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Random walks, dispersion and trapping
Nuclear Physics B (Proc Suppl ) 5A (1988) 229-233 North-Holland, Amsterdam
229
RANDOM WALKS, DISPERSION AND TRAPPING
Chrlst~an VAN DEN BROECK Center for Studies in Statistical Mechanics,
I. INTRODUCTION
The Un~verslty of Texas, Austin,
TX 78712, USA
where um is a "velocity profile" whose form
The theory of random walks has found many applications in physics and new developments,
such
depends on the physics of the problem.
Such
problems are encountered In the study of dls-
as random walks on fracta]s I, walks with inter-
perslon of particles In flows 6, Jn line width
nal states 2, doubly stochastic walks 3, and In-
calculations 7, and so on 8.
teracting walks 4 continue to stimulate intensive research.
One of the central quantities In th]s
theory ]s the cond2tlona]
probability P(m]m0,t)
We will show that both problems are related through a simple mathematical Moreover,
the condltJona]
transformation.
probability P pro-
to go from the slte ~0 at time zero to the site
vldes a partial solution to the dispersion
m at time t. Apart from be2ng of direct Impor-
problem. We also review some recent results for
tance in many applications,
the trapping problem.
other quantltles,
such as the mean first passage time and the expected number of distinct sites vlslted Jn a walk of a given duration,
terms of P (or rather, of ~ts Laplace transform ]n tlme, ~)5.
Consider the evolution equation (I). We will suppose that the random walk, performed by the variable m~ is stationary, but ]t need not be
Here, we will ~nvestlgate two e]osely-re]ated problems,
2. DISPERSION EXPRESSED IN TERMS OF
can be expressed in
trapping and dispersion,
gives, at least, a partial
Markov]an or homogeneous.
Obviously, one has
and show that
answer to the ques-
tions that arise Jn thls context.
d_ Z UmP m = u dt m mM ~ -
The trapping
,
(2)
problem ~s, in fact, a generalization of the concept of first passage times.
A particle per-
forms a random walk over a set of states m E M,
while Jt has a probability km per unit tlme to
where ~ is the average velocity, weighted by the stationary one-time probability P(m,~) Pm"
leave the lattice while residing in state m.
Furthermore,
one has (6u : u-u):
"Leaving" may represent here d~fferent physical mechanisms such as trapping, de-excitation, genuine chemlca]
a
6x(t)
= x(t)
reaction or the actual exlt
from the set M. In the dispersion problem,
the
random walk variable m determines the rate of
- = ft6u[m(~)]d~ 0
Hence, < 6 x 2 ( t ) > = ft d'~ ft d ~0< 6u[m(r) ] 6u[m("~0) ]> 0
change of another variable x:
0
= 2f t d~ f~ d~ 0 o
dE
d---t : u m
'
(l)
0920-5632/88/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshmg Division)
o
Z
Z
m
mo
(3)
C Van den Broeck / Random walks, dmperslon and trapping
230
Since the conditional
probabl]Ity P(m,~Im0,~0)
is a function of the time difference finds by Laplace
transformation
z-t0, one
of (4):
after a total
time t. For any given rea]Izatlon
%m' m£M, the partlc]e wl]l have covered a distance
=
Hence
sity P(x,t)-to
<6x(s)> = f~ 0 2
at position x
start at x=0,t=0)
reads: Pm0~m6Um0~(m
i os)
(5) P(x,t)
where P(m[m0,s) conditlona]
observe a particle
(given that all partlc]es
e-St<6x2(t)>dt
the probability d e n
is t h e Laplace
= f...f
{d~}~({T~},t)6(X-mEeMUm~).
transform of the On the other hand, we consider
probabl]ity P(m[m0,t).
a (Markovian)
Similar trapping process.
results can be obtained
(8)
For a cumulative
residence
for higher order moments time ~m at site m, the probability
but the Markov property
is then required
that a par-
to ticle has not decayed at this site is
obtain results
in terms of P only. exp(-~m~m).
Often, one is interested
in the long-time
havior of the dispersion <6x2(t)>,
be-
particle
Hence the probability
F(t), that a
has not exited the system at tJme t,
and one is given by:
defines
the effective
dispersion
coefflc~ent:
= f...f
F(t) K = llm <6x2(t)> - llm s2<6~2(s)> t÷~ 2t s+0 2
{d%m} ~({zm},t)exp(-mZeM %~m) . ( 9 )
(6)
' Note that in (8), u m stands change, while in (9),
provided
these ]imlts exist.
Moreover,
usually prove that the probability P(x,t) asymptotlca]]y
one can
decay)
rate.
tlve.
It is, however,
The latter are necessarl]y
pos~-
density
attains a Gausslan
always posslb]e
to con-
form: sider positive ve]ocJtles
P(x,t)
for a rate of
km denotes an exit (or
u m > 0, such that:
~ [2~<6x2(t)>] - t / 2 um =
exp[-(x-)2/2<~2(t)>]
~mli
,
(I0)
(7) with
This result then, combined with (2) and (6), completely ior.
describes
the asymptotic
Note also that the evaluation
requires
the "low frequency"
which can be obtained
i =
Z
behavior
of P,
For this choice of the ve]ocltles easily verifies
cumulative
f~
(12)
dx e_~X P(x,t)
0
this relation we will restrict
to Markovlan
the probability
In (8), one
that:
in many cases of interest. V(t) =
To establish
(ll)
time behavof K only
3. RELATION BETWEEN TRAPPING AND DISPERSION
ourselves
~Pm
density
residence
walks and we introduce ~({~m'~
a M},t) for
times ~m at the sites m,
Hence, Laplace
the life time distribution
F is the
transform in space of P(x,t).
that P(x,t) is zero for negative
(Note
values of x
231
C Van den Broeck / Random walks, dlsperszon and trappmg
since all the velocities tunately,
are positive).
Unfor-
P(x,t) is only known In a few particu-
lar cases, such as a dlchotom~c a Kubo-Anderson
(15)
type of random walk I0. Moreover,
in many cass, a direct ca]cu]atlon out to be much simpler.
in the dispersion
of F(t) turns
Nevertheless,
(12) allows one to translate
results,
the result
The result (12) is, in this context,
obtained
helpful
problem to the trapping prob-
lem (and vice versa). approaches
= f~ dt t(- d ~ t t )) = f ¢o dt F(t) 0 0
state space 9 or
For example,
if P(x,t)
the Gaussian form (7) in the long-
time limit with the first two moments given by
not very
since P(x,t) is usually not known for
all times.
In [12], we have therefore
engaged
in a more direct approach of the trapping problem, and we want to report here a few results. We again consider a Markovlan nearest neighbor random walk on a set of states m=1,...,N,
(2) and (6), one finds for F(t):
transition
with
rates Wm,m± 1 (and WN, 1 = WI, N = 0).
We could not flnd the value of • for the
(t3)
F(t) ~ exp[-(~ - Ki2)t]
genera]
case of decay rates km, m = I,...,N.
However, We conclude Markovlan
that the trapping problem becomes
in the long time limit,
exponentla]
form, with a trapping
smaller than the weighted
set of N states m=],...,N,wlth the transition
i.e., F has an rate somewhat
1
average ~. In the case
of a Markovlan nearest neighbor
km*Pm*
random walk on a Wm, m±l denoting
rates per unit time between m and
m+l or m-l, (with reflecting
for a]] the rates equal to zero,
except one of them, say km, ¢ 0, we found:
boundary
conditions
+
N iSr+-(Pirm*) i • j=r+i
Z r=l
~ ~"
,
(16)
Wr,r+l Pr
WN, 1 = WI, N = 0) the asymptotic Gausslan nature of P was rigorously
proven in reference
[11],
while the value of K was derived in reference
[5]:
0 where pj Js the probability
that the particle
starts at site j at time t : 0. Another case for which an exact solution could be obtained is the gambler's r
K=
n-1 [s_-E1 (Us-U)Ps] 2 Z r=1
ruin problem with trapping
centers at both extremities
of the chain:
(14)
Wr,r+ 1 Pr Kr km : k I ~ , i + ~
Kr 6m,N
(~7)
One then has: In the corresponding
trapping problem,
the N-I I]-I = [klP1+kNPN+kl plkNpN r=Zl(Wr,r+iPr )-
result (13) is obtaJned.
4. FURTHER RESULTS FOR THE TRAPPING PROBLEM The mean exit time ~, i.e., the average time that a particle
spends ~n the set M before exit,
is given by the fol]owlng expression that F(~)=0)
(supposing
]
[l + ~ kip ikNPN
N-I • r=l
I
N-I 51
]
Wr,r+lPr q=l Wq,q+lPq
N N N N ( Z - Z IPI( ~ -' ~ +IIP~ i=r+l i=q+l j=r+l 3=q
232
C Van den Broeck / Random walks, dtsperston and trapping N-I ] N N E Pi Z pO Z (klPl l=r+l jffir+l J rffil Wr~r + Pr
+ ~PN
obtains (for p~ = Pm' we write superscript w for the weak coupling ]~mlt):
r r 0 Z el I pj)J . i=I j=~
(~8)
~w =
~. meM
This expression Is rather complicated,
but its
generality allows one to investigate several interesting cases.
For example,
(18) reduces in
the llmlt k I + 0, to the well-known result for
~Ipm . -
(21)
One can moreover prove that • is a monotonously decreasing function of the parameter C, starting from its upper limit
~w at C = 0 and ap-
proachlng its lower limit ~s when C ÷ =:
the mean first passage time in the usual onedimenslona] and ~
random walk 13.
In the limit k I ÷ < ~ < ~w
÷ =, ~ becomes the mean first passage
(22)
time to the site 1 or N, and (18) reduces to the simpler form
The equality
~ = ~s = ~w only holds ~f all the
rates km are equal. ~N- 1 ~N- 1 i( N =~q r= [--~1(Wr'r+IPr)-~-Z1(Wq'q+IPq)-~-~ ~-~ i=r+l-JZ +I ) N
N
~ . N-I
]
(19)
[12], we also
give the first order correction terms to ~w and ~s.
1
• pi(j=Zr+l-j=Zq+l)p~J/[mr__Zl(Wr,r+lPr)-
In reference
As a partlcu]ar case, a result consistent
with (13) and (14) is recovered.
For further
discussion and comparison with related results So far, we have considered genera]
InltJa]
in the literature,
we refer to reference
[12].
conditions p~ for the particle. A case of practlca] interest for which some further progress can be made is p~ = Pm' i.e., the particles are distributed according to the steady state dis-
slows down the transition rates W In a uniform
extreme cases are worth considering.
the N.F.W.O.,
Be]glum.
For correspondence
B-36|0 D1epenbeek,
Belgium.
Note
that Pm remains unchanged when one speeds up or
way, i.e., by replacing Wm,m± 1 by CWm,m± I.
C. V.d.B. is "Bevoegdverklaard Navorser" at
write to L.U.C.,
trlbutlon reached in absence of the decay or exit mechanism described by the rates km.
ACKNOWLEDGEMENT
Two
REFERENCES I. J.W. Haus and K.W. Kehr, Phys. Rep. 150 (1987) 265.
In the so-
ca]led strong coupling limit (or fast modu]atlon
2. v. Landman and M.F. BI9 (1979) 6207.
Sh]eslnger,
Phys. Rev.
limit), we let C + =, so that the time scale of the random walk transitions are much faster than the decay times ~ I .
In this case, one can prove
that ~ approaches the limit
~s:
rs = (mZeM kmPm) -I
3. S. Alexander, J. Be rnasconl, W.R. Schneider and R. Orbach, Rev. Mod. Phys. 5 3 (]98|) 175. 4. M.E. Fisher, J. Star. Phys. 34 (]983) 667.
(20)
5. E.W. Montroll and G.H. Weiss, J. Math. Phys. ! (]965) 167. 6. C. Van den Broeck and R.M. Mazo, J. Chem. Phys. 81 (1984) 3624.
On the other hand, the random walk dynamics are turned off in the limit C ÷ 0, so that one
7. R.M. Mazo and C. Van den Broeck, Phys. Rev. A34 (1986) 2364.
C Van den Broeck / Random walks, dtsper~ton and trapping
8. H. Brenner, Phys. Chem. Hydrodyn. I (1980) 91; N.G. Van Kampen, Physlca 96A (]-979); R. Axis, Intracellular Transport. 5 (1966) 167; R. Hersch, Rocky Mountain J. Math. 4 (1974) 443: D.W. McLaughlln, G.C. Papanlcolaou and O.R. Pironneau, SIAM J. Appl. Math. 4 5 (1985) 780. 9. S. Goldsteln, Quart. J. Mech. Appl. Math. 4 (1951) 129; J.C. Giddlngs, J. Chem. Phys. ~| (1959) 146. I0. P.W. Anderson, J. Phys. Soc. Jap. 9 (1954) 316; R. Kubo, J. Phys. Soc. Jap. 9y(1954) 935. ll.M. Plnsky, Z. Wahr. Verw. Geb. 9 (1968) 101. 12. C. Van den Broeck and M. Bouten, J. Star. Phys. 45 (1986) 1031. 13. G.H. Weiss, J. Star. Phys. 24 (1981) 587; D.T. Gillespie, Physlca A95 (1979) 69.
233
229
RANDOM WALKS, DISPERSION AND TRAPPING
Chrlst~an VAN DEN BROECK Center for Studies in Statistical Mechanics,
I. INTRODUCTION
The Un~verslty of Texas, Austin,
TX 78712, USA
where um is a "velocity profile" whose form
The theory of random walks has found many applications in physics and new developments,
such
depends on the physics of the problem.
Such
problems are encountered In the study of dls-
as random walks on fracta]s I, walks with inter-
perslon of particles In flows 6, Jn line width
nal states 2, doubly stochastic walks 3, and In-
calculations 7, and so on 8.
teracting walks 4 continue to stimulate intensive research.
One of the central quantities In th]s
theory ]s the cond2tlona]
probability P(m]m0,t)
We will show that both problems are related through a simple mathematical Moreover,
the condltJona]
transformation.
probability P pro-
to go from the slte ~0 at time zero to the site
vldes a partial solution to the dispersion
m at time t. Apart from be2ng of direct Impor-
problem. We also review some recent results for
tance in many applications,
the trapping problem.
other quantltles,
such as the mean first passage time and the expected number of distinct sites vlslted Jn a walk of a given duration,
terms of P (or rather, of ~ts Laplace transform ]n tlme, ~)5.
Consider the evolution equation (I). We will suppose that the random walk, performed by the variable m~ is stationary, but ]t need not be
Here, we will ~nvestlgate two e]osely-re]ated problems,
2. DISPERSION EXPRESSED IN TERMS OF
can be expressed in
trapping and dispersion,
gives, at least, a partial
Markov]an or homogeneous.
Obviously, one has
and show that
answer to the ques-
tions that arise Jn thls context.
d
The trapping
,
(2)
problem ~s, in fact, a generalization of the concept of first passage times.
A particle per-
forms a random walk over a set of states m E M,
while Jt has a probability km per unit tlme to
where ~ is the average velocity, weighted by the stationary one-time probability P(m,~) Pm"
leave the lattice while residing in state m.
Furthermore,
one has (6u : u-u):
"Leaving" may represent here d~fferent physical mechanisms such as trapping, de-excitation, genuine chemlca]
a
6x(t)
= x(t)
reaction or the actual exlt
from the set M. In the dispersion problem,
the
random walk variable m determines the rate of
-
Hence, < 6 x 2 ( t ) > = ft d'~ ft d ~0< 6u[m(r) ] 6u[m("~0) ]> 0
change of another variable x:
0
= 2f t d~ f~ d~ 0 o
dE
d---t : u m
'
(l)
0920-5632/88/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshmg Division)
o
Z
Z
m
mo
(3)
C Van den Broeck / Random walks, dmperslon and trapping
230
Since the conditional
probabl]Ity P(m,~Im0,~0)
is a function of the time difference finds by Laplace
transformation
z-t0, one
of (4):
after a total
time t. For any given rea]Izatlon
%m' m£M, the partlc]e wl]l have covered a distance
=
Hence
sity P(x,t)-to
<6x(s)> = f~ 0 2
at position x
start at x=0,t=0)
reads: Pm0~m6Um0~(m
i os)
(5) P(x,t)
where P(m[m0,s) conditlona]
observe a particle
(given that all partlc]es
e-St<6x2(t)>dt
the probability d e n
is t h e Laplace
= f...f
{d~}~({T~},t)6(X-mEeMUm~).
transform of the On the other hand, we consider
probabl]ity P(m[m0,t).
a (Markovian)
Similar trapping process.
results can be obtained
(8)
For a cumulative
residence
for higher order moments time ~m at site m, the probability
but the Markov property
is then required
that a par-
to ticle has not decayed at this site is
obtain results
in terms of P only. exp(-~m~m).
Often, one is interested
in the long-time
havior of the dispersion <6x2(t)>,
be-
particle
Hence the probability
F(t), that a
has not exited the system at tJme t,
and one is given by:
defines
the effective
dispersion
coefflc~ent:
= f...f
F(t) K = llm <6x2(t)> - llm s2<6~2(s)> t÷~ 2t s+0 2
{d%m} ~({zm},t)exp(-mZeM %~m) . ( 9 )
(6)
' Note that in (8), u m stands change, while in (9),
provided
these ]imlts exist.
Moreover,
usually prove that the probability P(x,t) asymptotlca]]y
one can
decay)
rate.
tlve.
It is, however,
The latter are necessarl]y
pos~-
density
attains a Gausslan
always posslb]e
to con-
form: sider positive ve]ocJtles
P(x,t)
for a rate of
km denotes an exit (or
u m > 0, such that:
~ [2~<6x2(t)>] - t / 2 um =
exp[-(x-
~mli
,
(I0)
(7) with
This result then, combined with (2) and (6), completely ior.
describes
the asymptotic
Note also that the evaluation
requires
the "low frequency"
which can be obtained
i =
Z
behavior
of P,
For this choice of the ve]ocltles easily verifies
cumulative
f~
(12)
dx e_~X P(x,t)
0
this relation we will restrict
to Markovlan
the probability
In (8), one
that:
in many cases of interest. V(t) =
To establish
(ll)
time behavof K only
3. RELATION BETWEEN TRAPPING AND DISPERSION
ourselves
~Pm
density
residence
walks and we introduce ~({~m'~
a M},t) for
times ~m at the sites m,
Hence, Laplace
the life time distribution
F is the
transform in space of P(x,t).
that P(x,t) is zero for negative
(Note
values of x
231
C Van den Broeck / Random walks, dlsperszon and trappmg
since all the velocities tunately,
are positive).
Unfor-
P(x,t) is only known In a few particu-
lar cases, such as a dlchotom~c a Kubo-Anderson
(15)
type of random walk I0. Moreover,
in many cass, a direct ca]cu]atlon out to be much simpler.
in the dispersion
of F(t) turns
Nevertheless,
(12) allows one to translate
results,
the result
The result (12) is, in this context,
obtained
helpful
problem to the trapping prob-
lem (and vice versa). approaches
= f~ dt t(- d ~ t t )) = f ¢o dt F(t) 0 0
state space 9 or
For example,
if P(x,t)
the Gaussian form (7) in the long-
time limit with the first two moments given by
not very
since P(x,t) is usually not known for
all times.
In [12], we have therefore
engaged
in a more direct approach of the trapping problem, and we want to report here a few results. We again consider a Markovlan nearest neighbor random walk on a set of states m=1,...,N,
(2) and (6), one finds for F(t):
transition
with
rates Wm,m± 1 (and WN, 1 = WI, N = 0).
We could not flnd the value of • for the
(t3)
F(t) ~ exp[-(~ - Ki2)t]
genera]
case of decay rates km, m = I,...,N.
However, We conclude Markovlan
that the trapping problem becomes
in the long time limit,
exponentla]
form, with a trapping
smaller than the weighted
set of N states m=],...,N,wlth the transition
i.e., F has an rate somewhat
1
average ~. In the case
of a Markovlan nearest neighbor
km*Pm*
random walk on a Wm, m±l denoting
rates per unit time between m and
m+l or m-l, (with reflecting
for a]] the rates equal to zero,
except one of them, say km, ¢ 0, we found:
boundary
conditions
+
N iSr+-(Pirm*) i • j=r+i
Z r=l
~ ~"
,
(16)
Wr,r+l Pr
WN, 1 = WI, N = 0) the asymptotic Gausslan nature of P was rigorously
proven in reference
[11],
while the value of K was derived in reference
[5]:
0 where pj Js the probability
that the particle
starts at site j at time t : 0. Another case for which an exact solution could be obtained is the gambler's r
K=
n-1 [s_-E1 (Us-U)Ps] 2 Z r=1
ruin problem with trapping
centers at both extremities
of the chain:
(14)
Wr,r+ 1 Pr Kr km : k I ~ , i + ~
Kr 6m,N
(~7)
One then has: In the corresponding
trapping problem,
the N-I I]-I = [klP1+kNPN+kl plkNpN r=Zl(Wr,r+iPr )-
result (13) is obtaJned.
4. FURTHER RESULTS FOR THE TRAPPING PROBLEM The mean exit time ~, i.e., the average time that a particle
spends ~n the set M before exit,
is given by the fol]owlng expression that F(~)=0)
(supposing
]
[l + ~ kip ikNPN
N-I • r=l
I
N-I 51
]
Wr,r+lPr q=l Wq,q+lPq
N N N N ( Z - Z IPI( ~ -' ~ +IIP~ i=r+l i=q+l j=r+l 3=q
232
C Van den Broeck / Random walks, dtsperston and trapping N-I ] N N E Pi Z pO Z (klPl l=r+l jffir+l J rffil Wr~r + Pr
+ ~PN
obtains (for p~ = Pm' we write superscript w for the weak coupling ]~mlt):
r r 0 Z el I pj)J . i=I j=~
(~8)
~w =
~. meM
This expression Is rather complicated,
but its
generality allows one to investigate several interesting cases.
For example,
(18) reduces in
the llmlt k I + 0, to the well-known result for
~Ipm . -
(21)
One can moreover prove that • is a monotonously decreasing function of the parameter C, starting from its upper limit
~w at C = 0 and ap-
proachlng its lower limit ~s when C ÷ =:
the mean first passage time in the usual onedimenslona] and ~
random walk 13.
In the limit k I ÷ < ~ < ~w
÷ =, ~ becomes the mean first passage
(22)
time to the site 1 or N, and (18) reduces to the simpler form
The equality
~ = ~s = ~w only holds ~f all the
rates km are equal. ~N- 1 ~N- 1 i( N =~q r= [--~1(Wr'r+IPr)-~-Z1(Wq'q+IPq)-~-~ ~-~ i=r+l-JZ +I ) N
N
~ . N-I
]
(19)
[12], we also
give the first order correction terms to ~w and ~s.
1
• pi(j=Zr+l-j=Zq+l)p~J/[mr__Zl(Wr,r+lPr)-
In reference
As a partlcu]ar case, a result consistent
with (13) and (14) is recovered.
For further
discussion and comparison with related results So far, we have considered genera]
InltJa]
in the literature,
we refer to reference
[12].
conditions p~ for the particle. A case of practlca] interest for which some further progress can be made is p~ = Pm' i.e., the particles are distributed according to the steady state dis-
slows down the transition rates W In a uniform
extreme cases are worth considering.
the N.F.W.O.,
Be]glum.
For correspondence
B-36|0 D1epenbeek,
Belgium.
Note
that Pm remains unchanged when one speeds up or
way, i.e., by replacing Wm,m± 1 by CWm,m± I.
C. V.d.B. is "Bevoegdverklaard Navorser" at
write to L.U.C.,
trlbutlon reached in absence of the decay or exit mechanism described by the rates km.
ACKNOWLEDGEMENT
Two
REFERENCES I. J.W. Haus and K.W. Kehr, Phys. Rep. 150 (1987) 265.
In the so-
ca]led strong coupling limit (or fast modu]atlon
2. v. Landman and M.F. BI9 (1979) 6207.
Sh]eslnger,
Phys. Rev.
limit), we let C + =, so that the time scale of the random walk transitions are much faster than the decay times ~ I .
In this case, one can prove
that ~ approaches the limit
~s:
rs = (mZeM kmPm) -I
3. S. Alexander, J. Be rnasconl, W.R. Schneider and R. Orbach, Rev. Mod. Phys. 5 3 (]98|) 175. 4. M.E. Fisher, J. Star. Phys. 34 (]983) 667.
(20)
5. E.W. Montroll and G.H. Weiss, J. Math. Phys. ! (]965) 167. 6. C. Van den Broeck and R.M. Mazo, J. Chem. Phys. 81 (1984) 3624.
On the other hand, the random walk dynamics are turned off in the limit C ÷ 0, so that one
7. R.M. Mazo and C. Van den Broeck, Phys. Rev. A34 (1986) 2364.
C Van den Broeck / Random walks, dtsper~ton and trapping
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