such that .
Thus, after sufficiently long times, the chain reaches stationarity
where every state has a nonzero dwelling probability.
Our objective is now to describe the spatiotemporal behavior
by studying the transitions between two predetermined states.

[p. 398l, §4]

Let denote the probability density that a walker
starting form site at time will reach the site
(boundary)
^{a} (This is a footnote:) ^{a}By a suitable modification of the Markov chain
can also describe transistions between subsets of points.
for the first time after time .
Let be the conditional probability density
for first reaching after a time under the restriction
that the starting point is not visited.
Let the density describe the regeneration time between two visits to the
point .
Analogously, is the first-passage-time density
from to conditioned on not reaching .
We call the conditional probability that the walker
after starting at returns to without having visited .
With probability the walker passes directly form to
without ever returning to .
In Markov chain theory is called the harmonic measure relative to the boundary .

[p. 398r, §5]

The probability density governing the time
between start at and the first visit to consists of two parts.
With probability the walker will visit its starting point
for a second or third time before reaching .
Upon such a visit he starts anew because of the Markov property.
Therefore, in this case the random transition time is the sum of the random time
for conditional regeneration governed by
and the unconditioned first-passage time .
On the other hand, the walker manages with probability
to pass directly to site without revisiting .
Thence, we find

(1a) |

where denotes convolution as is appropiate for sums of random variables. Analogously a regeneration at the origin takes place either without visiting or via a direct visit to and a subsequent transition from to . That implies a second relation,

(1b) |

Laplace transforming equations (1a) und (1b) we obtain

(2a) | |||

(2b) |

Inserting from eq. (2a) into (2b) yields

(3) |

In the limit we get, using and

[p. 399l, §0]

the desired result for the mean-first-passage times ,

(4) |

[p. 399l, §1]

This relation may be visualized as follows.
One out of walkers will arrive at without having revisited .
Therefore, launching successively walkers form ,
one has to wait on the average times the mean regeneration time
until one of them will return who has reached the prescribed point .

[p. 399l, §2]

We now argue that eq. (4) is indeed
a probabilistic analogue of the Einstein relation.
It is furthermore a generalization in the sense that it is valid
for arbitrary inhomogeneous (including fractal) geometries.
For this to be valid we have to identify the quantities analogous
to the diffusion constant and the conductivity.

[p. 399l, §3]

To identify the diffusion constant we note that the relation
for the mean square displacement of a random walk in Euclidean space
is also valid in the form .
Here is the mean first-exit time for the random walk
to leave a sphere of radius around its starting point [6].
This is a consequence of the invariance of the Wiener process
under the transformation with .
If is a realization of the random process then also is a realization.
Thus, the time when exits for the first time
a sphere of radius around its origin defines also
the first-exit time for the scaled trajectory
and a sphere of radius .
It follows that the mean-first-exit time scales as in regular geometries.
With this in mind we can thus take
as the definition of a generalized scale-dependent diffusion coefficient
in an arbitrary inhomogeneous structure of linear dimension .

[p. 399l, §4]

To identify the conductivity we have to look at a different physical situation.
We need to introduce an external potential into our random-walk picture.
This is done by assuming that the walker has a probability
of being absorbed at and subsequently being replaced at site .
This “voltage source” between and will establish a probability
current depending on the magnitude of the “potential” .
If walkers are starting from the origin
then of them will reach without having returned to .
On the average there will be

(5) |

walkers passing through the voltage source between and . In equilibrium the probability current is thus equal to and we recognize (5) as Ohm’s law if interpreted as the conductance. For a system of linear dimension and cross section the probabilistic conductivity is then defined as . This identifies the probability as the essential quantity for the conductivity.

[p. 399l, §5]

We can now return to the pure-random-walk picture without external potential.
Assuming for the mean-first-passage time
to the boundary at a distance form the starting point we get from eq. (4)

where is the corresponding volume. We remember that is the stationary regeneration time (in the absence of the external potential) and hence independent of . We thus arrive at the Einstein relation . Independent of us, Gefen and Goldhirsch [7] have recently developed a similar picture.

[p. 399r, §6]

We proceed to apply this result to a fractal structure.
Consider a finitely ramified fractal lattice such as the Sierpinski gasket or its extensions.
A finite order of ramification [8, 9] can be roughly
characterized by the following two neccessary conditions:
(1) The finite lattice obtained after steps
of the iterative construction of the fractal (called stage- structure)
is connected through only a finite number of “contact sites” with the infinite lattice.
(2) For every the contact sites of a stage--structure
can ba mapped bijectively to those of a stage- structure.
This implies that a random walker on a finitely ramified fractal
can leave or enter a stage- substructure only through a well-defined
finite set of boundary (contact) sites (“bottlenecks”).

[p. 399r, §7]

We now decompose the transition matrix of a stage- structure
according to its boundary sites and its interior sites as

(6) |

An index corresponds to interior points, to boundary sites. We then recall from Markov-chain theory that the mean-first-passage time for a random walker starting at the interior site in the stage- structure is the th component of the vector given by

(7) |

Here the second equality defines the matrix , called the Green’s kernel, is the identity matrix, and denotes a vector whose components are .

[p. 399r, §8]

Our goal is to calculate the dynamical critical exponent for the fractal.
The dynamical exponent governs the diffusive behavior on the fractal [10, 11]
according to or, equivalently,
in terms of the mean-first-passage time.
In the Euclidean case one has .
If the fractal dimension for the lattice is
then is related to the fracton (spectral) dimension by .
Here is the length scaling factor and is defined by

being the number of lattice points in a stage- structure. To calculate we wish to utilize our probabilistic Einstein relation, equation (4). For this we consider a random walk starting at a junction of stage- structures in the fractal lattice. If stage- structures meet at then the number of points in this finite sublattice is roughly . In the long-time limit the stationary probabilities are thus proportional to . If the walker makes one step per unit time he spends a fraction of roughly of his steps at the origin. Thus we have for the regeneration time on the stage- structure.

[p. 400l, §0] If we now compare a stage- with a stage- structure we obtain from equation (4)

(8) |

Here is the mean-first-passage time to a boundary point and is the probability of reaching one of the boundary points without returning to . Since we have assumed dynamic scaling in the form it follows that exists and we can pass to the limit which yields

(9) |

where . Equation (9) has been called the fractal Einstein relation [5, 12]. It is sometimes written , where denotes the length-scaling exponent for the conductivity [1, 13]. This form is obtained from eq. (9) by taking logarithms and dividing by the logarithm of the length scaling factor .

[p. 400l, §1]

We pause to discuss the significance of these results.
First, we remark that for the case of finitely ramified fractals it can be shown [3]
that the probabilities obey a monotonicity property in the form .
This leads to the relation , expressing an interessting connection
between geometric and dynamic properties.
^{b} (This is a footnote:) ^{b}See [12, p. 33] for a different derivation.
Second, eqs. (8) und (9) give rise to a straightforward method
of calculating from numerical similations.
One simply measures directly the mean-first-passage times for two scaled structures.
Taking their ratios gives and thus .
While exact calculations are restricted to finitely ramified fractals
one can use eqs. (8) und (9) on any network as an approximate method.
We expect that this method will converge faster
than directly recording and deducing from .
However, care has to be exercised because is defined as the limit
and it is neccessary to check in any application whether a further increase
in the size of the structure will singificantly alter the value of .
Apart from being easily accessible in simulation and numerical calculations [14, 15]
mean-first-passage times can be measured directly in photoconductivity experiments
on amorphous materials [16], while any experimental determination
of has to be indirect.

[p. 400l, §2]

We conclude this paper with two applications.
First, we derive the probabilistic analogue
of the star-triangle transformation for resistor networks.
Second, we calculate the fracton dimension for a fractal tree.
Both calculations depend on the method of using mean-first-passage times.

[p. 400l, §3]

Consider the Markov chains for a star and a triangle as specified
through the transition matrices,

and

The mean-first-passage times to a site are obtained by eliminating the th row and column from and , respectively, and solving the linear system of equations , where denotes a reduced transition matrix. We demand that the mean-first-passage times between any two corresponding points and () are equal. Solving the resulting systems of linear equations one obtains the star-triangle relations

(10) | |||

(11) | |||

(12) |

Except for the factor this is exactly the star-triangle transformation
for resistor networks if one identifies with the resistance .
As a final application we determine the fracton dimension
of the fractal tree shown in Figure 1.
Because of its dangling ends this cannot be calculated exactly
via the usual real-space renormalization approach.
On the other hand, using the equations for mean-first-passage times,
mentioned above, we obtain [3] the exact scaling factor
already from stages .
From this the fracton dimension follows
^{c} (This is a footnote:) ^{c}[17] have recently obtained the same result.
as .

[p. 400r, §4]

In this paper we have exploited the intimate connections between the mathematical
foundations of the Einstein

[p. 401l, §0] relation and the theory of stochastic processes. We have exerted ourselves for a concise derivation of the mathematical result in order to focus on its interpretation and applicability. In a more systematic approach eq. (4) is found to follow from a general relation between generating functions for conditional first-passage probabilities [4]. Here we have established in a probabilistic framework the links between the mathematical approach and the physical picture, both for the ordinary and for the Einstein relation. These results could be used directly in simulation studies and experiments on transport in inhomogeneous media, regardless of whether the systems behave fractally or not.