2.1 Correlated Hopping in a Frozen Percolation Network
[179.1.1] Consider the random walk of a single particle of type A
in a percolating network on a regular lattice.
[179.1.2] We will always take the lattice constant of the
underlying lattice to be unity.
[179.1.3] For simplicity we consider the case of bond percolation
instead of site percolation.
[179.1.4] That is, the bonds of the regular lattice
are assumed to be blocked be B-particles (blockers) with probability p.
[179.1.5] If a bond is blocked by a B-particle
it cannot be crossed by the A-particle (walker).
[179.1.6] We assume that the walker has a memory of its previous step.
[179.1.7] It returns with a transition rate ωb, to the previously visited site,
and jumps with a rate ω to any other of the nearest neighbour sites.
[179.1.8] The ratio b=ωb/ω is a measure of the strength of the memory correlations.
[179.1.9] For b>1 the walker returns preferentially to its previously visited site,
and we will refer to this case as “enhanced reversals”.
[179.1.10] In the case b<1 the walker tends to avoid the previously visited site
and this will be termed “reduced reversals”.
[179.1.11] As usual, we are interested in the autocorrelation function P(i,t)=P(r→i,t|r0,0),
i. e. the probability density to find the walker at site i at time t
if it started from site 0 at time 0.
[179.1.12] We will show below that the problem can be formulated
as a system of second order equations for the Pi,t which reads
| d2dt2Pi,t+γi+ωb-ωddtPi,t=ω∑jiAijddtPj,t-Pi,t+ωγi∑jiAijPj,t-Pi,t | | (2.1) |
where γi=ωb+ωzi-1 and zi is the coordination number of site i.
[179.1.13] The symmetric quantities Aij=Aji represent the bond disorder and are defined as
| Aij=1if the bond ij is vacant,0if the bond ij is blocked. | | (2.2) |
[page 180, §0]
[180.0.1] The summation in eq. (2.1a)
runs over the nearest neighbour sites j of site i.
[180.0.2] Note that in the uncorrelated case, b=1, eq. (2.1)
reduces to the usual master equation for a random walk
on a bond percolation network if one replaces Pi,t
by the sum of P and its derivative.
[180.1.1] Equation (2.1) has to be supplemented by initial conditions
for Pi,t and its derivative.
[180.1.2] Special attention has to be paid
to the condition on ddtPi,t and its derivative.
[180.1.3] The correct choice is
| Pi,0+ | =δi0 | | (2.3a) |
| ddtPi,0+ | =[ωb+ω(zi-1)]1zi∑jiAij[P(j,0+)-P(i,0+)]=γi/zi∑jiAij(δj0-δi0) | | (2.3b) |
where the symbol 0+ stands for the limit t→0 from above.
[180.1.4] Note that γi/zi is the average transition rate
out of the starting point.
[180.2.1] We now derive eq. (2.1) as the equations of motion
for our correlated random walk.
[180.2.2] This will be done by a suitable reformulation of the equations
for the correlated random walk on the regular lattice[19],
and subsequent generalization to the disordered case.
[180.2.3] Consider therefore the random walker on a regular lattice.
[180.2.4] The random walker has a memory of its previous step
and as a consequence its walk is not markovian,
i. e. the transition probabilities are not completely determined
by the currently occupied site.
[180.2.5] However a markovian description can be obtained
by introducing an enlarged state space with internal states
which correspond to the previously occupied sites[21].
[180.2.6] Therefore the central quantity is the probability density Pi,j,t
to find the walker at site i at time t
given that it arrived at i via a direct transition from site j.
[180.2.7] Thus j labels the previously occupied site or history.
[page 181, §0]
[181.0.1] Then the symmetric probablity density Pi,t
is obtained from Pi,j,t by a summation over all possible histories
where the sum runs over all nearest neighbour sites j of site i.
[181.0.2] The conditional probability densities Pi,j,t obey the master equation
| ddtPi,j,t=ωbPj,i,t-Pi,j,t+ω∑k≠iPj,k,t-Pi,j,t | | (2.5) |
where the sum runs over all nearest neighbours k of site j
except for site i on the regular lattice.
[181.0.3] This is the starting point for deriving eq. (2.1).
[181.1.1] Equation (2.5) can now be reformulated
by first writing it in a more symmetric form.
[181.1.2] Using eq. (2.4) we can rewrite eq. (2.5) as
| ddtPi,j,t=ωb-ωPj,i,t+ωPj,t-γPi,j,t | | (2.6) |
where γ=ωb+z-1ω, and z denotes the coordination number of the lattice.
[181.1.3] Note that eq. (2.6) reduces to the master equation
for a random walk on a regular lattice if one sets b=1
and sums over all sites j which are nearest neighbours of site i.
[181.1.4] Next we differentiate eq. (2.6) and sum over j.
[181.1.5] We then employ it for i and j interchanged
to eliminate the term ddtPj,i,t and find
| d2dt2Pi,t+γddtPi,t=ωb-ωγPi,t+ω∑jiddtPj,t-ωb-ωγ∑jiPj,i,t. | | (2.7) |
[181.1.6] Solving eq. (2.6) for Pj,i,t
and inserting the result into eq. (2.7)
one obtains a closed second order equation for Pi,t
| d2dt2Pi,t+γ+ωb-ωddtPi,t=ω∑jiddtPj,t-Pi,t+ωγ∑jiPj,t-Pi,t | | (2.8) |
where the summations, as before,
run over all nearest neighbour sites j of site i.
[page 182, §0]
[182.0.1] Eq. (2.8) contains the same information as eq. (2.5)
but no longer involves the directional quantities Pi,j,t.
[182.0.2] This form can now be used to introduce disorder
and it leads directly to eq. (2.1).
[182.0.3] We now turn to the introduction of a time dependent network.
2.2 Correlated Hopping in a Dynamic Percolation Network
[182.1.1] Consider a system where the configuration of accessible sites fluctuates in time.
[182.1.2] We are interested in the case where the B-particles perform a random walk.
[182.1.3] Because we are dealing with bond percolation
this random walk occurs on the dual lattice.
[182.1.4] In an elementary step a blocking bond swings around either one of its end points
through an angle ±2π/z where z is the coordination number of the underlying lattice.
[182.1.5] It then occupies the new bond position if it is vacant.
[182.1.6] This process is repeated on the average after a time τB
which is the characteristic time scale for the blocker motion.
[182.1.7] In Figure 1 we depict the possible rotations
of a B-particle for the case of a hexagonal lattice.
[182.1.8] This model has been termed “dynamic bond percolation” model[22].
[182.1.9] The characteristic hopping time for a single B-particle is called τB.
[182.1.10] The ratio τ=τB/τA between the typical hopping time
of the blockers and the walker
will be the main variable characterizing the dynamics of the environment.
[182.2.1] Equation (2.1) must be generalized
to allow for time dependent transition rates.
[182.2.2] Therefore we have to consider an equation of the form
| d2dt2Pi,t+γi+ωb-ωddtPi,t=ω∑jiAijtddtPj,t-Pi,t+ωγi∑jiAijtPj,t-Pi,t | | (2.9a) |
where now the coefficients A are time dependent,
| Aijt=1if the bond ij is vacant at time t,0if the bond is occupied by a blocker at time t. | | (2.10a) |
[page 183, §0]
[183.0.1] The time dependence of these coefficients could in principle be determined
from the many particle master equation for all blockers.
[183.0.2] However, because that equation is much too complicated
we will approximate the true time dependence
by a simple renewal model in Section 4.
[183.0.3] Equation (2.9) completes the formulation of the model.
[183.0.4] We remark here that other forms of a two step memory
are possible and may be useful for applications.
[183.0.5] For example one can consider enhanced or reduced transitions
continuing in the same direction as the last step.
[183.0.6] Such correlations lead to more complicated equations,
but they can be treated by the same general approach presented here.
[183.1.1] We conclude this section with the formulas that will be used to calculate
the frequency dependent conductivity σω from Pi,t.
[183.1.2] This is done via a generalized Einstein relation which reads
where ρ is the carrier density,
e their electric charge, kB, the Boltzmann constant,
T the absolute tempreature, and Dω
the generalized frequency dependent diffusion coefficient.
[page 184, §0]
[184.0.1] Dω will be calculated in standard fashion from[23]
| D(ω)=-ω2z∫0∞∑r→i,r→0(r→i-r→0)2e-iωtP(r→i,t|r→0,0)dt | | (2.12) |
where P(r→i,t|r→0,0) is the solution
to eq. (2.9) or eq. (2.1) for the frozen case.
[184.0.2] The latter will be determined in the next section.
