- Email: [email protected]
Renormalization group approach for random walks on a disordered bond-diluted lattice
PhysicsLettersA 181 (1993) 119—122 North-Holland
PHYSICS LETTERS A
Renormalization group approach for random walks on a disordered bond-diluted lattice Michael Schulz Abteilungfur Theoretische Physik, Universität Ulm, W- 7900 Ulm, Germany Received 9 June 1993; revised manuscript received 29 July 1993; accepted for publication 30 July 1993 Communicated by V.M. Agranovich
Random walks in a d-dimensional stochastic environment with a static structure factor S(q) —. const and S( q) — q —2, respectively, are analyzed by renormalization group techniques. In dependence on the strength of the disorder normal, anomalous or limited diffusion (finite diffusion radius) for t-.co was detected.
It is well known that the behavior ofrandom walks on a diluted lattice with static disorder differs from the situation in an external static stochastic force field. In the last case, an anomalous diffusion is expected for dimensions d<2 and an arbitrary small but nonzero static random force field [1] (for d=2 follows a diffusion with logarithmic corrections [2]), in the first case we get an anomalous diffusion, if the lattice has a fractal structure [3,41.Therefore, for a random diluted lattice an anomalous diffusion can be expected only at the percolation threshold. The cause for this is the different behaviorof the random walker in different environs. For a random walker in a random force field there exists a nonzero possibility to cross a barrier, whereas for a walker in the second case the transition over an obstacle is strictly forbidden. In principle, a reasonable starting point is a formulation like to the Fokker—Planck equation. The bond-diluted lattice is given by present (A~=1) and absent (A,~=0) bonds (i, ~ are the numbers of the sites, which are neighboring the bond (if)) and only occupied bonds can be crossed by the walker. Therefore, one can obtain a Markovian description by considering the history of the random walker. In the previous case the knowledge of the time evolution for the probability P(i, j, I) to find a walker at site i at time t, which has occupied the sitej immediately prior the last jump, is necessary and sufficient. This
time evolution is given by the one-step master equation [5] -~-P(i,j,t)=w ~ A~,A JkP(J, k, t)
dt
k —
w ~ A1kA~P(i,j,t)
(1)
k
Note, that A0~1 only if i and] are neighboring lattice sites and the bond between i and j is occupied. This formulation guarantees that a transition over nonoccupied bonds is strictly forbidden. A wellknown generalization ofeq. (1) is the backwardjump model [6—81on a bond-diluted lattice [9]. Here, three types of jumps are possible: jumps to the previously occupied site with transition rate Wb, jumps from a site ito itself with rate w(z—z1) (z is the coordination number of the original regular lattice and z the coordination number of site i on the diluted lattice) and jumps in all other directions (transition rate w). From ref. [9] the time evolution of the probability P( i, t) = ~ P( i, j, t) Il i
(the sum includes all neighboring lattice sites] ofthe point i) is given by
0375-9601 /93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
119
Volume 181, number 2
PHYSICS LETrERS A
(d2 ~
d
p 0(p)= ~
=
[Wb
+ w( z—
w ~ A0 j
1)]
~~1) P(:, t)
p+Wb+W(Z1)
and z. Contrary the dimensionless to (3), thisdiffusion equationconstant reflects no D=longer / all details of the diluted lattice. While the information forcomplete, the long range scalesofthe of the dilution is conse~ed the details local structureare
P~,I)
+ [wb+w(z—
p+2(Wb—w)+Zw
m
+[2(Wb—W)+WZ,]~
+
4 October 1993
1)1w ~ AIJP(J, 1).
(2)
reduced to an irrelevant mean field description. The diffusion limit needs mainly information about the
3
Setting z1= + 8z1 and A(1 D,~ /Z+ 6A~,(Dii is the nearest neighbor function, D~,=1 ifthe lattice sites i and] are nearest neighbors and D~=0otherwise, is the averaged coordination number of the diluted lattice) with the connection &z~= >~ 6A~,,the Laplace transformation with respect to time and the Fourier transformation with
large scale behavior of the distribution of the lattice defects, i.e. the influence of the large scale lattice structure becomes important for the behavior of the particle for t—*oo. In the long time limit t—~oo,which is equivalent to p—~0, we get m0 ~ p and (4) corresponds to the usual Fokker—Planck equation type. In comparison to the Fokker—Planck equation with an external random drift force F,
SA~=D~~aqexp[~iq(rj+rj)]
(p+Dk2)+igoJdd1qk.F~_qP(q,p)=I*,
(5)
bfq
and P(i, t) =
~
eq. (4) has only a different structure in the integral term. In particular, this difference is evident, if the force Fhas a potential V. In this case the term k~qa~
P(q, t) exp(iq~r,)
in eq. (4) is replaced by ik (k—q) Vk_q. Note that g0 is an unessential prefactor, which results from the transition to the limit both cases (eqs. (4) and (5)). Incontinuum the following we in consider two types of bond-diluted lattices. (i) The correlation between the occupied bonds is random. Therefore, the correlation of the dilution
(q a first Brillouin zone of the lattice), give 2+2[(wb
w)+zw]p
—
{p +W[p+Wb
+ w(z— 1)]
[I —a(k)]}P(k, P)
+z ~ a
4[a(~q)—c(k —~q)]P(k—q,P)=Ik. (3) a(q) =
~ exp(iq~g.)
is the well-known lattice function (g are the vectors from a site to the nearest neighbors on a regular lattice) and I,. represents the initial conditions. For the determination of the long time behavior of the random walker, it is reasonable to use the continuum limit of eq. (3) for small k. Then, eq. (3) reduces to
becomes =4~5(q+q’). (ii) The bond dilution shows a 2o~5(q+q’). long range In correthis lation. Hence, =A/q particular casewe theget correlation of the lattice structure has the same form as the potential correlation =4/q2ô(q+q’) (which corresponds to a short range random potential force field =Aqaqfl/q2o( q + q’)). The correlation function can be identified with the static structure factor S(q), which is in the first case S(q)=const and in the second case S(q)—~q2.To investigate the long time behavior of the random walker for small ~= d~ d, it is natural to try to perform a dynamic renormalization group expansion in —
[m 2 ]P(k, ~o) 0(p) +Dk +g 0 ddq a~_4kqP(q, p) = ‘A,
J
with the mass 120
(4)
powers of the parameter A for sufficiently weak disorder [1,2,10,11]. The results of the renormalizalion group approach are restricted to the lowest non-
Volume 181, number 2
PHYSICS LETTERS A
trivial order of In principle, the dimensionless interaction constant u0 =A Ag~SdD2 (Sd is the unit surface of a d-dimensional sphere, A is the usual cutoff) behaves under the rescaling procedure k—~e~k like .
d/
(u
—
0) with the well-known /3 function. At least, the mean square by displacement the random walker is determined the scaling of law (6)
4 October 1993
collapse regime, e.g. the mean square displacement radius has a finite limit ~ const for t—~oo. This is the typical situation for a diffusion on a diluted lattice with occupation probability p, e.g. a dilution rate q= 1 —p. For p is determined 1.,~. p~, the lattice contains one infinite cluster, which has on large scales a homogeneous e.g.a dilution the dif2> structure, t. Only for fusion becomes normalcan
with the exponent C=2—y,,, y~=2+AaIn(Z 0)/8A (Z0 is the field renormalization constant). (i) Using the one-loop approximation (fig. 1) of perturbation theory we get in the first case l+4d U
)
/3(uo) =
_~0(~+ (2+d)d U0
0
c’
~
with the critical dimension d~= 0. That means e = d is always a negative value. The fixpoint u~=0 is stableforalld>0, whereas u~=IeId(d+2)/(ll +4d) is unstable, e.g. for a sufficient small uo u~.For large u0 > u~ the disorder becomes more and more important and tends to infinity. As a result of this behavior the exponent 2/C tends to zero and we get a —
r (2)
____________
-
I I
+
___________
The comparison of the anomalous diffusion coeffi2> t’~with the numerical results [3] allows of the values shown in tacient /1 the in determination
ponent is also calculable using the Potts model [12] for the percolation case [111. Here, from the wellknown critical exponents /3, v and t follows C= 2 + (1— /3)/v [4]. Consequently, the same exponents result from different approaches, which are hardly related to each other. (ii) In the case of long range disorder we get with
+
_______
r
(7)
~tI_[d(~2)]/(22+8~).
+
____________
Table I
d
MRG
l~rn,n,~rio
2
+.“ ____________
Fig. 1. Graphs ofthe one-loop approximation forthe vertex functions [‘(2) and [‘(4)~
0.79 0.71 0.67 0.56 0.56 0.42 5 0.44 0.37 6 0.31 0.33 ____________________________________________________
3 4
121
Volume 181, number 2
PHYSICS LETTERS A
the same method a critical dimension d~=2. For d< 2 there exists only the unstable fixpoint U~= 0, e.g. for each disorder strength A follows for t-+cc a finite mean diffusion radius, contrary to the situation in a random potential force field [2], where an anomabus diffusion is available for d< 2 and an arbitrary A. For d> 2 we have a situation analogous to the first case, e.g. a stable fixpoint U~= 0 and an unstable fixpoint ur = I ci d, from which follows a normal diffusion for a small disorder U 0 U~. An anomalous diffusion follows also only for one disorder strength. Only here, we get /\ X 2\/ ,,~,tI_(d_2)/8
.
‘8~ ‘ /
Note that for d> 2 in a random potential force field for each A follows a normal diffusion. A last special case is the behavior at the critical dimension d= 2. Here we get 2 8! —2U —
4 October 1993
different behaviorof the random walker for different environs. Characteristic is the behaviorofthe walker in d= 1: the exact solution of (3) gives a finite diffusion radius for t—* ~ in environs of random distributed obstacles and on the other hand a scaling law for the diffusion in a random force field [13] ln4i. Consequently, the difference between the two diffusion processes is very strong and we can expect a continuation of this behavior in higher dimensions, as shown in the present paper. Whereas the diffusion in a static stochastic force field leads to an anomalous diffusion below a critical dimension d =2 and a normal diffusion above d~ one gets in the case of static random distributed obC
stacles below d~only a limited diffusion in a finite region, whereas three regimes (depending on the disorder strength A) exist for d> d~:normal diffusion for weakly disorder, limited diffusion in a finite region for strong disorder and anomalousasdiffusion for one value A, which can be interpreted the strength of the disorder at the percolation threshold.
and consequently, using the renormalization relation for the characteristic frequency (mass) scale 0m (1) / 81= zm (1), it follows that
References
lfl(m(l)’~= 2l_~lfl[l_2U0(O)l].
(9)
\m(0)J
At sufficiently long times t, with corresponding bar mass m(0) l/t, the mean displacement res21. Therefore we square get from eq. (9) with cales as e m (1) const 2~ \X / /
2> ) ~
(1—u
(10)
0 ln
From this equation we conclude that the mean diffusion radius at long times is Rma,, e’,‘2~~• In principle, we find a remarkable difference between the diffusion in random force fields (or random potential fields) on the one hand and the diffusion in a system with random distributed obstacles on the other hand. The cause is the abovementioned
122
[I] D.S. Fisher, Phys. Rev. A 30 (1984) 960. [2] D.S. Fisher, D. Friedan, Z. Qui, S.J. Shenker and S.H. Shenker, Phys. Rev. A 31(1985) 384. [3] Stauffer, Phys. 54 (1979) [41D. S. Alexander and R.Rep. Orbach, J. Phys.1.(Paris) Lett. 43 (1982) [5] C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1983). [6]R. Fuerth, Z. Phys. 2 (1920) 244. [7] J.W. Haus and K.W. Kehr, Phys. Rep. 150 (1987) 265. [8] M.H. Ernst,J. Stat. Phys. 53 (1988) 191. [9] R. Hilfer, Phys. Rev. B 44 (1991) 638. [10] E. Medina, T. Hwa, M. Kardar and Y.C. Zhang, Phys. Rev. A 39 (1989) 3053. [11] D.J. Amit, J. Phys. A 9 (1976) 1441. [12]R.B. Potts, Proc. Cambridge Philos. Soc. 48(1952)106. [13] G. Sinai, in: Proc. Berlin Conference on Mathematical problems in theoretical physics, eds. R. Schrader, R. Seiler and D.A. Ohlenbrock (Springer, Berlin, 1982).
PHYSICS LETTERS A
Renormalization group approach for random walks on a disordered bond-diluted lattice Michael Schulz Abteilungfur Theoretische Physik, Universität Ulm, W- 7900 Ulm, Germany Received 9 June 1993; revised manuscript received 29 July 1993; accepted for publication 30 July 1993 Communicated by V.M. Agranovich
Random walks in a d-dimensional stochastic environment with a static structure factor S(q) —. const and S( q) — q —2, respectively, are analyzed by renormalization group techniques. In dependence on the strength of the disorder normal, anomalous or limited diffusion (finite diffusion radius) for t-.co was detected.
It is well known that the behavior ofrandom walks on a diluted lattice with static disorder differs from the situation in an external static stochastic force field. In the last case, an anomalous diffusion is expected for dimensions d<2 and an arbitrary small but nonzero static random force field [1] (for d=2 follows a diffusion with logarithmic corrections [2]), in the first case we get an anomalous diffusion, if the lattice has a fractal structure [3,41.Therefore, for a random diluted lattice an anomalous diffusion can be expected only at the percolation threshold. The cause for this is the different behaviorof the random walker in different environs. For a random walker in a random force field there exists a nonzero possibility to cross a barrier, whereas for a walker in the second case the transition over an obstacle is strictly forbidden. In principle, a reasonable starting point is a formulation like to the Fokker—Planck equation. The bond-diluted lattice is given by present (A~=1) and absent (A,~=0) bonds (i, ~ are the numbers of the sites, which are neighboring the bond (if)) and only occupied bonds can be crossed by the walker. Therefore, one can obtain a Markovian description by considering the history of the random walker. In the previous case the knowledge of the time evolution for the probability P(i, j, I) to find a walker at site i at time t, which has occupied the sitej immediately prior the last jump, is necessary and sufficient. This
time evolution is given by the one-step master equation [5] -~-P(i,j,t)=w ~ A~,A JkP(J, k, t)
dt
k —
w ~ A1kA~P(i,j,t)
(1)
k
Note, that A0~1 only if i and] are neighboring lattice sites and the bond between i and j is occupied. This formulation guarantees that a transition over nonoccupied bonds is strictly forbidden. A wellknown generalization ofeq. (1) is the backwardjump model [6—81on a bond-diluted lattice [9]. Here, three types of jumps are possible: jumps to the previously occupied site with transition rate Wb, jumps from a site ito itself with rate w(z—z1) (z is the coordination number of the original regular lattice and z the coordination number of site i on the diluted lattice) and jumps in all other directions (transition rate w). From ref. [9] the time evolution of the probability P( i, t) = ~ P( i, j, t) Il i
(the sum includes all neighboring lattice sites] ofthe point i) is given by
0375-9601 /93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
119
Volume 181, number 2
PHYSICS LETrERS A
(d2 ~
d
p 0(p)= ~
=
[Wb
+ w( z—
w ~ A0 j
1)]
~~1) P(:, t)
p+Wb+W(Z1)
and z. Contrary the dimensionless to (3), thisdiffusion equationconstant reflects no D=longer
P~,I)
+ [wb+w(z—
p+2(Wb—w)+Zw
m
+[2(Wb—W)+WZ,]~
+
4 October 1993
1)1w ~ AIJP(J, 1).
(2)
reduced to an irrelevant mean field description. The diffusion limit needs mainly information about the
3
Setting z1=
large scale behavior of the distribution of the lattice defects, i.e. the influence of the large scale lattice structure becomes important for the behavior of the particle for t—*oo. In the long time limit t—~oo,which is equivalent to p—~0, we get m0 ~ p and (4) corresponds to the usual Fokker—Planck equation type. In comparison to the Fokker—Planck equation with an external random drift force F,
SA~=D~~aqexp[~iq(rj+rj)]
(p+Dk2)+igoJdd1qk.F~_qP(q,p)=I*,
(5)
bfq
and P(i, t) =
~
eq. (4) has only a different structure in the integral term. In particular, this difference is evident, if the force Fhas a potential V. In this case the term k~qa~
P(q, t) exp(iq~r,)
in eq. (4) is replaced by ik (k—q) Vk_q. Note that g0 is an unessential prefactor, which results from the transition to the limit both cases (eqs. (4) and (5)). Incontinuum the following we in consider two types of bond-diluted lattices. (i) The correlation between the occupied bonds is random. Therefore, the correlation of the dilution
(q a first Brillouin zone of the lattice), give 2+2[(wb
w)+zw]p
—
{p +W[p+Wb
+ w(z— 1)]
[I —a(k)]}P(k, P)
+z ~ a
4[a(~q)—c(k —~q)]P(k—q,P)=Ik. (3) a(q) =
~ exp(iq~g.)
is the well-known lattice function (g are the vectors from a site to the nearest neighbors on a regular lattice) and I,. represents the initial conditions. For the determination of the long time behavior of the random walker, it is reasonable to use the continuum limit of eq. (3) for small k. Then, eq. (3) reduces to
becomes
[m 2 ]P(k, ~o) 0(p) +Dk +g 0 ddq a~_4kqP(q, p) = ‘A,
J
with the mass 120
(4)
powers of the parameter A for sufficiently weak disorder [1,2,10,11]. The results of the renormalizalion group approach are restricted to the lowest non-
Volume 181, number 2
PHYSICS LETTERS A
trivial order of In principle, the dimensionless interaction constant u0 =A Ag~SdD2 (Sd is the unit surface of a d-dimensional sphere, A is the usual cutoff) behaves under the rescaling procedure k—~e~k like .
d/
(u
—
0) with the well-known /3 function. At least, the mean square by displacement the random walker is determined the scaling of law (6)
4 October 1993
collapse regime, e.g. the mean square displacement radius has a finite limit
with the exponent C=2—y,,, y~=2+AaIn(Z 0)/8A (Z0 is the field renormalization constant). (i) Using the one-loop approximation (fig. 1) of perturbation theory we get in the first case l+4d U
)
/3(uo) =
_~0(~+ (2+d)d U0
0
c’
~
with the critical dimension d~= 0. That means e = d is always a negative value. The fixpoint u~=0 is stableforalld>0, whereas u~=IeId(d+2)/(ll +4d) is unstable, e.g. for a sufficient small uo
r (2)
____________
-
I I
+
___________
The comparison of the anomalous diffusion coeffi2> t’~with the numerical results [3] allows of the values shown in tacient /1 the in determination
ponent is also calculable using the Potts model [12] for the percolation case [111. Here, from the wellknown critical exponents /3, v and t follows C= 2 + (1— /3)/v [4]. Consequently, the same exponents result from different approaches, which are hardly related to each other. (ii) In the case of long range disorder we get with
+
_______
r
(7)
+
____________
Table I
d
MRG
l~rn,n,~rio
2
+.“ ____________
Fig. 1. Graphs ofthe one-loop approximation forthe vertex functions [‘(2) and [‘(4)~
0.79 0.71 0.67 0.56 0.56 0.42 5 0.44 0.37 6 0.31 0.33 ____________________________________________________
3 4
121
Volume 181, number 2
PHYSICS LETTERS A
the same method a critical dimension d~=2. For d< 2 there exists only the unstable fixpoint U~= 0, e.g. for each disorder strength A follows for t-+cc a finite mean diffusion radius, contrary to the situation in a random potential force field [2], where an anomabus diffusion is available for d< 2 and an arbitrary A. For d> 2 we have a situation analogous to the first case, e.g. a stable fixpoint U~= 0 and an unstable fixpoint ur = I ci d, from which follows a normal diffusion for a small disorder U 0
.
‘8~ ‘ /
Note that for d> 2 in a random potential force field for each A follows a normal diffusion. A last special case is the behavior at the critical dimension d= 2. Here we get 2 8! —2U —
4 October 1993
different behaviorof the random walker for different environs. Characteristic is the behaviorofthe walker in d= 1: the exact solution of (3) gives a finite diffusion radius for t—* ~ in environs of random distributed obstacles and on the other hand a scaling law for the diffusion in a random force field [13]
stacles below d~only a limited diffusion in a finite region, whereas three regimes (depending on the disorder strength A) exist for d> d~:normal diffusion for weakly disorder, limited diffusion in a finite region for strong disorder and anomalousasdiffusion for one value A, which can be interpreted the strength of the disorder at the percolation threshold.
and consequently, using the renormalization relation for the characteristic frequency (mass) scale 0m (1) / 81= zm (1), it follows that
References
lfl(m(l)’~= 2l_~lfl[l_2U0(O)l].
(9)
\m(0)J
At sufficiently long times t, with corresponding bar mass m(0) l/t, the mean displacement res21. Therefore we square get from eq. (9) with cales as e m (1) const 2~ \X / /
2> ) ~
(1—u
(10)
0 ln
From this equation we conclude that the mean diffusion radius at long times is Rma,, e’,‘2~~• In principle, we find a remarkable difference between the diffusion in random force fields (or random potential fields) on the one hand and the diffusion in a system with random distributed obstacles on the other hand. The cause is the abovementioned
122
[I] D.S. Fisher, Phys. Rev. A 30 (1984) 960. [2] D.S. Fisher, D. Friedan, Z. Qui, S.J. Shenker and S.H. Shenker, Phys. Rev. A 31(1985) 384. [3] Stauffer, Phys. 54 (1979) [41D. S. Alexander and R.Rep. Orbach, J. Phys.1.(Paris) Lett. 43 (1982) [5] C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1983). [6]R. Fuerth, Z. Phys. 2 (1920) 244. [7] J.W. Haus and K.W. Kehr, Phys. Rep. 150 (1987) 265. [8] M.H. Ernst,J. Stat. Phys. 53 (1988) 191. [9] R. Hilfer, Phys. Rev. B 44 (1991) 638. [10] E. Medina, T. Hwa, M. Kardar and Y.C. Zhang, Phys. Rev. A 39 (1989) 3053. [11] D.J. Amit, J. Phys. A 9 (1976) 1441. [12]R.B. Potts, Proc. Cambridge Philos. Soc. 48(1952)106. [13] G. Sinai, in: Proc. Berlin Conference on Mathematical problems in theoretical physics, eds. R. Schrader, R. Seiler and D.A. Ohlenbrock (Springer, Berlin, 1982).