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Lyapunov exponent of random walkers on a bond-disordered lattice
ELSEVIER
Physica A 247 (1997) 91-107
Lyapunov exponent of random walkers on a bond-disordered lattice L. Acedo a, M.H. Ernst b,* a Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz, Spain b Institute for Theoretical Physics, Utrecht University, P.O. Box 80006, 3508 TA Utrecht, The Netherlands
Received 23 May 1997
Abstract The chaotic properties of a random walker in a quenched random environment are studied analytically, following the work of Gaspard et al. on Lorentz gases, for systems with closed (periodic) or open (absorbing) boundaries. The model of interest describes random walkers hopping on a disordered lattice, on which the hopping probabilities across bonds are quenched random variables. For closed systems an exact expression for the Lyapunov exponent is derived, which not only depends on the composition, but also on the number of clusters of certain type. For open systems escape rates and Lyapunov exponents are calculated from a mean-field approximation. The theoretical predictions are compared with the results of extensive computer simulations, based on the thermodynamic formalism. For large systems the agreement is excellent. P A C S : 05.45+b; 05.20Dd; 05.60+w Keywords: Lyapunov exponents; Thermodynamic formalism; Discrete Markov process;
Hopping models on disordered lattices
1. Introduction The complex macroscopic behavior o f fluids out o f equilibrium appears to be intimately connected to chaos properties o f the nonlinear microscopic dynamics o f the underlying many body system. Indeed macroscopic transport properties, such as diffusion coefficients, have been succesfully related to fundamental dynamic quantifies such as Lyapunov exponents and dynamical entropies both for nonequilibrium fluids [ 1 - 5 ] as well as for Lorentz gases [ 6 - 1 3 ] . * Corresponding author. E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved P H S0378-4371(97)00386-5
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L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
These problems have been studied most extensively for simple models of nonequilibrium statistical mechanics, the so-called Lorentz gases either in continuous space [1-7] or on lattices [8-13] where independent particles move diffusively in a random static environment, characterized by spatial inhomogeneities in local hopping rates or transition probabilities, local scattering laws, etc. The lattice models are all realizations of discrete Markov processes in quenched random environments. In case the domain, in which the particles are moving, has open or absorbing boundaries particles can escape from the system. The system constitutes a dynamical repeller. The rate V at which trajectories escape from this repeller, was found - perhaps not surprisingly - to be related to the macroscopic diffusion coefficient D of the system, i.e. 7 = Dq g where qo "~ 1/L is the wave number of the slowest decaying mode in the system of length L. Similarly, the Lyapunov exponent for a lattice Lorentz gas was found [12] in the form 2 = 20 + 21q g + . . . , where 20 is the size-independnet Lyapunov exponent of the corresponding closed system. The coefficients {D, 20, 21, . . .} depend on characteristic properties of the disorder, such as the density of these inhomogeneities and are related to macroscopic properties of the system. Using the thermodynamic formalism [10,14,15] one is able to derive exact or approximate analytical expressions for chaotic properties. Moreover, this formalism is well suited for numerical simulations of chaos properties of diffusive systems with static (bond or site) disorder on lattices by numerically calculating the largest eigenvalue of a large and sparse random matrix [12,13]. The goal of this paper is to extend the physical understanding and intuition about such microscopic models, and to analyze the chaotic dynamics of another basic model of nonequilibrium statistical mechanics: the one-dimensional random bond model (RBM) [ 16-18]. In a RBM, a particle moves along the bonds of a regular lattice jumping from a site to a nearest-neighbor site at discrete times according to a quenched set of probabilities associated with the bonds. If the sum of these probabilities on a site is less than unity, there is a nonvanishing (waiting) probability of remaining on the same site at the next time step. The occurrence of waiting probabilities in the RBM forms a key difference with the Lattice Lorentz gases and we expect a richer chaotic behavior in the RBMs because of the larger divergence of trajectories at every site. In the Lorentz lattice gases the trajectory of the moving particle in a quenched random configuration is free ballistic motion on nonscattering sites. So, only scattering sites contribute to the exponential divergence of initially nearby phase space trajectories. The corresponding Lyapunov exponent depends only on the total number N =- pL of scatterers, but not on further details of the configuration. Then, the basic idea for a mean-field approximation in one-dimensional Lattice Lorentz gases is to rescale the typical random configurations of L sites, which are characterizable by a mean-free path 10 and mean-free time to, to a uniform lattice with N lattice points, lattice distance 10 and unit time step to. For Lorentz gases on higherdimensional lattices a good mean-field theory has yet to be developed. In this case the naive meanfield theory (see also Section 5) is expected to give only quantitatively
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correct escape rates at low densities, but its predictions are very poor at higher densities [19]. In RBM and other hopping models on regular lattices with lattice distance a, on the other hand, free ballistic motion is absent, the mean-free path 10 equals a, and every site contributes to the divergence of the realizable phase space trajectories. Here the Lyapunov exponent depends on more detailed properties of the quenched random bond configurations. For instance, in a closed RBM with a random mixture of N = pL a-bonds and ( L - N ) b-bonds, the Lyapunov exponent in a quenched configuration not only depends on N, but also on the number o f clusters, consisting of adjacent bonds of the same type (as will be discussed in Section 4). In the present paper we study closed RBMs with periodic boundary conditions (PBC), and open RBMs with absorbing boundary conditions (ABC). In the first case an exact expression for the Lyapunov exponent 2 is obtained. In the second case exact resuits are not available, but we propose a mean-field theory in which spatial correlations in the quenched disorder are neglected. The plan of the paper is as follows. Section 2 defines the model and describes its relevant properties. Section 3 summarizes the thermodynamic formalism. Section 4 presents exact results for the Lyapunov exponent in a closed RBM, for a single quenched configuration, and for its ensemble average. In Section 5 we develope a mean field approximation for the Lyapunov exponent in open RBM, which is compared with the results of numerical simulations in Section 6. We present some conclusions in Section 7. Appendix A offers an alternative formula for calculating Lyapunov exponents in closed and open systems, and Appendix B presents a method of calculating the determinant of a large sparse random L × L matrix through an L-step operation, based on a two-step linear recursion relation.
2. Random-bond model We consider a random walker hopping at discrete times t = 0, 1, 2 . . . . from site to site on a one-dimensional lattice with unit lattice distance. The sites are labeled r with r E .~e -- {1, 2 . . . . . L} and the hopping rate ~ ( r ) across the bond (r,r + 1) is a quenched random variable in the interval (0, ½], drawn from a site-independent probability distribution n(w). The time evolution of the probability p ( r , t ) of finding the random walker at site r at time t is described by the Chapman-Kolmogorov (CK) equation p(r, t + 1 ) --- rP(r)p(r + 1, t) + ~ ( r - 1 ) p ( r - 1, t)
+(1 - ~ ( r ) - v~(r - 1) ) p ( r , t ) = ~
rV(rlr')p(r',t),
(1)
r t
with r I E La. The terms following the first equality sign represent, respectively, the probabilities for jumping to the right, jumping to the left and for staying at site r. The second equality defines a random symmetric transition matrix W. We are interested in
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
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the solution of the CK-equation for closed systems with PBC and for open systems with ABC. In the PBC case we identify site r = L with site r --- 0, so that if(0) = ~(L), and we impose the boundary conditions
p(r + L, t) = p(r, t).
(2)
In the ABC case, where p(r,t) represents the survival probability at site r, the boundary sites r = {1,L} are bordered by absorbers, placed on sites r = {0,L + 1}. This implies the ABC:
p(L + 1, t) = 0,
p(0, t) = 0,
(3)
where the r'-summation in Eq. (1) runs again from 1 to L. The CK-equations at the boundary sites r = {1,L} take the form
p(1,t + 1) --- @(1)p(2,t) + (1 - i f ( l ) - f f ( 0 ) ) p ( 1 , / ) , p(L,t + 1 ) = f f ( L - 1 ) p ( L - 1 , t ) + (1 - ~ ( L ) -
~ ( L - 1))p(L,t).
(4)
An ABC configuration, ~b = {ff(r)lr = 0, 1. . . . . L}, contains (L + 1 ) independent random variables; a PBC configuration ~ = {ff(r)lr : 1,... ,L} contains L such variables, as lb(0) -- if(L). The transition matrix satisfies the following conditions:
W(rlr') = ~ r
W(r'l r)
-- 1 (PBC), ~<1 (ABC).
(5)
r
In the PBC case the stationary solution of Eq. (1) in any fixed configuration ff --{~,(r), r = 1, 2, ..., L} is p(r, oo) = Po = 1/L. Moreover, the total probability is conserved, ff'~r p(r,t) = 1. In the ABC case the total probability satisfies the inequality, ~ r p(r,t) < 1, and keeps decreasing with increasing t. The solution of Eq. (1) depends on the quenched configuration of random-bond variables, ~b = {if(r)}. The transport properties and other macroscopic quantities of interest will be obtained by finally averaging over the quenched disorder, denoted by (...). The transport properties of this model have been studied extensively in the literature [16,18]. For our present purpose only the diffusion coefficient D is required, which is given by the exact result [16], =
.
(6)
3. The thermodynamic formalism The chaos properties of stochastic models, described by the CK-equation can be calculated [14,15] from the dynamical partition function Z(H, tlro). It is defined in a dynamic phase space whose points I2(t) represent trajectories of t time steps in a system of size L:
ZL(fl, tlro ) = y ~ [P(f2,tlro)] ~ = y~(W~)(rlro ) , f~
r
(7)
L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
95
where P(f2, t]ro) is the probability that the system follows a trajectory g2(t) = {r0, rh..., rt}, starting at ro at t = 0. The inverse temperature-like parameter fl allows one to scan the structure of the probability distribution P, where large positive and negative fl values select, respectively, the most probable and the most improbable trajectories [20]. The sum in Eq. (7) is over all possible trajectories of t time steps and the probability for a given trajectory can be expressed in terms of the transition probabilities W(rtr' ), i.e. t
P( ,ttro) = ]-I rvCr.l ._l).
(8)
n=l
The partition function is determined by the properties of the random matrix
W~(r[r') = [W(rlr')] ~, and Wt# in Eq. (7) represents the tth power of the matrix W& For large times the partition function becomes independent of the point of origin r0 (ergodicity), and it is determined by the laroest positive eigenvalue AL(/~) of the matrix W/~,which is assumed to be non-degenerate. Associated with the partition function there is a free-energy like function, ~bL(/~lff,)called the Ruelle or topological pressure:
{lnZL([~,tlro) = lnaL(/~).
~bL(/~[~) = limt__.~
(9)
The Ruelle pressure depends on the quenched configuration ~ of random bonds. The limit t ~ cx~ is analogous to the thermodynamic limit in statistical mechanics. The exponential increase of ZL with t is proportional to Tr W~ where, Tr denotes the trace of the tth power of the matrix W~, which is proportional to AL(/~)t for large times. The chaos properties of interest, such as the Lyapunov exponent 2, the escape-rate 7, and the Kolmogorov-Sinai entropy per unit time, hKs = 2 -- y, can be derived from the Ruelle pressure, as ~L(~) = --~L(ll~), 2L(~') = --q~(ll~),
(10)
where the prime denotes a /~-derivative. For a closed system AL(IIff) = 1, so that 7L(ff) = 0. The quantities of macroscopic interest are the averaoe Ruelle pressure, ~'L(/~) = <¢L(/~I~)) =
(1 1)
the average escape rate ~L and average Lyapunov exponent 2t. In the appendix we have derived an alternative expression for the Lyapunov exponent in a quenched configuration ~ of scatterers. It is more convenient for analytical calculations and reads 1 2L(W) --
AL
~ r,
u°(rlC')u°(r'lw)W(rlr')ln W(rlr') "
(12)
rt
It is expressed in terms of the largest eigenvalue AL -~ AL(fl =- 1) and the corresponding normalized eigenvector u0(rl~') of the transition matrix W(rlr' ) in the CKequation, normalized such that )-'it u~(rl~) = 1. It does not involve the matrix W#(r[r')
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L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
used in the thermodynamic formalism. The expression applies to open as well as closed systems. In the next section these expressions will be evaluated analytically for the closed random-bond model. Apart from the analytical approach we use in Section 6 the thermodynamic formalism for a numerical evaluation of chaos properties. The method is referred to as simulations. The procedure is to calculate the largest eigenvalue AL(/~Iff) of the random matrix W#(rlr t) in a fixed configuration if, and to evaluate the chaos quantities from Eqs. (9) and (10). The procedure for calculating the largest eigenvalue of the large sparse random matrix W~(rlr' ) is described in Appendix B.
4. Closed random-bond model (PBC) In closed systems the total probability is conserved. Consequently, the largest eigenvalue of W(rlr' ) is equal to unity and the corresponding eigenfunction uo(r) = 1/x/-L is proportional to the stationary solution P0 of the CK-equation. Hence, the escape rate ?o(ff) in Eq. (10) vanishes and the KS-entropy h°(~) equals the Lyapunov exponent 2°(if), which simplifies to 1 2L°Ob) = - - ~ ~
(13)
W(rlr')ln W(rlr' ) .
r, r t
We note that the thermodynamic formalism requires only the largest eigenvalue AL(fll)v) of the matrix W~(rlr ~) = (W(rlr~)) ~ , whereas the alternative method requires both the largest cigenvalue AL(ll)~ ) as well as the corresponding eigenvector u0(rl~ ) of the simpler transition matrix W(rlr ~) = Wl(rlr' ). The result (13) is exact for a closed system in any quenched configuration )~. We analyze this expression for a random mixture of a- and b-bonds, characterized by the hopping rates, ~= and Wb, respectively, and containing N~ = N a-bonds and Nb = L - N b-bonds. For this case the Lyapunov exponent reads in full 2o()~) = --~1 {2Nawa In Wa + 2NbWb In Wb +(NabO b) + Nb~(~))(1 -- W# -- wb)ln(1 -- Wa -- Wb) +Naa(~)(1 - 2w~)ln(1 - 2wa) +Nbb(~)(1 -- 2wb)ln(1 -- 2Wb)} •
(14)
The coefficients Nab, etc. represent the number of sites, connected on the left to an a-bond and on the right to a b-bond, etc. We refer to a set of l connected a-bonds (l = 1,2 .... ) as an a-cluster. As the system obeys PBC, the number Nab also equals the number of a-clusters (which equals the number of b-clusters on a closed ring). Then Nab(~) = Nba(~) = k Naa(~) = N - ~
(k = 1 , 2 , " ' ) , Nbb(~) = L - N -
k.
(15)
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97
This implies that the Lyapunov exponent 2°(if,) of the closed random-bond model in a quenched configuration ~ depends only on the number of a-clusters k(ffO (k = 1,2 . . . . . N ) in that configuration, and it assumes only N distinct values. This is in sharp contrast to the lattice Lorentz gas [12], where 2°(ff) is the same for all configurations of scatterers, and depends only on the number of scatterers, and not on their distribution. The macroscopic quantity of interest is the average over all possible configurations with a fixed number N of a-bonds and L sites. For sufficiently large systems the average number of clusters is given by (k(ffO) = (Nab(frO) = Lp(1 - p ) ,
(16)
where p = N/L, is the fraction of a-bonds. The average Lyapunov exponent follows then from (15) and (14) as 1
2o = (2°(if0) = - Z Z
(W(rlr')lnW(rlr'))
r,r t
= - 2 p w a lnwa - 2(1 - p)Wb lnwb -- p2(1 -- 2Wa) In (1 -- 2Wa)
--(1 -- p)2(1 -- 2wb)ln(1 -- 2wb) --2p(1 -- p)(1 -- Wa -- W6) In (1 -- Wa -- Wb).
(17)
The previous discussion shows that there exist two methods for calculating the discrete distribution P(2) of the N Lyapunov exponents in a closed random-bond model (PBC) with N a-bonds and L sites: (i) Determine the probability distribution F ( k ) for finding k a-clusters in a configuration, containing N a-bonds on a ring consisting of L sites. This distribution may be estimated analytically or measured in a Monte Carlo simulation by counting the frequency distribution for the occurrence of exactly k clusters in a large number of runs, in which configurations are generated with fixed {N,L}. As 2 = a + b k is a linear function of k, the desired probability distribution for large { N , L } is P(2) = bF((2 - a)/b). (ii) Calculate the Lyapunov exponent 2L(ff,) for a fixed configuration ff numerically, using the thermodynamic formalism (see Appendix B), and determine P(2) by measuring the frequency distribution of 2L(ff) over a sufficiently large ensemble of configurations if,. The results of both methods are compared in Fig. 1 for a system of length L = 100 and a density p = 0.5 of a-bonds where 3 x 10s runs have been used in either method. The agreement is excellent, which also serves as a reliability test for the computer simulations using the thermodynamic formalism. The largest value of the Lyapunov exponent in Eq. (14) for all allowed values of {Wa, Wb} is found in configurations with the maximum number of clusters (kma~ = N ) , where each cluster consists of an isolated single a-bond. The smallest Lyapunov exponent is found in a configuration with the minimum number of clusters (kmi, = 1), where all a-bonds form a single cluster. The average Lyapunov exponent, obtained from either distribution is in excellent agreement with the result for 20 in Eq. (17), except for a small difference, which decreases with L
L. Acedo, M.II. ErnstlPhysica A 247 (1997) 91-107
98 0.16
i
I
I
0.05
0.10
I
0.14 0.12 0.I0
P(~)
0.08 0.06 0.04 tr
0.02 0 -0.15
-O,lO
-0.05
0.15
A - Ao
Fig. 1. Distribution function P(2) for the Lyapunov exponent in a closed system with L sites, N = pL a-bonds, and hopping rates wa = 0.1 and Wb = 0.5, obtained from 3 × l0 s runs. The simulation results are based on the thermodynamic formalism, and denoted by an asterisk. The exact results, Eqs. (13) and (14), indicated by open circles, assumes exactly N discrete values. and reaches a m a x i m u m for p = 0.5. It is a finite-size effect. The difference is positive and less than 0.4% for L 1> 50.
5. Open random bond model (ABC) The eigenvalue problem for the open system is rather complicated, and cannot be solved exactly. W e therefore try to estimate the chaos quantities b y developing a mean-field theory, in which spatial correlations between fluctuations are being neglected. The most naive form o f a mean-field theory is obtained b y averaging the eigenvalue equation Wl~u(~@) = A ( ~ l ~ ) u ( ~ l ~ ), and then factorizing averages o f products into products o f averages. The eigenvalue equation becomes then, in explicit form,
A(fl)u(fl) =- (Wfl) u(fl) = b(fl) [u(r + 1) + u(r - 1)] + a ( f l ) u ( r ) ,
(18)
with
(1
p )W~b ,
a(]~) -- {(1 -- ~ ( r ) -- ~ ( r -- 1))/~} = p2(1 -- 2Wa) # + 2 p ( 1 -- p)(1 -- Wa -- Wb) ~ + (1 -- p)2(1 - 2Wb) ~ ,
(19)
where p = NIL is the fraction o f sites occupied b y a-bonds. For later reference the averages a and b have been written out explicitly for the random mixture o f a- and b-bonds.
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L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
The eigenfunctions of (18) are linear combinations of exp[+iqr]. To satisfy the ABC, {u(r = O) = u(r = L + 1) = 0}, we take u(r) = sinqr. The smallest wave number to satisfy the ABC, is q0 = 7~/(L + 1). The corresponding largest eigenvalue A(fl) in this naive mean-field approximation follows by inserting the eigenfunctions into Eq. (18), yielding AMF(fl) = a + 2bcosq0 ~- a + 2b - bq 2 + . . . .
(20)
For large systems q0 is small. The relevant chaos properties can then be calculated from Eq. (20) in combination with Eqs. (9) and (10). The result is AMF(1) = 1 --qo2b(1) = 1 - O o ~ , 7MF : --InAMF(1 ) ~ Doq 2 , ~'MF--
A~tF(1 ) AMy( 1) '
-~2~ +qo2 [Oo2~ + (fflnff)].
(21)
Here we have introduced the mean-field approximation for the diffusion coefficient Do -- b(1) = (if) = pwa + (1 - p)Wb,
(22)
and the Lyapunov exponent 2~ for the infinite system (qo = 0) 2oo = (2oo0b)) = - 2 (~(r) In lb(r)) -
((1 -
if(r)
-
~(r
-
1))ln(1
-
if(r)
-
~(r
-
1)))
.
(23)
The average value 2oo equals 20 in Eq. (17) for a closed system. The second equality defines 2oo(ff). Comparison of 2oo(ff) for ABC with 20(~) for PBC in a given configuration ~,, shows that these quantities are almost equal, except for two differences: (i) The matrix elements W~(LI1 ) are equal to 0, ~(L) for ABC, PBC, respectively. (see Appendix B, Eq. (B.3)). (ii) The matrix Wa contains (L + 1) independent random variables {ff(r)lr = 0, 1,2 ..... L} for ABC, and only L for PBC, where ~,(0) = if(L). The quantity 20(ff) for PBC assumes only N discrete values over all possible configurations {~,(r)} with N a-bonds. Consequently, 2oo(ff) for ABC assumes twice as many discrete values over all possible configurations {~,(r)lr = 0, 1,2 ..... L}, as if(0) is either wa or Wb. The mean-field value Do in (22) deviates for general parameter values substantially from the exact diffusion coefficient D, which is given by Eq. (6) [16], i.e. l D
( 1 ) p
+l-p Wa
(24)
Wb
Table 1 shows values of D, Do and 20 for some typical parameter values {p, Wa = as used in the simulations. The slowest decaying mode of the average CK-equation, which determines the average decay rate 7, is of course determined by W, Wb = 12} ,
100
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107 Table 1 Numerical values of analytic result, in closed systems for average Lyapunov exponent 20, Eq. (17), exact diffusion coefficientD, Eq. (24), and ratio Do/D where Do is the mean-field approximation to D at different fractions p = NIL of a-bonds, and at different hopping rates wa = w, as used in the simulations Lyapunov exponent 2o
w/p
0.25
0.5
0.4 0.25 0.1
0.8096 0.8448 0.7836
0.9087 0.9905 0.9531 1.0181 0.8047 0.7565
0.4706 0.4000 0.2500
0.4444 0.4211 0.3333 0.2857 0.1667 0.1250
1.009 1.094 1.600
1.013 1.125 1.800
0.75
Diffusion coefficient D 0.4 0.25 0.1 Ratio Do/D 0.4 0.25 0.1
1.009 1.094 1.600
the macroscopic diffusion coefficient D. By replacing the mean-field Do by the exact D we obtain more realistic mean-field estimates for escape rate and Lyapunov exponent
~MF = Dq~, 2MF = "~oo -'[-q2 [D2o~ + ( f f l n f f ) ] .
(25)
As q2 = n/(L + 1) 2 ~-- 10 -3 for L = 100, the Lyapunov exponent 2Mr for an open system is very close to 2o~, which is equal to 2o for the corresponding closed system. In Table 3 the mean-field predictions (25) for the correction A2 = ).MF- ).oo are listed for some typical parameter values, as used in our computer simulations. In the next section A2 will be compared with the results o f computer simulations on the random mixture o f a- and b-bonds, as obtained b y numerical simulations o f the thermodynamic formalism.
6. Numerical simulations for ABC In the previous section, mean-field results for the chaos properties (25) have been derived for the escape rate 7 and Lyapunov exponent 2 in the open random-bond model. In the present section these results will be compared with the results obtained b y numerical simulations based on the thermodynamic formalism. In these simulations one determines the largest eigenvalue A(fl) o f a large L x L sparse matrix W~ corresponding
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
101
Table 2 Simulation results in open systems for /2L(~))am compared with the mean-field result 2o0 in Eq. (23) for an infinite system, at different p and w {p;w;L}
(2L(ffO)sim )t~
{0.2; 0.3; 50} {0.4; 0.2; 50} {0.6; 0.4; 50} {0.8; 0.3; 50} (0.3; 0.4; 100} (0.9; 0.4; 100} {0.2;0.45;200}
0.81540 0.89484 0.94040 1.05131 0.82988 1.02916 0.75515
0.81445 0.89311 0.93953 1.05044 0.82955 1.02897 0.75510
to a quenched random configuration {ff(r)lr = 0, 1,2 . . . . . L} with L typically 50, 100 or 200. The secular determinant DL(A) = detlW # - AI, is calculated in a number of steps, linear in the system size L. This can be done by solving a two-step recursion relation, derived in Appendix B. Then the largest eigenvalue is found as the largest root of DL(A) = 0, using the Newton method, and direct application of Eq. (10) yields y and 2. The simulations are all carried out on a random mixture of (L + 1) a- and b-bonds with a fixed fraction o f a-bonds p = N/(L + 1 ). Moreover, the hopping rates are taken as w6 = 1 and Wa = w = {0.1;0.25;0.4}. We first consider the escape rates. The agreement between simulation averages of the escape rate and the values predicted by Eq. (25) is excellent with relative differences around 0.5% for systems with L = 50 and around 0.04% for systems with L = 400. This relative difference increases with the disorder as p tends to 0.5 and as Wa and Wb ~ 0). They can be considered as finite-size are as different as possible ( W b ~--- i,Wa 1 effects. The results for Do/D in Table 1 show convincingly that the naive mean-field theory in Eq. (21) o f Section 5, with D replaced by its mean-field approximation is totally inadequate. The distribution of escape rates is nearly Gaussian with a typical width of 5% of the mean. Testing the mean-field prediction for the Lyapunov exponent (25) involves two levels of accuracy: (i) the dominant, L-independent term 2 ~ and (ii) the small correction A2 ~ 0(L-2). The quantity 2 ~ = ( 2 ~ ( ~ ) ) = (2o(~)) is given by the analytic expression (17), whose numerical value is listed in Table 2. The simulation results for (,;t0b))si,n, obtained from 3 x 106 runs are compared with the dominant term 2 ~ in the same Table 2. The agreement is excellent; with a typical accuracy of 1 part in 103-104 for w = 0.2 to w = 0.45, respectively, at densities p = {0.2 . . . . . 0.9}. To test the validity o f our mean-field results (25) beyond the leading term 2 ~ , we need a very high statistical accuracy. The reason is that the theoretical prediction A2 = A2*n2/(L + 1) 2 is typically 5 x 10 -5 for L = 100, as can be seen in Table 3. Hence, the statistical accuracy for a quantitative verification should typically be I x 10 -5
102
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
Table 3 Simulation results for the average Lyapunov exponent A2* = A2(L+ 1 )2/•2 in the ABC ease at different size L, density of a-bonds p = N/(L + 1) and hopping rates Wa = w, where wb = ½. An estimate of the statistical error in the last digit is included in parenthesis w
L
0.4
50 100
p = 0.3
50 100
MFT
0.1
50 100
MFT
0.4
200
MFT
0.25
200
MFT
0.1
200
MFT
p = 0.6
0.39(2) 0.37(5)
0.51 (2) 0.49(4)
0.53 (2) 0.49(4)
0.34
0.46
0.50
-0.04(5) --0.12(6)
-0.30(2) --0.37(6)
-0.44(2) --0.45(5)
-0.12
-0.28
-0.38
-
-1.38(1) 1.46(4)
-1.81(1) - 1.79(4)
-1.89(2) - 1.91(4)
- 1.28
- 1.51
- 1.58
p = 0.25
p = 0.5
p = 0.75
MFT
0.25
p = 0.5
0.33( 1)
0.49( 1)
0.44(7)
0.29
0.46
0.54
-0.2(3)
-0.34(2)
-0.69(8)
-0.09
-0.28
-0.54
-1.5(1)
-1.8(1)
-1.9(I)
- 1.29
- 1.51
- 1.63
to 5 × 10 -6. This a c c u r a c y is usually not reached in 107 runs, because the small correction, A2 ~ d~(L - 2 ) , is o v e r s h a d o w e d by fluctuations in the d o m i n a n t t e r m around its average (2oo(v~)). To suppress the fluctuations in 2oo(ff), it is necessary to subtract f r o m 2L(ff) the quantity 2 ~ ( f f ) , w h i c h is evaluated in e v e r y configuration, {v~(r)}, f r o m its definition in Eq. ( 2 3 ) and w h i c h assumes only a discrete set o f values, proportional to the n u m b e r o f a-bonds in the system. So, w e calculate (A,~(~,))
= (,ZL(,~) - ,loo(~')>.
(26)
The results are displayed in Table 3 and c o m p a r e d with the mean-field predictions in Eq. (25); the reduced quantity A2* = A 2(L + 1)2/n2 has b e e n used to eliminate explicit d e p e n d e n c e on the system size L. The error bars h a v e b e e n calculated supposing that
103
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107 45
i
I
40 35
30 25 20 15 10 5 0 -0.06
"-_::I i
-0.04
-0.02
0
A~
0.02
0.04
0.06
Fig. 2. Probability distribution P(A2) for A2 = 2L(ff) - 2o~(~) (see Eq. (23)) in an open system with L = 100, and N = p(L+ !) a-bonds with Wa = 0.1 and Wb= ½. The mean (32) _~ -1.43 x 10-4. the distributions are normal. They are usually large, even averaging over 107 random configurations does not substantially reduce the error bars, because the ratio between width and mean is as large as 103 in some cases. In Fig. 2 a typical distribution with a large relative width is shown. The agreement with mean-field theory to d)(1/L2)-terms included is good for w = 0.4 over the whole range of densities p and system sizes L. Significant negative discrepancies in O(1/L2)-terms between mean-field theory and simulations can be observed for small hopping probabilities w. The conclusion is that the predictions of mean-field theory have the right sign and correct order of magnitude, but are definitely not exact. On the other hand, the correction A2 is in general much too small, to be observed in standard simulations and 2 ~ is an excellent estimate for (2L(ff)) for all system sizes with L ~>50, for all densities p of a-bonds and for all transition probabilites w.
7. Conclusions
( I ) In the present paper we have studied the chaotic dynamical quantities of a one-dimensional hopping model with bond disorder and PBC or ABC, using analytical methods and numerical simulations. The explicit results refer to a random mixture of aand b-bonds. For such a mixture and PBC an exact result for the Lyapunov exponent in a fixed configuration is obtained which not only depends on the number N of a-bonds, but also on the number k of a-clusters. As a consequence, Eqs. (13) and (14) imply that the Lyapunov exponent in an ensemble of quenched random configurations, each containing exactly N a-bonds, takes on only a discrete set of N possible values. The simplicity of this result is a direct consequence of the one-dimensional geometry of the model.
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104
(2) The mean-field predictions in the ABC case for the average escape rate y : Dq 2 and Lyapunov exponent, 2 = 2o0 +22q~ with qo = tel(L+ 1), are in excellent agreement with the simulation results for systems with L >_-50. For the Lyapunov exponent the accuracy is typically 1 part in 103, which is solely given by the dominant term 2~, as 22q02 is smaller than 10-5. To actually verify the mean-field prediction to the terms 22q~ included, we have increased the number of runs in the ensemble to 107, and we have observed that the expected value for A2 = 2 - 2 0 has the correct sign and order of magnitude, but there is no quantitative agreement at small hopping rates across a-bonds (Wb = 1, and Wa~<0.1) and at high densities of a-bonds (p~>0.6). (3) The dependence of the Lyapunov exponent in a closed random-bond model on the number of clusters of pure a-bonds indicates a richer chaotic behavior than in lattice Lorentz gases where the Lyapunov exponent is independent of the particular configuration. This is a consequence of the existence of waiting probabilities. (4) The relation between the escape-rate and the diffusion coefficient (24) has also been derived for lattice Lorentz gases [12] and for continuous Lorentz gas with a periodic array of scatterers [1]. This relation expresses the connection between the chaotic dynamical quantities and the transport in simple systems. (5) The extremely broad distribution for the Lyapunov exponent in open systems, also found in lattice Lorentz gases, implies important technical problems in performing computer simulations specially when extensions to two-dimensional systems are considered. The same problems arise in numerical studies of two-dimensional Lorentz lattice gases [19] An analysis of the moments of these distributions as a function of the system parameters {p, Wa,Wb} needs to be performed as well as a study of the configurations leading to the extreme values of the Lyapunov exponent and to a dynamical phase diagram, describing the different types of localization transitions [21]. Work along this line is currently in progress.
Acknowledgements L.A. gratefully acknowledges support from the Ministerio de Educaci6n y Ciencia (Spain) and the hospitality of the University of Utrecht during his visit in the fall of 1994 where this work was started.
Appendix A. Alternative expression for Lyapunov exponents In this appendix we briefly outline the derivation of Eq. (12) for the Lyapunov exponent starting from the thermodynamic formalism. By combining Eqs. (9) and (10) we obtain for the Lyapunov exponent in a fixed configuration k of random bonds 2(v~) = -
lira
1 ~
t---,c~ t
P(~2,tlro)lnP(f2,tlro ) Ef~ P(f2,tlro)
(A.1)
L. Acedo, M.H. ErnstlPhysicaA 247 (1997) 91-107
105
Before analyzing this expression we consider the eigenvalues An and the eigenvectors n = O, 1 . . . . . L - 1, of the symmetric transition matrix:
u.(r),
W(rlr')un(r') = Anun(r),
(A.2)
rt
and denote its largest nondegenerate eigenvalue by A0. For a closed system Ao = 1 and for an open system Ao < 1 on account of Eq. (5). We also define the scalar product of the eigenfunctions a and b as
(alb) = Z
a(r)b(r),
(A.3)
r
and take the eigenvectors un(r) as normalized according to this definition. The spectral decomposition of the matrix W has the form
W(rt/) = Z
un(r)Anun(r').
(A.4)
n
Next, we denote the denominator in (A.1) by ~ ( t ) and insert Eqs. (8) and (A.4), which yields the result
~(t) = ~
Wt(rlro) = y~ Un(r)AtnUn(ro) ,
r
(A.5)
r,n
where W t is the t-th power of the matrix W. At large t Eq. (A.5) approaches ~ ( t ) "" A~ (llu0) uo(ro),
(A.6)
where (l[uo) = )--~ruo(r). Next we consider the numerator of (A.1), denoted by X ( t ) and combine it with Eqs. (8) and (A.4) to find t
JV(t)=Z
Z
z=l
Wt-~(r"lr') [W(r'lr)lnW(/[r)] W*-'(r[r°)
r, r t , r tt
t
= Z Z At~-~A*~I (l[Un) (Un[Wln Wlum)um(ro). 3=1
(A.7)
n,m
By observing that the dominant long-time contribution in these sums come from An = Am = A0, the above expression reduces in the limit t ~ oo to tiff(t) '~ (t -- 1)A~-1 (lluo) (uolWln Wluo) uo(ro).
(A.8)
Substitution of Eqs. (A.6) and (A.8) in Eq. (A.1) yields an expression [11] for the Lyapunov exponent 20b) =
1
Ao (uo[WlnW[uo),
(A.9)
in terms of the largest eigenvalue Ao and corresponding eigenvector uo(r) of the transition matrix W. This formula is valid for open and closed systems, and applies to any quenched configuration ff of impurity bonds. In a closed system Ao = 1, and
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
106
uo(r) = 1/v/L equals the stationary solution P0 o f the CK-equation (1), apart from a normalization constant. Then we have for a closed system: 2(~) =
-~1 ~
W(rlr')ln W(rlr' ) .
(A.10)
r, r /
The formulae (A.9) and (A.10) for open and closed systems can be directly generalized to higher-dimensional systems.
Appendix B. Recursion relations for the determinant of Wp In the ABC case the configuration ~ = {~(r)[r = 0,1,2 . . . . . L} contains (L + 1) independent random variables; in the PBC case ~ = { ~ ( r ) ] r = 1,2,... ,L} contains L such variables as ~ ( 0 ) = ~(L). The random matrix, defined below Eq. (8), has the form
WlJ(r[r') =
{
br-1
(r' = r - 1),
ar br
(~' = O , (r' = r + 1),
(B.1)
where r = 1,2 . . . . . L and b~ = ( ~ ( r ) ) ~ , ar = (1 -- if(r) - ff(r -- 1)) fl .
(B.2)
Moreover, the determinant, detlW~[, in the ABC and PBC case is denoted by DL (0, 1,2 . . . . . L) and EL(I,2 .... ,L) respectively, where the arguments refer to the set o f random variables ft. To derive the recursion relation we write EL explicitly
bl bl a2 b2 b2 a3 b3
al
EL(1,2,...,L) :
bL
b3 a4 b4
bL
(B.3) az-1 bL-1 bL-1 aL
and DL(1,2,... ,L) follows from (B.3) by setting bz = 0. We calculate DL by developing Eq. (B.3) (with bL = 0) with respect to the last column. This yields at once the recursion relation for the ABC ease:
Dr(O, 1,2 . . . . . L) = aLDL--I(O, 1,2 . . . . . L - 1) -- b2_IDL_2(0, 1,2 .... ,L - 2)
(B.4) with initial conditions DI(1) = al,
Do = 1.
(B.5)
L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
107
The determinant EL is slightly more complicated. Developing EL with respect to the last column yields EL(l,2 ..... L) = a~DL_I(L, 1,2 ..... L - 1) - bL-1ML-1,L + (--1)L-IbLMI,L. (B.6) The two minors are developed with respect to the last row and yield ML-1,L = bL-IDL-2(L, 1,2 .... ,L - 2) + (-1)LbLb~b2 . . . b~-2, M1,L = blb2 " " " bL-1 + (-1)LbLDI.-2(2 . . . . . L -
1).
(B.7)
Combination of (B.6), (B.7) and (B.4) yields the relation EL(l, 2 ..... L) = DL(L, 1,2,... ,L) - bEDL_2(1,2 ..... L - l) - 2 ( - 1 )Lblb2... bL.
(B.8) The recursion relation (B.4) provides a very efficient numerical algorithm for calculating the determinants DL and EL for a given set of quenched variables. The secular determinants, det] WI~- A] = 0, which is a polynomial in A, of degree L, can be calculated by replacing ar by a r - A. The largest zero AL(fl) of this polynomial is calculated numerically, and provides all necessary information required in the thermodynamic formalism.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
P. Gaspard, G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693. H.A. Posch, W.G. Hoover, Phys. Rev. Lett. A 123 (1987) 227; Phys. Rev. A 39 (1989) 2175. D.J. Evans, E.G.D. Cohen, G.P. Mordss, Phys. Rev. A 42 (1990) 5990. N.I. Chernov, G.L. Eyink, J.L. Lebowitz, Ya. G. Sinai, Commun. Math. Phys. 154 (1993) 569. G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74 (1995); 2694; J. Stat. Phys. 80 (1995) 931. P. Gaspard, F. Baras, Phys. Rev. E 51 (1995) 5332. H. van Beijeren, J. R. Dorfman, Phys. Rev. Lett. 74 (1995) 4412. J.R. Dorfman, P. Gaspard, Phys. Rev. E 51 (1995) 28. P. Gaspard, J.R. Dorfman, Phys. Rev. E 52 (1995) 3525. J.R. Dorfinan, M.H. Ernst, D. Jacobs, J. Star. Phys. 81 (1995) 497. M.H. Ernst, J.R. Dorfman, Chaos and Diffusion in Periodic Random Walks, (unpublished). M.H. Ernst, J.R. Dorfman, R. Nix, D. Jacobs, Phys. Rev. Lett. 74 (1995) 4416. C. Appert, M.H. Ernst, chao-dyn/9705011. D. Ruelle, Thermodynamic Formalism, (Addison-Wesley, Reading, MA, 1978) C. Beck and F. Schl6gl, Thermodynamics of chaotic systems, (Cambridge University Press, Cambridge, 1993). J.W. Hans, K. Kehr, Phys. Pep. 150 (1987) 263. J.W. Haus, K. Kehr, J.W. Lyklcma, Phys. Rev. B 25 (1982) 2905. P.J.H. Denteneer, M.H. Ernst, Phys. Rev. B 29 (1984) 1755. C. Appert, private communication. C. Appert, H. van Beijeren, M.H. Ernst, J.R. Dorfman, J. Stat. Phys. 87 (1997) June issue. L. Acedo, M.H. Ernst, in preparation.
Physica A 247 (1997) 91-107
Lyapunov exponent of random walkers on a bond-disordered lattice L. Acedo a, M.H. Ernst b,* a Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz, Spain b Institute for Theoretical Physics, Utrecht University, P.O. Box 80006, 3508 TA Utrecht, The Netherlands
Received 23 May 1997
Abstract The chaotic properties of a random walker in a quenched random environment are studied analytically, following the work of Gaspard et al. on Lorentz gases, for systems with closed (periodic) or open (absorbing) boundaries. The model of interest describes random walkers hopping on a disordered lattice, on which the hopping probabilities across bonds are quenched random variables. For closed systems an exact expression for the Lyapunov exponent is derived, which not only depends on the composition, but also on the number of clusters of certain type. For open systems escape rates and Lyapunov exponents are calculated from a mean-field approximation. The theoretical predictions are compared with the results of extensive computer simulations, based on the thermodynamic formalism. For large systems the agreement is excellent. P A C S : 05.45+b; 05.20Dd; 05.60+w Keywords: Lyapunov exponents; Thermodynamic formalism; Discrete Markov process;
Hopping models on disordered lattices
1. Introduction The complex macroscopic behavior o f fluids out o f equilibrium appears to be intimately connected to chaos properties o f the nonlinear microscopic dynamics o f the underlying many body system. Indeed macroscopic transport properties, such as diffusion coefficients, have been succesfully related to fundamental dynamic quantifies such as Lyapunov exponents and dynamical entropies both for nonequilibrium fluids [ 1 - 5 ] as well as for Lorentz gases [ 6 - 1 3 ] . * Corresponding author. E-mail: [email protected]. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved P H S0378-4371(97)00386-5
92
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
These problems have been studied most extensively for simple models of nonequilibrium statistical mechanics, the so-called Lorentz gases either in continuous space [1-7] or on lattices [8-13] where independent particles move diffusively in a random static environment, characterized by spatial inhomogeneities in local hopping rates or transition probabilities, local scattering laws, etc. The lattice models are all realizations of discrete Markov processes in quenched random environments. In case the domain, in which the particles are moving, has open or absorbing boundaries particles can escape from the system. The system constitutes a dynamical repeller. The rate V at which trajectories escape from this repeller, was found - perhaps not surprisingly - to be related to the macroscopic diffusion coefficient D of the system, i.e. 7 = Dq g where qo "~ 1/L is the wave number of the slowest decaying mode in the system of length L. Similarly, the Lyapunov exponent for a lattice Lorentz gas was found [12] in the form 2 = 20 + 21q g + . . . , where 20 is the size-independnet Lyapunov exponent of the corresponding closed system. The coefficients {D, 20, 21, . . .} depend on characteristic properties of the disorder, such as the density of these inhomogeneities and are related to macroscopic properties of the system. Using the thermodynamic formalism [10,14,15] one is able to derive exact or approximate analytical expressions for chaotic properties. Moreover, this formalism is well suited for numerical simulations of chaos properties of diffusive systems with static (bond or site) disorder on lattices by numerically calculating the largest eigenvalue of a large and sparse random matrix [12,13]. The goal of this paper is to extend the physical understanding and intuition about such microscopic models, and to analyze the chaotic dynamics of another basic model of nonequilibrium statistical mechanics: the one-dimensional random bond model (RBM) [ 16-18]. In a RBM, a particle moves along the bonds of a regular lattice jumping from a site to a nearest-neighbor site at discrete times according to a quenched set of probabilities associated with the bonds. If the sum of these probabilities on a site is less than unity, there is a nonvanishing (waiting) probability of remaining on the same site at the next time step. The occurrence of waiting probabilities in the RBM forms a key difference with the Lattice Lorentz gases and we expect a richer chaotic behavior in the RBMs because of the larger divergence of trajectories at every site. In the Lorentz lattice gases the trajectory of the moving particle in a quenched random configuration is free ballistic motion on nonscattering sites. So, only scattering sites contribute to the exponential divergence of initially nearby phase space trajectories. The corresponding Lyapunov exponent depends only on the total number N =- pL of scatterers, but not on further details of the configuration. Then, the basic idea for a mean-field approximation in one-dimensional Lattice Lorentz gases is to rescale the typical random configurations of L sites, which are characterizable by a mean-free path 10 and mean-free time to, to a uniform lattice with N lattice points, lattice distance 10 and unit time step to. For Lorentz gases on higherdimensional lattices a good mean-field theory has yet to be developed. In this case the naive meanfield theory (see also Section 5) is expected to give only quantitatively
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correct escape rates at low densities, but its predictions are very poor at higher densities [19]. In RBM and other hopping models on regular lattices with lattice distance a, on the other hand, free ballistic motion is absent, the mean-free path 10 equals a, and every site contributes to the divergence of the realizable phase space trajectories. Here the Lyapunov exponent depends on more detailed properties of the quenched random bond configurations. For instance, in a closed RBM with a random mixture of N = pL a-bonds and ( L - N ) b-bonds, the Lyapunov exponent in a quenched configuration not only depends on N, but also on the number o f clusters, consisting of adjacent bonds of the same type (as will be discussed in Section 4). In the present paper we study closed RBMs with periodic boundary conditions (PBC), and open RBMs with absorbing boundary conditions (ABC). In the first case an exact expression for the Lyapunov exponent 2 is obtained. In the second case exact resuits are not available, but we propose a mean-field theory in which spatial correlations in the quenched disorder are neglected. The plan of the paper is as follows. Section 2 defines the model and describes its relevant properties. Section 3 summarizes the thermodynamic formalism. Section 4 presents exact results for the Lyapunov exponent in a closed RBM, for a single quenched configuration, and for its ensemble average. In Section 5 we develope a mean field approximation for the Lyapunov exponent in open RBM, which is compared with the results of numerical simulations in Section 6. We present some conclusions in Section 7. Appendix A offers an alternative formula for calculating Lyapunov exponents in closed and open systems, and Appendix B presents a method of calculating the determinant of a large sparse random L × L matrix through an L-step operation, based on a two-step linear recursion relation.
2. Random-bond model We consider a random walker hopping at discrete times t = 0, 1, 2 . . . . from site to site on a one-dimensional lattice with unit lattice distance. The sites are labeled r with r E .~e -- {1, 2 . . . . . L} and the hopping rate ~ ( r ) across the bond (r,r + 1) is a quenched random variable in the interval (0, ½], drawn from a site-independent probability distribution n(w). The time evolution of the probability p ( r , t ) of finding the random walker at site r at time t is described by the Chapman-Kolmogorov (CK) equation p(r, t + 1 ) --- rP(r)p(r + 1, t) + ~ ( r - 1 ) p ( r - 1, t)
+(1 - ~ ( r ) - v~(r - 1) ) p ( r , t ) = ~
rV(rlr')p(r',t),
(1)
r t
with r I E La. The terms following the first equality sign represent, respectively, the probabilities for jumping to the right, jumping to the left and for staying at site r. The second equality defines a random symmetric transition matrix W. We are interested in
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the solution of the CK-equation for closed systems with PBC and for open systems with ABC. In the PBC case we identify site r = L with site r --- 0, so that if(0) = ~(L), and we impose the boundary conditions
p(r + L, t) = p(r, t).
(2)
In the ABC case, where p(r,t) represents the survival probability at site r, the boundary sites r = {1,L} are bordered by absorbers, placed on sites r = {0,L + 1}. This implies the ABC:
p(L + 1, t) = 0,
p(0, t) = 0,
(3)
where the r'-summation in Eq. (1) runs again from 1 to L. The CK-equations at the boundary sites r = {1,L} take the form
p(1,t + 1) --- @(1)p(2,t) + (1 - i f ( l ) - f f ( 0 ) ) p ( 1 , / ) , p(L,t + 1 ) = f f ( L - 1 ) p ( L - 1 , t ) + (1 - ~ ( L ) -
~ ( L - 1))p(L,t).
(4)
An ABC configuration, ~b = {ff(r)lr = 0, 1. . . . . L}, contains (L + 1 ) independent random variables; a PBC configuration ~ = {ff(r)lr : 1,... ,L} contains L such variables, as lb(0) -- if(L). The transition matrix satisfies the following conditions:
W(rlr') = ~ r
W(r'l r)
-- 1 (PBC), ~<1 (ABC).
(5)
r
In the PBC case the stationary solution of Eq. (1) in any fixed configuration ff --{~,(r), r = 1, 2, ..., L} is p(r, oo) = Po = 1/L. Moreover, the total probability is conserved, ff'~r p(r,t) = 1. In the ABC case the total probability satisfies the inequality, ~ r p(r,t) < 1, and keeps decreasing with increasing t. The solution of Eq. (1) depends on the quenched configuration of random-bond variables, ~b = {if(r)}. The transport properties and other macroscopic quantities of interest will be obtained by finally averaging over the quenched disorder, denoted by (...). The transport properties of this model have been studied extensively in the literature [16,18]. For our present purpose only the diffusion coefficient D is required, which is given by the exact result [16], =
.
(6)
3. The thermodynamic formalism The chaos properties of stochastic models, described by the CK-equation can be calculated [14,15] from the dynamical partition function Z(H, tlro). It is defined in a dynamic phase space whose points I2(t) represent trajectories of t time steps in a system of size L:
ZL(fl, tlro ) = y ~ [P(f2,tlro)] ~ = y~(W~)(rlro ) , f~
r
(7)
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95
where P(f2, t]ro) is the probability that the system follows a trajectory g2(t) = {r0, rh..., rt}, starting at ro at t = 0. The inverse temperature-like parameter fl allows one to scan the structure of the probability distribution P, where large positive and negative fl values select, respectively, the most probable and the most improbable trajectories [20]. The sum in Eq. (7) is over all possible trajectories of t time steps and the probability for a given trajectory can be expressed in terms of the transition probabilities W(rtr' ), i.e. t
P( ,ttro) = ]-I rvCr.l ._l).
(8)
n=l
The partition function is determined by the properties of the random matrix
W~(r[r') = [W(rlr')] ~, and Wt# in Eq. (7) represents the tth power of the matrix W& For large times the partition function becomes independent of the point of origin r0 (ergodicity), and it is determined by the laroest positive eigenvalue AL(/~) of the matrix W/~,which is assumed to be non-degenerate. Associated with the partition function there is a free-energy like function, ~bL(/~lff,)called the Ruelle or topological pressure:
{lnZL([~,tlro) = lnaL(/~).
~bL(/~[~) = limt__.~
(9)
The Ruelle pressure depends on the quenched configuration ~ of random bonds. The limit t ~ cx~ is analogous to the thermodynamic limit in statistical mechanics. The exponential increase of ZL with t is proportional to Tr W~ where, Tr denotes the trace of the tth power of the matrix W~, which is proportional to AL(/~)t for large times. The chaos properties of interest, such as the Lyapunov exponent 2, the escape-rate 7, and the Kolmogorov-Sinai entropy per unit time, hKs = 2 -- y, can be derived from the Ruelle pressure, as ~L(~) = --~L(ll~), 2L(~') = --q~(ll~),
(10)
where the prime denotes a /~-derivative. For a closed system AL(IIff) = 1, so that 7L(ff) = 0. The quantities of macroscopic interest are the averaoe Ruelle pressure, ~'L(/~) = <¢L(/~I~)) =
(1 1)
the average escape rate ~L and average Lyapunov exponent 2t. In the appendix we have derived an alternative expression for the Lyapunov exponent in a quenched configuration ~ of scatterers. It is more convenient for analytical calculations and reads 1 2L(W) --
AL
~ r,
u°(rlC')u°(r'lw)W(rlr')ln W(rlr') "
(12)
rt
It is expressed in terms of the largest eigenvalue AL -~ AL(fl =- 1) and the corresponding normalized eigenvector u0(rl~') of the transition matrix W(rlr' ) in the CKequation, normalized such that )-'it u~(rl~) = 1. It does not involve the matrix W#(r[r')
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L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
used in the thermodynamic formalism. The expression applies to open as well as closed systems. In the next section these expressions will be evaluated analytically for the closed random-bond model. Apart from the analytical approach we use in Section 6 the thermodynamic formalism for a numerical evaluation of chaos properties. The method is referred to as simulations. The procedure is to calculate the largest eigenvalue AL(/~Iff) of the random matrix W#(rlr t) in a fixed configuration if, and to evaluate the chaos quantities from Eqs. (9) and (10). The procedure for calculating the largest eigenvalue of the large sparse random matrix W~(rlr' ) is described in Appendix B.
4. Closed random-bond model (PBC) In closed systems the total probability is conserved. Consequently, the largest eigenvalue of W(rlr' ) is equal to unity and the corresponding eigenfunction uo(r) = 1/x/-L is proportional to the stationary solution P0 of the CK-equation. Hence, the escape rate ?o(ff) in Eq. (10) vanishes and the KS-entropy h°(~) equals the Lyapunov exponent 2°(if), which simplifies to 1 2L°Ob) = - - ~ ~
(13)
W(rlr')ln W(rlr' ) .
r, r t
We note that the thermodynamic formalism requires only the largest eigenvalue AL(fll)v) of the matrix W~(rlr ~) = (W(rlr~)) ~ , whereas the alternative method requires both the largest cigenvalue AL(ll)~ ) as well as the corresponding eigenvector u0(rl~ ) of the simpler transition matrix W(rlr ~) = Wl(rlr' ). The result (13) is exact for a closed system in any quenched configuration )~. We analyze this expression for a random mixture of a- and b-bonds, characterized by the hopping rates, ~= and Wb, respectively, and containing N~ = N a-bonds and Nb = L - N b-bonds. For this case the Lyapunov exponent reads in full 2o()~) = --~1 {2Nawa In Wa + 2NbWb In Wb +(NabO b) + Nb~(~))(1 -- W# -- wb)ln(1 -- Wa -- Wb) +Naa(~)(1 - 2w~)ln(1 - 2wa) +Nbb(~)(1 -- 2wb)ln(1 -- 2Wb)} •
(14)
The coefficients Nab, etc. represent the number of sites, connected on the left to an a-bond and on the right to a b-bond, etc. We refer to a set of l connected a-bonds (l = 1,2 .... ) as an a-cluster. As the system obeys PBC, the number Nab also equals the number of a-clusters (which equals the number of b-clusters on a closed ring). Then Nab(~) = Nba(~) = k Naa(~) = N - ~
(k = 1 , 2 , " ' ) , Nbb(~) = L - N -
k.
(15)
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97
This implies that the Lyapunov exponent 2°(if,) of the closed random-bond model in a quenched configuration ~ depends only on the number of a-clusters k(ffO (k = 1,2 . . . . . N ) in that configuration, and it assumes only N distinct values. This is in sharp contrast to the lattice Lorentz gas [12], where 2°(ff) is the same for all configurations of scatterers, and depends only on the number of scatterers, and not on their distribution. The macroscopic quantity of interest is the average over all possible configurations with a fixed number N of a-bonds and L sites. For sufficiently large systems the average number of clusters is given by (k(ffO) = (Nab(frO) = Lp(1 - p ) ,
(16)
where p = N/L, is the fraction of a-bonds. The average Lyapunov exponent follows then from (15) and (14) as 1
2o = (2°(if0) = - Z Z
(W(rlr')lnW(rlr'))
r,r t
= - 2 p w a lnwa - 2(1 - p)Wb lnwb -- p2(1 -- 2Wa) In (1 -- 2Wa)
--(1 -- p)2(1 -- 2wb)ln(1 -- 2wb) --2p(1 -- p)(1 -- Wa -- W6) In (1 -- Wa -- Wb).
(17)
The previous discussion shows that there exist two methods for calculating the discrete distribution P(2) of the N Lyapunov exponents in a closed random-bond model (PBC) with N a-bonds and L sites: (i) Determine the probability distribution F ( k ) for finding k a-clusters in a configuration, containing N a-bonds on a ring consisting of L sites. This distribution may be estimated analytically or measured in a Monte Carlo simulation by counting the frequency distribution for the occurrence of exactly k clusters in a large number of runs, in which configurations are generated with fixed {N,L}. As 2 = a + b k is a linear function of k, the desired probability distribution for large { N , L } is P(2) = bF((2 - a)/b). (ii) Calculate the Lyapunov exponent 2L(ff,) for a fixed configuration ff numerically, using the thermodynamic formalism (see Appendix B), and determine P(2) by measuring the frequency distribution of 2L(ff) over a sufficiently large ensemble of configurations if,. The results of both methods are compared in Fig. 1 for a system of length L = 100 and a density p = 0.5 of a-bonds where 3 x 10s runs have been used in either method. The agreement is excellent, which also serves as a reliability test for the computer simulations using the thermodynamic formalism. The largest value of the Lyapunov exponent in Eq. (14) for all allowed values of {Wa, Wb} is found in configurations with the maximum number of clusters (kma~ = N ) , where each cluster consists of an isolated single a-bond. The smallest Lyapunov exponent is found in a configuration with the minimum number of clusters (kmi, = 1), where all a-bonds form a single cluster. The average Lyapunov exponent, obtained from either distribution is in excellent agreement with the result for 20 in Eq. (17), except for a small difference, which decreases with L
L. Acedo, M.II. ErnstlPhysica A 247 (1997) 91-107
98 0.16
i
I
I
0.05
0.10
I
0.14 0.12 0.I0
P(~)
0.08 0.06 0.04 tr
0.02 0 -0.15
-O,lO
-0.05
0.15
A - Ao
Fig. 1. Distribution function P(2) for the Lyapunov exponent in a closed system with L sites, N = pL a-bonds, and hopping rates wa = 0.1 and Wb = 0.5, obtained from 3 × l0 s runs. The simulation results are based on the thermodynamic formalism, and denoted by an asterisk. The exact results, Eqs. (13) and (14), indicated by open circles, assumes exactly N discrete values. and reaches a m a x i m u m for p = 0.5. It is a finite-size effect. The difference is positive and less than 0.4% for L 1> 50.
5. Open random bond model (ABC) The eigenvalue problem for the open system is rather complicated, and cannot be solved exactly. W e therefore try to estimate the chaos quantities b y developing a mean-field theory, in which spatial correlations between fluctuations are being neglected. The most naive form o f a mean-field theory is obtained b y averaging the eigenvalue equation Wl~u(~@) = A ( ~ l ~ ) u ( ~ l ~ ), and then factorizing averages o f products into products o f averages. The eigenvalue equation becomes then, in explicit form,
A(fl)u(fl) =- (Wfl) u(fl) = b(fl) [u(r + 1) + u(r - 1)] + a ( f l ) u ( r ) ,
(18)
with
(1
p )W~b ,
a(]~) -- {(1 -- ~ ( r ) -- ~ ( r -- 1))/~} = p2(1 -- 2Wa) # + 2 p ( 1 -- p)(1 -- Wa -- Wb) ~ + (1 -- p)2(1 - 2Wb) ~ ,
(19)
where p = NIL is the fraction o f sites occupied b y a-bonds. For later reference the averages a and b have been written out explicitly for the random mixture o f a- and b-bonds.
99
L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
The eigenfunctions of (18) are linear combinations of exp[+iqr]. To satisfy the ABC, {u(r = O) = u(r = L + 1) = 0}, we take u(r) = sinqr. The smallest wave number to satisfy the ABC, is q0 = 7~/(L + 1). The corresponding largest eigenvalue A(fl) in this naive mean-field approximation follows by inserting the eigenfunctions into Eq. (18), yielding AMF(fl) = a + 2bcosq0 ~- a + 2b - bq 2 + . . . .
(20)
For large systems q0 is small. The relevant chaos properties can then be calculated from Eq. (20) in combination with Eqs. (9) and (10). The result is AMF(1) = 1 --qo2b(1) = 1 - O o ~ , 7MF : --InAMF(1 ) ~ Doq 2 , ~'MF--
A~tF(1 ) AMy( 1) '
-~2~ +qo2 [Oo2~ + (fflnff)].
(21)
Here we have introduced the mean-field approximation for the diffusion coefficient Do -- b(1) = (if) = pwa + (1 - p)Wb,
(22)
and the Lyapunov exponent 2~ for the infinite system (qo = 0) 2oo = (2oo0b)) = - 2 (~(r) In lb(r)) -
((1 -
if(r)
-
~(r
-
1))ln(1
-
if(r)
-
~(r
-
1)))
.
(23)
The average value 2oo equals 20 in Eq. (17) for a closed system. The second equality defines 2oo(ff). Comparison of 2oo(ff) for ABC with 20(~) for PBC in a given configuration ~,, shows that these quantities are almost equal, except for two differences: (i) The matrix elements W~(LI1 ) are equal to 0, ~(L) for ABC, PBC, respectively. (see Appendix B, Eq. (B.3)). (ii) The matrix Wa contains (L + 1) independent random variables {ff(r)lr = 0, 1,2 ..... L} for ABC, and only L for PBC, where ~,(0) = if(L). The quantity 20(ff) for PBC assumes only N discrete values over all possible configurations {~,(r)} with N a-bonds. Consequently, 2oo(ff) for ABC assumes twice as many discrete values over all possible configurations {~,(r)lr = 0, 1,2 ..... L}, as if(0) is either wa or Wb. The mean-field value Do in (22) deviates for general parameter values substantially from the exact diffusion coefficient D, which is given by Eq. (6) [16], i.e. l D
( 1 ) p
+l-p Wa
(24)
Wb
Table 1 shows values of D, Do and 20 for some typical parameter values {p, Wa = as used in the simulations. The slowest decaying mode of the average CK-equation, which determines the average decay rate 7, is of course determined by W, Wb = 12} ,
100
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107 Table 1 Numerical values of analytic result, in closed systems for average Lyapunov exponent 20, Eq. (17), exact diffusion coefficientD, Eq. (24), and ratio Do/D where Do is the mean-field approximation to D at different fractions p = NIL of a-bonds, and at different hopping rates wa = w, as used in the simulations Lyapunov exponent 2o
w/p
0.25
0.5
0.4 0.25 0.1
0.8096 0.8448 0.7836
0.9087 0.9905 0.9531 1.0181 0.8047 0.7565
0.4706 0.4000 0.2500
0.4444 0.4211 0.3333 0.2857 0.1667 0.1250
1.009 1.094 1.600
1.013 1.125 1.800
0.75
Diffusion coefficient D 0.4 0.25 0.1 Ratio Do/D 0.4 0.25 0.1
1.009 1.094 1.600
the macroscopic diffusion coefficient D. By replacing the mean-field Do by the exact D we obtain more realistic mean-field estimates for escape rate and Lyapunov exponent
~MF = Dq~, 2MF = "~oo -'[-q2 [D2o~ + ( f f l n f f ) ] .
(25)
As q2 = n/(L + 1) 2 ~-- 10 -3 for L = 100, the Lyapunov exponent 2Mr for an open system is very close to 2o~, which is equal to 2o for the corresponding closed system. In Table 3 the mean-field predictions (25) for the correction A2 = ).MF- ).oo are listed for some typical parameter values, as used in our computer simulations. In the next section A2 will be compared with the results o f computer simulations on the random mixture o f a- and b-bonds, as obtained b y numerical simulations o f the thermodynamic formalism.
6. Numerical simulations for ABC In the previous section, mean-field results for the chaos properties (25) have been derived for the escape rate 7 and Lyapunov exponent 2 in the open random-bond model. In the present section these results will be compared with the results obtained b y numerical simulations based on the thermodynamic formalism. In these simulations one determines the largest eigenvalue A(fl) o f a large L x L sparse matrix W~ corresponding
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
101
Table 2 Simulation results in open systems for /2L(~))am compared with the mean-field result 2o0 in Eq. (23) for an infinite system, at different p and w {p;w;L}
(2L(ffO)sim )t~
{0.2; 0.3; 50} {0.4; 0.2; 50} {0.6; 0.4; 50} {0.8; 0.3; 50} (0.3; 0.4; 100} (0.9; 0.4; 100} {0.2;0.45;200}
0.81540 0.89484 0.94040 1.05131 0.82988 1.02916 0.75515
0.81445 0.89311 0.93953 1.05044 0.82955 1.02897 0.75510
to a quenched random configuration {ff(r)lr = 0, 1,2 . . . . . L} with L typically 50, 100 or 200. The secular determinant DL(A) = detlW # - AI, is calculated in a number of steps, linear in the system size L. This can be done by solving a two-step recursion relation, derived in Appendix B. Then the largest eigenvalue is found as the largest root of DL(A) = 0, using the Newton method, and direct application of Eq. (10) yields y and 2. The simulations are all carried out on a random mixture of (L + 1) a- and b-bonds with a fixed fraction o f a-bonds p = N/(L + 1 ). Moreover, the hopping rates are taken as w6 = 1 and Wa = w = {0.1;0.25;0.4}. We first consider the escape rates. The agreement between simulation averages of the escape rate and the values predicted by Eq. (25) is excellent with relative differences around 0.5% for systems with L = 50 and around 0.04% for systems with L = 400. This relative difference increases with the disorder as p tends to 0.5 and as Wa and Wb ~ 0). They can be considered as finite-size are as different as possible ( W b ~--- i,Wa 1 effects. The results for Do/D in Table 1 show convincingly that the naive mean-field theory in Eq. (21) o f Section 5, with D replaced by its mean-field approximation is totally inadequate. The distribution of escape rates is nearly Gaussian with a typical width of 5% of the mean. Testing the mean-field prediction for the Lyapunov exponent (25) involves two levels of accuracy: (i) the dominant, L-independent term 2 ~ and (ii) the small correction A2 ~ 0(L-2). The quantity 2 ~ = ( 2 ~ ( ~ ) ) = (2o(~)) is given by the analytic expression (17), whose numerical value is listed in Table 2. The simulation results for (,;t0b))si,n, obtained from 3 x 106 runs are compared with the dominant term 2 ~ in the same Table 2. The agreement is excellent; with a typical accuracy of 1 part in 103-104 for w = 0.2 to w = 0.45, respectively, at densities p = {0.2 . . . . . 0.9}. To test the validity o f our mean-field results (25) beyond the leading term 2 ~ , we need a very high statistical accuracy. The reason is that the theoretical prediction A2 = A2*n2/(L + 1) 2 is typically 5 x 10 -5 for L = 100, as can be seen in Table 3. Hence, the statistical accuracy for a quantitative verification should typically be I x 10 -5
102
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
Table 3 Simulation results for the average Lyapunov exponent A2* = A2(L+ 1 )2/•2 in the ABC ease at different size L, density of a-bonds p = N/(L + 1) and hopping rates Wa = w, where wb = ½. An estimate of the statistical error in the last digit is included in parenthesis w
L
0.4
50 100
p = 0.3
50 100
MFT
0.1
50 100
MFT
0.4
200
MFT
0.25
200
MFT
0.1
200
MFT
p = 0.6
0.39(2) 0.37(5)
0.51 (2) 0.49(4)
0.53 (2) 0.49(4)
0.34
0.46
0.50
-0.04(5) --0.12(6)
-0.30(2) --0.37(6)
-0.44(2) --0.45(5)
-0.12
-0.28
-0.38
-
-1.38(1) 1.46(4)
-1.81(1) - 1.79(4)
-1.89(2) - 1.91(4)
- 1.28
- 1.51
- 1.58
p = 0.25
p = 0.5
p = 0.75
MFT
0.25
p = 0.5
0.33( 1)
0.49( 1)
0.44(7)
0.29
0.46
0.54
-0.2(3)
-0.34(2)
-0.69(8)
-0.09
-0.28
-0.54
-1.5(1)
-1.8(1)
-1.9(I)
- 1.29
- 1.51
- 1.63
to 5 × 10 -6. This a c c u r a c y is usually not reached in 107 runs, because the small correction, A2 ~ d~(L - 2 ) , is o v e r s h a d o w e d by fluctuations in the d o m i n a n t t e r m around its average (2oo(v~)). To suppress the fluctuations in 2oo(ff), it is necessary to subtract f r o m 2L(ff) the quantity 2 ~ ( f f ) , w h i c h is evaluated in e v e r y configuration, {v~(r)}, f r o m its definition in Eq. ( 2 3 ) and w h i c h assumes only a discrete set o f values, proportional to the n u m b e r o f a-bonds in the system. So, w e calculate (A,~(~,))
= (,ZL(,~) - ,loo(~')>.
(26)
The results are displayed in Table 3 and c o m p a r e d with the mean-field predictions in Eq. (25); the reduced quantity A2* = A 2(L + 1)2/n2 has b e e n used to eliminate explicit d e p e n d e n c e on the system size L. The error bars h a v e b e e n calculated supposing that
103
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107 45
i
I
40 35
30 25 20 15 10 5 0 -0.06
"-_::I i
-0.04
-0.02
0
A~
0.02
0.04
0.06
Fig. 2. Probability distribution P(A2) for A2 = 2L(ff) - 2o~(~) (see Eq. (23)) in an open system with L = 100, and N = p(L+ !) a-bonds with Wa = 0.1 and Wb= ½. The mean (32) _~ -1.43 x 10-4. the distributions are normal. They are usually large, even averaging over 107 random configurations does not substantially reduce the error bars, because the ratio between width and mean is as large as 103 in some cases. In Fig. 2 a typical distribution with a large relative width is shown. The agreement with mean-field theory to d)(1/L2)-terms included is good for w = 0.4 over the whole range of densities p and system sizes L. Significant negative discrepancies in O(1/L2)-terms between mean-field theory and simulations can be observed for small hopping probabilities w. The conclusion is that the predictions of mean-field theory have the right sign and correct order of magnitude, but are definitely not exact. On the other hand, the correction A2 is in general much too small, to be observed in standard simulations and 2 ~ is an excellent estimate for (2L(ff)) for all system sizes with L ~>50, for all densities p of a-bonds and for all transition probabilites w.
7. Conclusions
( I ) In the present paper we have studied the chaotic dynamical quantities of a one-dimensional hopping model with bond disorder and PBC or ABC, using analytical methods and numerical simulations. The explicit results refer to a random mixture of aand b-bonds. For such a mixture and PBC an exact result for the Lyapunov exponent in a fixed configuration is obtained which not only depends on the number N of a-bonds, but also on the number k of a-clusters. As a consequence, Eqs. (13) and (14) imply that the Lyapunov exponent in an ensemble of quenched random configurations, each containing exactly N a-bonds, takes on only a discrete set of N possible values. The simplicity of this result is a direct consequence of the one-dimensional geometry of the model.
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
104
(2) The mean-field predictions in the ABC case for the average escape rate y : Dq 2 and Lyapunov exponent, 2 = 2o0 +22q~ with qo = tel(L+ 1), are in excellent agreement with the simulation results for systems with L >_-50. For the Lyapunov exponent the accuracy is typically 1 part in 103, which is solely given by the dominant term 2~, as 22q02 is smaller than 10-5. To actually verify the mean-field prediction to the terms 22q~ included, we have increased the number of runs in the ensemble to 107, and we have observed that the expected value for A2 = 2 - 2 0 has the correct sign and order of magnitude, but there is no quantitative agreement at small hopping rates across a-bonds (Wb = 1, and Wa~<0.1) and at high densities of a-bonds (p~>0.6). (3) The dependence of the Lyapunov exponent in a closed random-bond model on the number of clusters of pure a-bonds indicates a richer chaotic behavior than in lattice Lorentz gases where the Lyapunov exponent is independent of the particular configuration. This is a consequence of the existence of waiting probabilities. (4) The relation between the escape-rate and the diffusion coefficient (24) has also been derived for lattice Lorentz gases [12] and for continuous Lorentz gas with a periodic array of scatterers [1]. This relation expresses the connection between the chaotic dynamical quantities and the transport in simple systems. (5) The extremely broad distribution for the Lyapunov exponent in open systems, also found in lattice Lorentz gases, implies important technical problems in performing computer simulations specially when extensions to two-dimensional systems are considered. The same problems arise in numerical studies of two-dimensional Lorentz lattice gases [19] An analysis of the moments of these distributions as a function of the system parameters {p, Wa,Wb} needs to be performed as well as a study of the configurations leading to the extreme values of the Lyapunov exponent and to a dynamical phase diagram, describing the different types of localization transitions [21]. Work along this line is currently in progress.
Acknowledgements L.A. gratefully acknowledges support from the Ministerio de Educaci6n y Ciencia (Spain) and the hospitality of the University of Utrecht during his visit in the fall of 1994 where this work was started.
Appendix A. Alternative expression for Lyapunov exponents In this appendix we briefly outline the derivation of Eq. (12) for the Lyapunov exponent starting from the thermodynamic formalism. By combining Eqs. (9) and (10) we obtain for the Lyapunov exponent in a fixed configuration k of random bonds 2(v~) = -
lira
1 ~
t---,c~ t
P(~2,tlro)lnP(f2,tlro ) Ef~ P(f2,tlro)
(A.1)
L. Acedo, M.H. ErnstlPhysicaA 247 (1997) 91-107
105
Before analyzing this expression we consider the eigenvalues An and the eigenvectors n = O, 1 . . . . . L - 1, of the symmetric transition matrix:
u.(r),
W(rlr')un(r') = Anun(r),
(A.2)
rt
and denote its largest nondegenerate eigenvalue by A0. For a closed system Ao = 1 and for an open system Ao < 1 on account of Eq. (5). We also define the scalar product of the eigenfunctions a and b as
(alb) = Z
a(r)b(r),
(A.3)
r
and take the eigenvectors un(r) as normalized according to this definition. The spectral decomposition of the matrix W has the form
W(rt/) = Z
un(r)Anun(r').
(A.4)
n
Next, we denote the denominator in (A.1) by ~ ( t ) and insert Eqs. (8) and (A.4), which yields the result
~(t) = ~
Wt(rlro) = y~ Un(r)AtnUn(ro) ,
r
(A.5)
r,n
where W t is the t-th power of the matrix W. At large t Eq. (A.5) approaches ~ ( t ) "" A~ (llu0) uo(ro),
(A.6)
where (l[uo) = )--~ruo(r). Next we consider the numerator of (A.1), denoted by X ( t ) and combine it with Eqs. (8) and (A.4) to find t
JV(t)=Z
Z
z=l
Wt-~(r"lr') [W(r'lr)lnW(/[r)] W*-'(r[r°)
r, r t , r tt
t
= Z Z At~-~A*~I (l[Un) (Un[Wln Wlum)um(ro). 3=1
(A.7)
n,m
By observing that the dominant long-time contribution in these sums come from An = Am = A0, the above expression reduces in the limit t ~ oo to tiff(t) '~ (t -- 1)A~-1 (lluo) (uolWln Wluo) uo(ro).
(A.8)
Substitution of Eqs. (A.6) and (A.8) in Eq. (A.1) yields an expression [11] for the Lyapunov exponent 20b) =
1
Ao (uo[WlnW[uo),
(A.9)
in terms of the largest eigenvalue Ao and corresponding eigenvector uo(r) of the transition matrix W. This formula is valid for open and closed systems, and applies to any quenched configuration ff of impurity bonds. In a closed system Ao = 1, and
L. Acedo, M.H. ErnstlPhysica A 247 (1997) 91-107
106
uo(r) = 1/v/L equals the stationary solution P0 o f the CK-equation (1), apart from a normalization constant. Then we have for a closed system: 2(~) =
-~1 ~
W(rlr')ln W(rlr' ) .
(A.10)
r, r /
The formulae (A.9) and (A.10) for open and closed systems can be directly generalized to higher-dimensional systems.
Appendix B. Recursion relations for the determinant of Wp In the ABC case the configuration ~ = {~(r)[r = 0,1,2 . . . . . L} contains (L + 1) independent random variables; in the PBC case ~ = { ~ ( r ) ] r = 1,2,... ,L} contains L such variables as ~ ( 0 ) = ~(L). The random matrix, defined below Eq. (8), has the form
WlJ(r[r') =
{
br-1
(r' = r - 1),
ar br
(~' = O , (r' = r + 1),
(B.1)
where r = 1,2 . . . . . L and b~ = ( ~ ( r ) ) ~ , ar = (1 -- if(r) - ff(r -- 1)) fl .
(B.2)
Moreover, the determinant, detlW~[, in the ABC and PBC case is denoted by DL (0, 1,2 . . . . . L) and EL(I,2 .... ,L) respectively, where the arguments refer to the set o f random variables ft. To derive the recursion relation we write EL explicitly
bl bl a2 b2 b2 a3 b3
al
EL(1,2,...,L) :
bL
b3 a4 b4
bL
(B.3) az-1 bL-1 bL-1 aL
and DL(1,2,... ,L) follows from (B.3) by setting bz = 0. We calculate DL by developing Eq. (B.3) (with bL = 0) with respect to the last column. This yields at once the recursion relation for the ABC ease:
Dr(O, 1,2 . . . . . L) = aLDL--I(O, 1,2 . . . . . L - 1) -- b2_IDL_2(0, 1,2 .... ,L - 2)
(B.4) with initial conditions DI(1) = al,
Do = 1.
(B.5)
L. Acedo, M.H. Ernstl Physica A 247 (1997) 91-107
107
The determinant EL is slightly more complicated. Developing EL with respect to the last column yields EL(l,2 ..... L) = a~DL_I(L, 1,2 ..... L - 1) - bL-1ML-1,L + (--1)L-IbLMI,L. (B.6) The two minors are developed with respect to the last row and yield ML-1,L = bL-IDL-2(L, 1,2 .... ,L - 2) + (-1)LbLb~b2 . . . b~-2, M1,L = blb2 " " " bL-1 + (-1)LbLDI.-2(2 . . . . . L -
1).
(B.7)
Combination of (B.6), (B.7) and (B.4) yields the relation EL(l, 2 ..... L) = DL(L, 1,2,... ,L) - bEDL_2(1,2 ..... L - l) - 2 ( - 1 )Lblb2... bL.
(B.8) The recursion relation (B.4) provides a very efficient numerical algorithm for calculating the determinants DL and EL for a given set of quenched variables. The secular determinants, det] WI~- A] = 0, which is a polynomial in A, of degree L, can be calculated by replacing ar by a r - A. The largest zero AL(fl) of this polynomial is calculated numerically, and provides all necessary information required in the thermodynamic formalism.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
P. Gaspard, G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693. H.A. Posch, W.G. Hoover, Phys. Rev. Lett. A 123 (1987) 227; Phys. Rev. A 39 (1989) 2175. D.J. Evans, E.G.D. Cohen, G.P. Mordss, Phys. Rev. A 42 (1990) 5990. N.I. Chernov, G.L. Eyink, J.L. Lebowitz, Ya. G. Sinai, Commun. Math. Phys. 154 (1993) 569. G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74 (1995); 2694; J. Stat. Phys. 80 (1995) 931. P. Gaspard, F. Baras, Phys. Rev. E 51 (1995) 5332. H. van Beijeren, J. R. Dorfman, Phys. Rev. Lett. 74 (1995) 4412. J.R. Dorfman, P. Gaspard, Phys. Rev. E 51 (1995) 28. P. Gaspard, J.R. Dorfman, Phys. Rev. E 52 (1995) 3525. J.R. Dorfinan, M.H. Ernst, D. Jacobs, J. Star. Phys. 81 (1995) 497. M.H. Ernst, J.R. Dorfman, Chaos and Diffusion in Periodic Random Walks, (unpublished). M.H. Ernst, J.R. Dorfman, R. Nix, D. Jacobs, Phys. Rev. Lett. 74 (1995) 4416. C. Appert, M.H. Ernst, chao-dyn/9705011. D. Ruelle, Thermodynamic Formalism, (Addison-Wesley, Reading, MA, 1978) C. Beck and F. Schl6gl, Thermodynamics of chaotic systems, (Cambridge University Press, Cambridge, 1993). J.W. Hans, K. Kehr, Phys. Pep. 150 (1987) 263. J.W. Haus, K. Kehr, J.W. Lyklcma, Phys. Rev. B 25 (1982) 2905. P.J.H. Denteneer, M.H. Ernst, Phys. Rev. B 29 (1984) 1755. C. Appert, private communication. C. Appert, H. van Beijeren, M.H. Ernst, J.R. Dorfman, J. Stat. Phys. 87 (1997) June issue. L. Acedo, M.H. Ernst, in preparation.