- Email: [email protected]
Dynamical localization of trajectories on a bond-disordered lattice
Physica A 262 (1999) 129–143
Dynamical localization of trajectories on a bond-disordered lattice a Departamento
b Institute
L. Acedoa , M.H. Ernst b; ∗
de Fsica, Universidad de Extremadura, E-06071 Badajoz, Spain for Theoretical Physics, Utrecht University, P.O. Box 80006, 3508 TA Utrecht, The Netherlands Received 27 April 1998
Abstract The chaotic dynamics of a random walker in a quenched environment is studied via the thermodynamic formalism of Ruelle, Sinai, and Bowen, in which chaotic properties are expressed in terms of a free energy-type function, ( ), of an inverse temperature-like parameter, . Localization phenomena in this system are elucidated both analytically and numerically. The in nite system limit of the Ruelle pressure at ¿ 1 and ¡ 1 is shown to be controlled by rare con gurations of the bond disorder, and this is related, respectively, to the extreme con gurations associated to the minimum and maximum Lyapunov exponent in nite systems. These extreme values and the corresponding con gurations are obtained numerically from Monte Carlo simulac 1999 Elsevier Science B.V. All rights reserved. tions based on the thermodynamic formalism. PACS: 05.45.+b; 05.20Dd; 05.60+W Keywords: Thermodynamic formalism; Lyapunov exponents; Nonlinear dynamics; Chaos theory
Dedicated to Bob Dorfman on the occasion of his 60th birthday
1. Introduction In the present decade a great interest has developed around the problem of the connection between macroscopic transport properties, such as the di usion coecient, and chaos properties of the microscopic dynamics. Enormous research has been done in order to elucidate such connection both for nonequilibrium elds [1–5,20,21] as well as for Lorentz gases [6–13]. The dynamical quantities of interest in our case are the Lyapunov exponents, dynamical entropies, and escape rates. ∗
Corresponding author. Tel.: +30-2533060=2284; fax: +30-253-1137; e-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 8 5 - 9
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A detailed and extensive analysis of this dynamics has only been performed for simple models of nonequilibrium statistical mechanics: Lorentz gases in continuous space [1–7,20,21] or on lattices [8–13], and bond-disordered lattices [19]. In these basic models, a set of independent particles move di usively in a random static environment, with a quenched con guration of the disorder, and the macroscopic transport and dynamical properties are de ned as an average over all trajectories the particle can describe in a given quenched con guration, followed by an average over all these con gurations of the disorder. As these systems are governed by a discrete dynamics, a trajectory between the initial instant and time t consists of a set of t successive states
= {x1 ; x2 ; : : : ; xt }. Every trajectory is a point of a dynamical phase space. A probability distribution P( ; t|x0 ) is de ned over this space, as well as, a dynamic partition P function, Z( ) = P( ; t|x0 ) , where is an inverse temperature-like parameter. The topological or Ruelle pressure, ( ), is then de ned in a way similar to that of the free energy per particle in an equilibrium canonical ensemble with t playing the role of the volume. The chaotic properties of physical interest are related with this function and its rst derivative at = 1 [15,16]. This formalism has been used in analytical and numerical studies of the chaotic quantities. In the case of systems with absorbing boundaries, the relation = Dq02 between the escape rate, , and the di usion coecient, D, of the system have been found. Here, q0 ∼ 1=L, is the wave number of the slowest decaying mode in a system of length L. In a similar way, the Lyapunov exponent admits a perturbative expansion = 0 + 1 q02 + · · ·, 0 being the Lyapunov exponent of an in nite system. Both 0 and 1 depend upon some average properties of the disorder. The behaviour of the Ruelle pressure at values di erent from one is also of interest because this allows us to scan the structure of the probability distribution of trajectories, as the large and small values enhance the most probable and more unlikely trajectories, respectively. A detailed analysis of this kind for a lattice Lorentz gas has been recently carried out [13,14]. These authors have rigorously proved that the Ruelle pressure becomes independent of the density of scatterers for di erent from unity in the thermodynamic limit, L → ∞. The origin of this remarkable phenomenon is the localization of trajectories in rare con gurations of the disorder. It was found, both numerically and analytically, that the trajectories are localized on the “most chaotic” (highest density of scatterers) region, and the “most deterministic” (lowest density of scatterers) regions for ¡ 1 and ¿ 1, respectively. Our aim in this paper is to discuss the dependence of the dynamical partition function of a one-dimensional random bond model (RBM) [17,18,22], so as to complete the analysis of the dynamical behaviour of this model initiated with the study of the chaotic quantities presented in Ref. [19]. The extension of the previous studies on the typically used one-dimensional lattice Lorentz gas to another basic model of nonequilibrium statistical mechanics will provide us with a more profound insight on the structure of the dynamical phase space and its relation with the nature of the model. Moreover, the bond model presents a more interesting and richer chaotic dynamics than the lattice Lorentz gas. In a Lorentz gas the particle moves freely between the
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scatterers where the trajectories diverge, while in a bond model this divergence takes place at every site as the particles have three di erent alternatives: jumping to the right adjacent site, jumping to the left one, or remain at the same site, according to a set of probabilities associated to the corresponding bonds. This peculiarity of the bond model is the origin of the Lyapunov exponent dependence, not only with the density but also with the number of clusters, as was proved for a closed model with a mixture of two species of bonds in Ref. [19]. This is also manifested in the existence of three chaotic dynamical phases as it will be shown in Section 3. The structure of the paper is as follows: in Section 2 the model is de ned and a brief summary of the thermodynamic formalism is given. In Section 3 the dynamical phase diagram of a bond model with a mixture of two species of bonds, a- and b-bonds, is discussed. It will be shown that the trajectories localize on rare con gurations, as it occurs in the lattice Lorentz gas [13,14], depending on 6= 1 and the hopping probabilities wa ; wb , so the Ruelle pressure carries very little information on the global structure of the static bond disorder. The rare con gurations corresponding to the extreme values of the chaotic properties are derived in Section 4 from a numerical Monte Carlo method. The Lyapunov exponent will be used as an order parameter which may be tuned in order to observe a segregation of the chaotic phases predicted by the analysis of Section 3. The paper is ended with a set of conclusions in Section 5 and with an appendix where a numerical method for the calculation of the Ruelle pressure, based on the thermodynamic formalism, is presented.
2. Random bond model and the thermodynamic formalism As many basic models of nonequilibrium statistical mechanics, the random bond model (RBM) is de ned on a regular lattice with a quenched disorder and a particle that moves between its sites at integer times t =0; 1; 2; : : : . In our case the lattice is one dimensional with unit lattice distance. The sites are labelled with r ∈ L ≡ {1; 2; : : : ; L} and every bond (r; r + 1) is characterized by a hopping rate w(r) ˆ ∈ (0; 12 ], drawn from a site-independent probability distribution (w). The random walker at site r will jump to the right adjacent site with probability w(r), ˆ to the left one with probability w(r ˆ − 1), or will remain at the same site r with probability 1 − w(r ˆ − 1) − w(r). ˆ The time evolution of the probability p(r; t) of nding the random walker at site r at time t is then described by the Chapman–Kolmogorov or CK-equation: X W (r|r 0 )p(r 0 ; t) ; (1) p(r; t + 1) = r0
with r 0 ∈ L. This equality de nes the random symmetric transition matrix W (r|r 0 ), given by ˆ − 1)r; r 0 +1 + w(r) ˆ ˆ − w(r ˆ − 1))r; r 0 ; W (r|r 0 ) = w(r r; r 0 −1 + (1 − w(r)
(2)
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where wˆ = {w(r) ˆ | r = 0; 1; : : : ; L} is a quenched con guration of bonds. To x ideas, we consider a bimodal distribution, (w) = (w − wa ) + (1 − )(w − wb ) ;
(3)
where wa is the hopping rate associated to the a-bond type and wb is the one assigned to the b-bond type. We usually assume that wb = 12 and wa 6 12 . In this case we refer to the b-bonds as normal and the a-bonds as impurities. A quenched con guration is better expressed in the form ˆ + wb (1 − n(r)); ˆ w(r) ˆ = wa n(r)
r = 0; 1; 2; : : : ; L ;
(4)
where n(r) ˆ is a random variable which takes the value 1 with probability , and the value 0 with probability 1−. So, is the fraction of a-bonds in the system. Depending on the boundary conditions, absorbing (ABC) or periodic (PBC), the transition matrix satis es one of the following conditions: X
W (r|r 0 ) =
r
X
X
W (r 0 |r) = 1
(PBC) ;
W (r 0 |r)61
(ABC) :
r
W (r|r 0 ) =
X
r
(5)
r
In this paper we will use mainly the open system case (ABC) with L + 1 bonds {(0; 1); (1; 2); : : : ; (L; L + 1)}, L sites {1; 2; : : : ; L} and two absorbing borders {0; L + 1}. Every nonescaping trajectory of t time steps is represented by a sequence of t + 1 positions (t) = {r0 ; : : : ; rt }, corresponding to a particle which starts at site r0 at t = 0 and arrives at site rt at time t. We can think of (t) as a point of a dynamical phase space. The probability for a given trajectory, (t), can be expressed in terms of the transition probabilities W (r|r 0 ), P( ; t|r0 ) =
t Y
W (rn |rn−1 ) :
(6)
n=1
In parallel with the standard procedure in the canonical ensemble, a partition function is de ned as X X [P( ; t|r0 )] = (W t )(r|r0 ) : (7) Z L ( ; t|r0 ) =
r
Here, the generalized transition matrix whose elements are W (r|r0 )=W (r|r0 ) has been used. The inverse temperature-like parameter provides a simple way of scanning the structure of the probability distribution P( ; t|r0 ) via a smooth partition function [14]. This is the starting point of the so-called thermodynamic formalism [15,16]. For large times, the partition function becomes independent of the point of origin r0 (ergodicity) and it is completely determined by the largest positive eigenvalue, L ( ), of the matrix W , which is assumed to be nondegenerate.
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The Ruelle or topological pressure is a free-energy-like function de ned in terms of the partition function in the form ˆ L ( |w)
= lim
L→∞
1 ln Z L ( ; t|r0 ) = ln L ( ) ; t
(8)
which depends on the quenched con guration wˆ of random bonds. This function contains all the necessary information to derive the chaotic quantities of interest, such as the Lyapunov exponent, , the escape rate, , and the Kolmogorov–Sinai entropy per unit time, hKS = − , through the relations ˆ =−
L (w)
ˆ L (1|w)
;
ˆ =− L (w)
0 ˆ L (1|w)
;
(9)
where the prime denotes a -derivative. The Eqs. (8) and (9), together with the numerical algorithm for the calculation of the largest eigenvalue of the large sparse matrix W , supply us with a very ecient method in order to obtain these chaotic properties. We must also notice that, according to Eq. (9), the Ruelle pressure behaviour at = 1 gives us the global chaotic properties of any quenched con guration. The quantities of macroscopic interest are obtained as con gurational averages represented by h· · ·i, so L ( )
= hln L ( )i ;
(10)
L = −h
ˆ L (1|w)i
;
(11)
L = −h
0 ˆ L (1|w)i
:
(12)
A numerical evaluation of the last two quantities, L and L , in nite open and closed systems with L of order 100 has been already compared with mean eld theory estimations both for the lattice Lorentz gas [12] and the bond model [19] in one dimension.
3. Localization and the dynamical phase diagram In this section we discuss the limiting behaviour for large L of the Ruelle pressure, ( ), in a bond model with a random mixture of a- and b-bonds at a xed density . We consider a b-bond hopping rate, wb = 12 , and di erent values for the a-bond hopping rate, wa 6 12 , and the fraction of bonds occupied by a-bonds, = N=(L + 1). It has been recently established that, in the thermodynamic limit, the Ruelle pressure of random walkers on disordered lattices becomes independent of the density at 6= 1, and it is completely controlled by a rare con guration of the disorder. So, it does not carry any information on the average structure of this disorder. This has been rigorously proved in the case of a Lorentz lattice gas with identical point-like scatterers [14]. The rare con gurations which control the Ruelle pressure on a lattice Lorentz gas are those
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with a large void for ¿ 1 or a large cluster of scatterers for ¡ 1, which are the “most deterministic” and “most chaotic”, respectively. This phenomenon is also present in the random bond model, and probably in any di usion model with static disorder, but with a richer and more complex dynamical phase diagram. In other words, the trajectories may localize in three di erent kinds of clusters: sequences of a-bonds (aa-clusters), sequences of b-bonds (bb-clusters) or sequences of pairs formed by an a-bond and a b-bond (ab-clusters). We will follow the arguments in Ref. [14] in order to sketch the corresponding derivation of these results for the random bond model. For a closed system it follows from the de nition of the generalized transition matrix W (r 0 |r) and Eq. (2) that X W (r 0 |r) = [w(r)] ˆ + [w(r ˆ − 1)] + [1 − w(r) ˆ − w(r ˆ − 1)] ; (13) r0
where w(r ˆ − 1) and w(r) ˆ are the hopping rates of the bond whose right end is site r, and the bond whose left end is site r, respectively. In our case, Eq. (13) can also be written in the following form: X W (r 0 |r) = Waa ( )n(r ˆ − 1)n(r) ˆ r0
ˆ − 1))(1 − n(r)) ˆ +Wbb ( )(1 − n(r ˆ − 1)(1 − n(r)) ˆ + n(r)(1 ˆ − n(r ˆ − 1))] ; +Wab ( )[n(r
(14)
with Waa ( ) = 2wa + (1 − 2wa ) ;
(15)
Wbb ( ) = 2wb + (1 − 2wb ) ;
(16)
Wab ( ) = wa + wb + (1 − wa − wb ) :
(17)
For open systems the equality sign in Eq. (15) is replaced by a ‘less than’ sign in case r is a site adjacent to the borders, that is r = {1; L}. So, in general, we can write X
W (r 0 |r)6Waa ( )
if (r − 1; r) and (r; r + 1) are a-bonds ;
W (r 0 |r)6Wbb ( )
if (r − 1; r) and (r; r + 1) are b-bonds ;
W (r 0 |r)6Wab ( )
if (r − 1; r) is an a-bond and
r0
X r0
X r0
(r; r + 1) is a b-bond; or vice versa :
(18)
We now de ne W ( ) = max{Waa ( ); Wbb ( ); Wab ( )} as the maximum of the three functions given in Eqs. (15) for a given set of parameter values { ; wa ; wb }. According to this de nition and Eq. (18) it follows that the sum of all terms of any column of
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the matrix W satis es the inequality with Eqs. (6) and (7) yields
P r0
135
W (r 0 |r)6W ( ). This inequality combined
Z L ( ; t|x0 )6W ( )Z L ( ; t − 1|x0 )6(W ( ))t :
(19)
Then, an upper bound of the Ruelle pressure (8) is given by ˆ W ( ) L ( |w)6ln
:
(20)
A lower bound to Z L ( ; t|x0 ) is constructed by considering the largest cluster of a-bonds, b-bonds or that in which the a- and b-bonds appear alternatively, if W ( ) = Waa ( ); W ( ) = Wbb ( ) or W ( ) = Wab ( ), respectively. We suppose that this cluster has length M , so it contains exactly M bonds. A partition function ZM ( ; t|x0 ) corresponding to the subset of trajectories that remain con ned for t time steps to this cluster is de ned. As all terms in the sum (7) are nonnegative, any sum over a subset of trajectories will give an exact lower bound for Z L and, consequently, we can write Z L ( ; t|x0 )¿ZM ( ; t|x0 ) :
(21)
This partition function ZM ( ; t|x0 ) is that of an open system of length M in a rare con guration {a; a; : : : ; a}; {b; b; : : : ; b} or, {a; b; a; b; : : : ; a; b}. The Ruelle pressure of such a system is then given by the logarithm of the largest eigenvalue of the matrix W [19]: M ( ) = W ( ) − ( )q02 + O(q03 ) ;
(22)
where q0 = =M is the wave number of the slowest decaying mode in the largest dynamics dominant cluster. By using the mean eld theory for the dynamical properties of Ref. [19], it can be easily found that the function ( ) is approximately given by pq ( ) = (wp + wq )=2, with p; q = {a; b}. Starting from Eqs. (8), (21) and (22) the following lower bound of the Ruelle pressure follows: 2 : (23) L ( )¿hln M ( )i = ln W ( ) − ( ) M2 It can be shown that the average h1=M 2 i tends to zero in the thermodynamic limit, L → ∞, so the upper and the lower bound of L ( ) coincide. This means that, for a given set of parameter values { 6= 1; wa ; wb } the con gurational average of the Ruelle pressure is given by L ( )
= ln W ( );
L→∞;
(24)
with W ( ) ≡ Wpq ( ) = max{Waa ( ); Wbb ( ); Wab ( )}. Consequently, ( ) = limL→∞ L ( ) is independent of the proportion between a- and b-bonds. Nevertheless, this result is only approximately valid for very large systems as the moment h1=M 2 i tends to zero very slowly with the system length. It has been shown that h1=M 2 i ∼ (ln L)− , where is a positive exponent dependent on the model [14]. The scheme of ( ) values in the parameter space { ; wa ; wb } is called dynamical phase diagram. For the sake of simplicity, we take wb = 12 , so this diagram is reduced to a plane. In Fig. 1 the physical region ¿ 0; 06wa 6 12 of this plane is represented and
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Fig. 1. Dynamical phase diagram in a bond model with wb = 12 . The Ruelle pressure is the logarithm of the largest value W ( )=max{Waa ( ); Wbb ( ); Wab ( )} in every region separated from the others by coexistence lines.
divided in six di erent portions according to the numerical order of the three quantities Waa ( ); Wbb ( ), and Wab ( ) in Eq. (15). The Ruelle pressure in every region of the -wa plane is given by the logarithm of the largest of these functions (24). These regions are separated by three coexistence curves: Waa ( ) = Wbb ( ) = Wab ( ) ( = 1); Wab ( ) = Waa ( ) (wa = 14 ) and Waa ( ) = Wbb ( ). The last one intersects the = 1 line at wa = wac = 0:113(1), which plays the role of a triple point, and approaches asymptotically to wa = 14 following the law = ln 2=(1 − 4wa ). The Ruelle pressure is a continuous function at these coexistence curves but its rst derivative has a nite jump. This means that L ( ) becomes nonanalytic in the thermodynamic limit.
4. Numerical results on rare conÿgurations In the last section we have concluded that in a bond model at 6= 1, the trajectories localize on rare con gurations of the disorder which depends upon the set of parameter values { ; wa ; wb }. Here, we intend to perform a numerical analysis of this phenomenon by means of the chaotic dynamical quantities distribution over the con gurational space. In particular, we are interested in the extreme values of these quantities and the corresponding con gurations, especially that of the Lyapunov exponent. The reasons for this choice are clear. The Ruelle pressure, L ( ), is an analytic function of { ; wa ; wb } in any nite system but it becomes nonanalytic in the thermodynamic limit, L → ∞, and its rst derivative changes abruptly at the coexistence curves of the dynamical phase diagram (Fig. 1). For example at the curve = 1, this phenomenon occurs as a consequence of the localization of trajectories on the “most chaotic” and the “most deterministic” clusters at the ¡ 1 and ¿ 1 branches of the diagram, respectively. From this result and Eq. (9), we can argue that the maximum, max (L), and minimum, min (L), Lyapunov exponent of a nite system relate with the left, 0 ( − ), and right
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derivative, 0 ( + ), of the Ruelle pressure at = 1 in the thermodynamic limit through the following expressions: 1 0 ; (25) ( − ) = −max (L) + O ln L
0
( + ) = −min (L) + O
1 ln L
:
(26)
These equations are the base to the numerical inspection of the neighbourhood of = 1 in the dynamical phase diagram. According to them, the extreme values of the Lyapunov exponent in an open bond model with length L and a xed set of parameters {N; wa ; wb } are given by the derivatives of the Ruelle pressure of an in nite system at ± =1±, with 1, save logarithmical corrections in the system length. The presence of these corrections was already established in Section 3. Nevertheless, we can expect that the localization phenomena may already be visible in the small systems attainable in simulations (L ' 100), at least in an incipient way. In the following subsections it will be shown that this assumption is true. 4.1. The Monte Carlo method The chaotic properties depend strongly on the bond con gurations, wˆ = {w(r) ˆ |r = 0; : : : ; L}, in the case of the open system (ABC), so the Lyapunov exponent and the escape rate assumes a number of order 2L of di erent values in a relatively broad interval. The situation is very di erent in the closed system (PBC). Here, the escape rate is null, = 0, and the Lyapunov exponent depends only on the number of clusters of a-bonds, or clusters of b-bonds (these numbers coincide). So, if the number of a-bonds is Na 6Nb , the Lyapunov exponent takes only Na di erent values over the con gurational space [19]. This degeneracy precludes the use of the closed system in a numerical analysis of the dynamical phase transition. In the rest of the paper we consider absorbing boundary conditions (ABC). For a given quenched con guration, the Ruelle pressure, and the chaotic dynamical quantities linked with it through the relations in Eq. (9), can be calculated from a standard method based on the numerical evaluation of the largest eigenvalue of the random sparse matrix W (see appendix). This method has already been used in the calculation of the con gurational average of the chaotic properties [12,19], but we are now interested in the extreme values of these quantities. The Monte Carlo method we have used with these purpose consists of three steps performed successively in a loop: (i) A con guration wˆ = {w(r) ˆ | r = 0; 1; : : : ; L}, with a given N number of a-bonds, is chosen at random and the dynamical quantity of interest is calculated numerically by the method discussed in the appendix. (ii) The positions of an a-bond and a b-bond selected at random are interchanged and the mentioned dynamical quantity is calculated again.
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L. Acedo, M.H. Ernst / Physica A 262 (1999) 129–143 Table 1 The average, h (w)i, ˆ minimum, min , and maximum, max , escape rate values in a system of length L = 99 with hopping rates wa = 0:1; 0:2; 0:3; 0:4; wb = 12 and a density =N=(L+1) of a-bonds. These values have been scaled with the factor 104 = 0:3
= 0:7
wa
h (w)i ˆ
min
max
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
2.2431 3.4033 4.1123 4.5905 1.2986 2.4072 3.3646 4.1998
1.5144 2.6627 3.5963 4.3424 1.033 2.0453 3.0265 3.9849
3.8735 4.5728 4.7596 4.8729 1.6823 2.8966 3.7543 4.4208
(iii) If the dynamical quantity is increased (or decreased if we are seeking the smallest values), we take the new con guration as the starting point of the algorithm, otherwise the previous one is reset. This method provides a fast convergence to the extreme regions of con guration space. 4.2. Extreme values of the dynamical properties. Rare conÿgurations As a test of the method explained in the last section we have applied it to the numerical calculation of the extreme values of the escape rate, and the con gurations associated to them, in an open system with length L = 99. The results for two typical a-bond densities and hopping rates, = 0:3; 0:7; wa = 0:1; 0:2; 0:3, and 0.4 are displayed in Table 1. The corresponding con gurations are best detailed in terms of the random variable n(r) ˆ de ned by Eq. (4). As these con gurations are very regular, it is convenient to abridge them by grouping the patterns {a; b; c; : : : ; a; b; c; : : :} which repeats n times by n(a; b; c; : : :). The following results are found for the rare con guration corresponding to the largest escape rates: = 0:3
{35(0); 30(1); 35(0)} ;
= 0:7
{15(0); 70(1); 15(0)} :
(27)
These results are valid for arbitrary values of the hopping rates in the interval (0; 12 ) whenever the condition wa ¡ wb is satis ed. Similarly, the largest escape rates are attained at the con gurations = 0:3
{15(1); 70(0); 15(1)} ;
= 0:7
{35(1); 30(0); 35(1)} :
(28)
The generalization of these results to any system of length L with N a-bonds is obvious. The largest (smallest) escape rate is attained at a con guration with only one aa-cluster
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139
Table 2 The maximum Lyapunov exponent and the corresponding rare con guration in the random bond model with L = 99 and wb = 12 at three di erent a-bond densities, , and for several a-bond hopping rates, wa . In the last column the theoretical size independent prediction of Eq. (25) is shown wa
Con guration
max (L)
max (L → ∞)
{25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {20(0); 20(1); 10(0; 1); 20(1); 20(0)} {16(0); 16(1); 18(0; 1); 16(1); 16(0)} {15(0); 14(1); 22(0; 1); 14(1); 13(0)} {7(0); 13(1); 24(0; 1); 13(1); 19(0)}
0.9184 1.0244 1.0781 1.0958 1.0796 1.0250 0.9803 0.8985 0.7877
0.9489 1.0549 1.0961 1.0889 1.0397 1.0296 0.9986 0.9433 0.8557
{32(0); 3(0; 1); 23(1); 4(0; 1); 31(0)} {28(0); 7(0; 1); 17(1); 6(0; 1); 29(0)} {26(0); 6(1); 18(0; 1); 6(1); 26(0)} {26(0); 5(1); 20(0; 1); 5(1); 24(0)}
0.9242 0.9914 0.9778 0.8958
1.0549 1.0889 1.0296 0.9433
{15(0); 70(1); 15(0)} {15(0); 70(1); 15(0)} {15(0); 70(1); 15(0)} {21(1); 2(0); 28(0; 1); 21(1)}
1.0666 1.1290 1.0300 0.8512
1.0549 1.0889 1.0296 0.9433
= 0:5 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 = 0:3 0.4 0.3 0.2 0.1 = 0:7 0.4 0.3 0.2 0.1
(bb-cluster) whose central site coincides with the middle of the system: nˆ = {(L − N )=2(0); N (1); (L − N )=2(0)}
for the maximum escape rate ;
(29)
nˆ = {(L − N )=2(1); N (0); (L − N )=2(1)} for the minimum escape rate :
(30)
As the escape rate depends only on the value of the Ruelle pressure strictly at =1 (9), these con gurations have no relation with the localization phenomena discussed in Section 3. Its interpretation is simple: the trajectories are trapped on those con gurations with clusters of a-bonds adjacent to the absorbing boundaries as we suppose that the hopping rate associated to them is smaller than the one corresponding to b-bonds. The aa-clusters adjacent to the boundaries are identical in order to provide the most e ective trapping of the moving particle. The con gurations corresponding to the largest escape rate can be given similar interpretation. The maximum Lyapunov exponent values, max (L), and the rare con gurations associated to them, calculated by the same numerical method as those of the escape rate, are shown in Table 2. The same results for the minimum Lyapunov exponent are displayed in Table 3. In the last column of these tables the prediction given by Eq. (25) is shown as a comparison. The relative di erence between the numerical value and the theoretical estimation based on the in nite system limit of the Ruelle pressure
140
L. Acedo, M.H. Ernst / Physica A 262 (1999) 129–143 Table 3 The minimum Lyapunov exponent and the corresponding rare con guration in the random bond model with L = 99 and wb = 12 at three di erent a-bond densities, , and for several a-bond hopping rates wa . In the last column the theoretical size independent prediction of Eq. (25) is shown wa
Con guration
min (L)
min (L → ∞)
{25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {2(1); 4(0; 1); 20(1); 42(0); 20(1); 4(0; 1)} {2(1); 8(0; 1); 16(1); 34(0); 16(1); 8(0; 1)} {2(1); 11(0; 1); 13(1); 28(0); 13(1); 11(0; 1)} {13(1; 0); 12(1); 25(0); 11(1); 12(0; 1); 2(1)} {22(0); 5(0; 1); 39(1); 6(0; 1); 17(0)}
0.7627 0.8037 0.8337 0.8481 0.8597 0.8376 0.7925 0.7247 0.5333
0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.3944
{11(1); 4(0; 1); 62(0); 4(0; 1); 11(1)} {1; 7(0; 1); 7(1); 55(0); 7(1); 8(0; 1)} {1; 10(0; 1); 4(1); 50(0); 4(1); 10(0; 1); 1} {34(0); 29(1); 0; 1; 35(0)}
0.7442 0.7668 0.7675 0.5738
0.6931 0.6931 0.6931 0.3944
{35(1); 30(0); 35(1)} {35(1); 30(1); 35(1)} {1; 4(0; 1); 30(1); 22(0); 30(1); 4(0; 1); 1} {3(0); 11(1; 0); 47(1); 11(0; 1); 5(0); 1}
0.8908 0.9395 0.8990 0.4645
0.6931 0.6931 1.6931 0.3944
= 0:5 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 = 0:3 0.4 0.3 0.2 0.1 = 0:7 0.4 0.3 0.2 0.1
is usually large, specially for the minimum Lyapunov exponents, but in some cases it can be less than 5%. These large di erences were expected as a consequence of the logarithmical corrections in the system length. By direct inspection of the rare con gurations and Fig. 1, we can infer that the highest Ruelle pressure phase is always at the center of the system surrounded by clusters with lower Ruelle pressure. In summary, we have (i) The con gurations associated to the smallest Lyapunov exponent ( ¿ 1 branch of the dynamical phase diagram) are characterized by a centered bb-cluster in the range of hopping probabilities wa ∈ (wac ; 12 ), and an aa-cluster in the range wa ∈ (0; wac ). (ii) The con gurations corresponding to the largest Lyapunov exponent ( ¿ 1 branch) and characterized by an aa-cluster for systems with hopping rates wa ∈ ( 14 ; 12 ), and an ab-cluster in the wa ∈ (0; 14 ) range. The dominant cluster is always in the center of the system in order to prevent the particle from escaping for as long a time as possible. That way the fractal set of nonescaping trajectories is bigger and, consequently, the sum in Eq. (7) and the Ruelle pressure is maximized. This central cluster is an incipient version of the one which controls the dynamics of the in nite system, so the numerical results con rm, within the nite system errors, the localization phenomena in the bond model discussed in Section 3.
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5. Conclusions (1) In this paper we have studied the dynamical phase space of a random bond model with static disorder by means of the thermodynamic formalism [15,16]. The recent results which indicate that localization phenomenon occurs in lattice Lorentz gases and other di usive models in the in nite system limit have been extended to the random bond model. In particular, we have found that the Ruelle pressure, ( ), becomes a non-analytic function of the temperature-like parameter and the bond hopping rates in the thermodynamic limit. (2) The case with two species of bonds, a- and b-bonds, with associated hopping rates, wa and wb , has been explicitly considered. It has been shown that the thermodynamic limit of the Ruelle pressure, ( ), is a function independent of density (the fraction of a-bonds in the system) as a consequence of the localization of trajectories in rare con guration of the disorder. These rare con gurations have three di erent forms: clusters of a-bonds (aa-clusters), clusters of b-bonds (bb-clusters) or sequences {a; b; a; b; : : : ; a; b} (ab-clusters) depending on the set of parameter values { ; wa ; wb }. This is in contrast with the simpler dynamic behaviour of the lattice Lorentz gas where the localization occurs only in two rare con gurations: the largest void for ¿ 1 and the largest cluster of scatterers for ¡ 1 [13]. (3) The dynamical phase diagram of ( ) in the -wa plane (with wb = 12 ) has been deduced. This diagram consists of six di erent regions separated by three coexistence curves. (4) The extreme values of the chaotic dynamical properties have been calculated with a Monte Carlo method based on the thermodynamic formalism. The con gurations corresponding to the largest and smallest values of the Lyapunov exponent in a system with L = 99 have been related with the ¡ 1 and ¿ 1 branches of the diagram. These con gurations are characterized by its regularity and a conspicuous central cluster corresponding to the rare con guration which dominates the dynamics for a given set of parameter values { ; wa }. In spite of the small length of the system studied, transitions at = 1 − ; wa = 14 and = 1 + ; wa = w ac (with 1), predicted theoretically, are observed. Acknowledgements L.A. gratefully acknowledges support from the Ministerio de Educacion y Ciencia (Spain) and the hospitality of the University of Utrecht during a visit in the autumn of 1994 where this work was started. Appendix A. Recursion relations for the determinant of Wÿ In the ABC case, the con guration wˆ = {w(r) ˆ | r = 0; 1; 2; : : : ; L} contains (L + 1) independent random variables: in the PBC case, wˆ = {w(r) ˆ | r = 1; 2; : : : ; L} contains
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L such variables as w(0) ˆ = w(L). ˆ The random matrix, de ned below Eq. (6), has the form W (r|r 0 ) = br−1 ;
(r 0 = r − 1) ;
W (r|r 0 ) = ar ;
(r 0 = r) ;
W (r|r 0 ) = br ;
(r 0 = r + 1) ;
(A1)
where r = 1; 2; : : : ; L and ˆ ; br = (w(r)) ˆ − w(r ˆ − 1)) : ar = (1 − w(r)
(A2)
Moreover, the determinant, det |W |, in the ABC and PBC case is denoted by D L (0; 1; 2; : : : ; L) and E L (1; 2; : : : ; L) respectively, where the arguments refer to the set of random variables w. ˆ To derive the recursion relation we write E L explicitly a1 b1 bL b a b 1 2 2 b2 a3 b3 (A3) EL (1; 2; : : : ; L) = b3 a4 b4 ··· b a L−1 L−1 bL bL−1 aL and D L (0; 1; 2; : : : ; L) follows from Eq. (A3) by setting b L = 0. We calculate D L by developing Eq. (A3) (with b L = 0) with respect to the last column. This yields at once the recursion relation for the ABC case: D L (0; 1; 2; : : : ; L) = a L DL−1 (0; 1; 2; : : : ; L − 1) − b2L−1 DL−2 (0; 1; 2; : : : ; L − 2) (A4) with initial conditions D1 (1) = a1 ;
D0 = 1 :
(A5)
The determinant E L is slightly more complicated. Developing E L with respect to the last column yields E L (1; 2; : : : ; L) = a L DL−1 (L; 1; 2; : : : ; L − 1) − bL−1 ML−1; L + (−1)L−1 b L M1; L : (A6) The two minors are developed with respect to the last row and yield ML−1; L = bL−1 DL−2 (L; 1; 2; : : : ; L − 2) + (−1)L b L b1 b2 · · · bL−2 ; M1; L = b1 b2 · · · bL−1 + (−1)L b L DL−2 (2; : : : ; L − 1) :
(A7)
Combination of Eqs. (A6), (A7) and (A4) yields the relation E L (1; 2; : : : ; L) = D L (L; 1; 2; : : : ; L) − b2L DL−2 (1; 2; : : : ; L − 1) −2(−1)L b1 b2 · · · b L :
(A8)
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The recursion relation (A4) provides a very ecient numerical algorithm for calculating the determinants D L and E L for a given set of quenched variables. The secular determinant, det |W −|=0, which is a polynomial in , of degree L, can be calculated by replacing ar by ar − . The largest zero L ( ) 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] [22]
P. Gaspard, G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693. H.A. Posch, W.G. Hoover, Phys. Rev. Lett. A 123 (1987) 227. D.J. Evans, E.G.D. Cohen, G.P. Morriss, 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. 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. Dorfman, M.H. Ernst, D. Jacobs, J. Stat. Phys. 81 (1995) 497. M.H. Ernst, J.R. Dorfman, Chaos and Di usion 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, Phys. Rev. E 56 (1997) 5106. C. Appert, H. van Beijeren, M.H. Ernst, J.R. Dorfman, J. Stat. Phys. 87 (1997) 1253. D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, MA, 1978. C. Beck, F. Schlogl, Thermodynamics of Chaotic Systems, Cambridge University Press, Cambridge, 1993. J.W. Haus, K. Kehr, Phys. Rep. 150 (1987) 263. P.J.H. Denteneer, M.H. Ernst, Phys. Rev. B 29 (1984) 1755. L. Acedo, M.H. Ernst, Physica A 247 (1997) 91. H.A. Posch, W.G. Hoover, Phys. Rev. A 39 (1989) 2175. G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80 (1995) 931–970. J.W. Haus, K.W. Kehr, J.W. Lyklema, Phys. Rev. B 25 (1982) 2905.
Dynamical localization of trajectories on a bond-disordered lattice a Departamento
b Institute
L. Acedoa , M.H. Ernst b; ∗
de Fsica, Universidad de Extremadura, E-06071 Badajoz, Spain for Theoretical Physics, Utrecht University, P.O. Box 80006, 3508 TA Utrecht, The Netherlands Received 27 April 1998
Abstract The chaotic dynamics of a random walker in a quenched environment is studied via the thermodynamic formalism of Ruelle, Sinai, and Bowen, in which chaotic properties are expressed in terms of a free energy-type function, ( ), of an inverse temperature-like parameter, . Localization phenomena in this system are elucidated both analytically and numerically. The in nite system limit of the Ruelle pressure at ¿ 1 and ¡ 1 is shown to be controlled by rare con gurations of the bond disorder, and this is related, respectively, to the extreme con gurations associated to the minimum and maximum Lyapunov exponent in nite systems. These extreme values and the corresponding con gurations are obtained numerically from Monte Carlo simulac 1999 Elsevier Science B.V. All rights reserved. tions based on the thermodynamic formalism. PACS: 05.45.+b; 05.20Dd; 05.60+W Keywords: Thermodynamic formalism; Lyapunov exponents; Nonlinear dynamics; Chaos theory
Dedicated to Bob Dorfman on the occasion of his 60th birthday
1. Introduction In the present decade a great interest has developed around the problem of the connection between macroscopic transport properties, such as the di usion coecient, and chaos properties of the microscopic dynamics. Enormous research has been done in order to elucidate such connection both for nonequilibrium elds [1–5,20,21] as well as for Lorentz gases [6–13]. The dynamical quantities of interest in our case are the Lyapunov exponents, dynamical entropies, and escape rates. ∗
Corresponding author. Tel.: +30-2533060=2284; fax: +30-253-1137; e-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 8 5 - 9
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A detailed and extensive analysis of this dynamics has only been performed for simple models of nonequilibrium statistical mechanics: Lorentz gases in continuous space [1–7,20,21] or on lattices [8–13], and bond-disordered lattices [19]. In these basic models, a set of independent particles move di usively in a random static environment, with a quenched con guration of the disorder, and the macroscopic transport and dynamical properties are de ned as an average over all trajectories the particle can describe in a given quenched con guration, followed by an average over all these con gurations of the disorder. As these systems are governed by a discrete dynamics, a trajectory between the initial instant and time t consists of a set of t successive states
= {x1 ; x2 ; : : : ; xt }. Every trajectory is a point of a dynamical phase space. A probability distribution P( ; t|x0 ) is de ned over this space, as well as, a dynamic partition P function, Z( ) = P( ; t|x0 ) , where is an inverse temperature-like parameter. The topological or Ruelle pressure, ( ), is then de ned in a way similar to that of the free energy per particle in an equilibrium canonical ensemble with t playing the role of the volume. The chaotic properties of physical interest are related with this function and its rst derivative at = 1 [15,16]. This formalism has been used in analytical and numerical studies of the chaotic quantities. In the case of systems with absorbing boundaries, the relation = Dq02 between the escape rate, , and the di usion coecient, D, of the system have been found. Here, q0 ∼ 1=L, is the wave number of the slowest decaying mode in a system of length L. In a similar way, the Lyapunov exponent admits a perturbative expansion = 0 + 1 q02 + · · ·, 0 being the Lyapunov exponent of an in nite system. Both 0 and 1 depend upon some average properties of the disorder. The behaviour of the Ruelle pressure at values di erent from one is also of interest because this allows us to scan the structure of the probability distribution of trajectories, as the large and small values enhance the most probable and more unlikely trajectories, respectively. A detailed analysis of this kind for a lattice Lorentz gas has been recently carried out [13,14]. These authors have rigorously proved that the Ruelle pressure becomes independent of the density of scatterers for di erent from unity in the thermodynamic limit, L → ∞. The origin of this remarkable phenomenon is the localization of trajectories in rare con gurations of the disorder. It was found, both numerically and analytically, that the trajectories are localized on the “most chaotic” (highest density of scatterers) region, and the “most deterministic” (lowest density of scatterers) regions for ¡ 1 and ¿ 1, respectively. Our aim in this paper is to discuss the dependence of the dynamical partition function of a one-dimensional random bond model (RBM) [17,18,22], so as to complete the analysis of the dynamical behaviour of this model initiated with the study of the chaotic quantities presented in Ref. [19]. The extension of the previous studies on the typically used one-dimensional lattice Lorentz gas to another basic model of nonequilibrium statistical mechanics will provide us with a more profound insight on the structure of the dynamical phase space and its relation with the nature of the model. Moreover, the bond model presents a more interesting and richer chaotic dynamics than the lattice Lorentz gas. In a Lorentz gas the particle moves freely between the
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131
scatterers where the trajectories diverge, while in a bond model this divergence takes place at every site as the particles have three di erent alternatives: jumping to the right adjacent site, jumping to the left one, or remain at the same site, according to a set of probabilities associated to the corresponding bonds. This peculiarity of the bond model is the origin of the Lyapunov exponent dependence, not only with the density but also with the number of clusters, as was proved for a closed model with a mixture of two species of bonds in Ref. [19]. This is also manifested in the existence of three chaotic dynamical phases as it will be shown in Section 3. The structure of the paper is as follows: in Section 2 the model is de ned and a brief summary of the thermodynamic formalism is given. In Section 3 the dynamical phase diagram of a bond model with a mixture of two species of bonds, a- and b-bonds, is discussed. It will be shown that the trajectories localize on rare con gurations, as it occurs in the lattice Lorentz gas [13,14], depending on 6= 1 and the hopping probabilities wa ; wb , so the Ruelle pressure carries very little information on the global structure of the static bond disorder. The rare con gurations corresponding to the extreme values of the chaotic properties are derived in Section 4 from a numerical Monte Carlo method. The Lyapunov exponent will be used as an order parameter which may be tuned in order to observe a segregation of the chaotic phases predicted by the analysis of Section 3. The paper is ended with a set of conclusions in Section 5 and with an appendix where a numerical method for the calculation of the Ruelle pressure, based on the thermodynamic formalism, is presented.
2. Random bond model and the thermodynamic formalism As many basic models of nonequilibrium statistical mechanics, the random bond model (RBM) is de ned on a regular lattice with a quenched disorder and a particle that moves between its sites at integer times t =0; 1; 2; : : : . In our case the lattice is one dimensional with unit lattice distance. The sites are labelled with r ∈ L ≡ {1; 2; : : : ; L} and every bond (r; r + 1) is characterized by a hopping rate w(r) ˆ ∈ (0; 12 ], drawn from a site-independent probability distribution (w). The random walker at site r will jump to the right adjacent site with probability w(r), ˆ to the left one with probability w(r ˆ − 1), or will remain at the same site r with probability 1 − w(r ˆ − 1) − w(r). ˆ The time evolution of the probability p(r; t) of nding the random walker at site r at time t is then described by the Chapman–Kolmogorov or CK-equation: X W (r|r 0 )p(r 0 ; t) ; (1) p(r; t + 1) = r0
with r 0 ∈ L. This equality de nes the random symmetric transition matrix W (r|r 0 ), given by ˆ − 1)r; r 0 +1 + w(r) ˆ ˆ − w(r ˆ − 1))r; r 0 ; W (r|r 0 ) = w(r r; r 0 −1 + (1 − w(r)
(2)
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where wˆ = {w(r) ˆ | r = 0; 1; : : : ; L} is a quenched con guration of bonds. To x ideas, we consider a bimodal distribution, (w) = (w − wa ) + (1 − )(w − wb ) ;
(3)
where wa is the hopping rate associated to the a-bond type and wb is the one assigned to the b-bond type. We usually assume that wb = 12 and wa 6 12 . In this case we refer to the b-bonds as normal and the a-bonds as impurities. A quenched con guration is better expressed in the form ˆ + wb (1 − n(r)); ˆ w(r) ˆ = wa n(r)
r = 0; 1; 2; : : : ; L ;
(4)
where n(r) ˆ is a random variable which takes the value 1 with probability , and the value 0 with probability 1−. So, is the fraction of a-bonds in the system. Depending on the boundary conditions, absorbing (ABC) or periodic (PBC), the transition matrix satis es one of the following conditions: X
W (r|r 0 ) =
r
X
X
W (r 0 |r) = 1
(PBC) ;
W (r 0 |r)61
(ABC) :
r
W (r|r 0 ) =
X
r
(5)
r
In this paper we will use mainly the open system case (ABC) with L + 1 bonds {(0; 1); (1; 2); : : : ; (L; L + 1)}, L sites {1; 2; : : : ; L} and two absorbing borders {0; L + 1}. Every nonescaping trajectory of t time steps is represented by a sequence of t + 1 positions (t) = {r0 ; : : : ; rt }, corresponding to a particle which starts at site r0 at t = 0 and arrives at site rt at time t. We can think of (t) as a point of a dynamical phase space. The probability for a given trajectory, (t), can be expressed in terms of the transition probabilities W (r|r 0 ), P( ; t|r0 ) =
t Y
W (rn |rn−1 ) :
(6)
n=1
In parallel with the standard procedure in the canonical ensemble, a partition function is de ned as X X [P( ; t|r0 )] = (W t )(r|r0 ) : (7) Z L ( ; t|r0 ) =
r
Here, the generalized transition matrix whose elements are W (r|r0 )=W (r|r0 ) has been used. The inverse temperature-like parameter provides a simple way of scanning the structure of the probability distribution P( ; t|r0 ) via a smooth partition function [14]. This is the starting point of the so-called thermodynamic formalism [15,16]. For large times, the partition function becomes independent of the point of origin r0 (ergodicity) and it is completely determined by the largest positive eigenvalue, L ( ), of the matrix W , which is assumed to be nondegenerate.
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The Ruelle or topological pressure is a free-energy-like function de ned in terms of the partition function in the form ˆ L ( |w)
= lim
L→∞
1 ln Z L ( ; t|r0 ) = ln L ( ) ; t
(8)
which depends on the quenched con guration wˆ of random bonds. This function contains all the necessary information to derive the chaotic quantities of interest, such as the Lyapunov exponent, , the escape rate, , and the Kolmogorov–Sinai entropy per unit time, hKS = − , through the relations ˆ =−
L (w)
ˆ L (1|w)
;
ˆ =− L (w)
0 ˆ L (1|w)
;
(9)
where the prime denotes a -derivative. The Eqs. (8) and (9), together with the numerical algorithm for the calculation of the largest eigenvalue of the large sparse matrix W , supply us with a very ecient method in order to obtain these chaotic properties. We must also notice that, according to Eq. (9), the Ruelle pressure behaviour at = 1 gives us the global chaotic properties of any quenched con guration. The quantities of macroscopic interest are obtained as con gurational averages represented by h· · ·i, so L ( )
= hln L ( )i ;
(10)
L = −h
ˆ L (1|w)i
;
(11)
L = −h
0 ˆ L (1|w)i
:
(12)
A numerical evaluation of the last two quantities, L and L , in nite open and closed systems with L of order 100 has been already compared with mean eld theory estimations both for the lattice Lorentz gas [12] and the bond model [19] in one dimension.
3. Localization and the dynamical phase diagram In this section we discuss the limiting behaviour for large L of the Ruelle pressure, ( ), in a bond model with a random mixture of a- and b-bonds at a xed density . We consider a b-bond hopping rate, wb = 12 , and di erent values for the a-bond hopping rate, wa 6 12 , and the fraction of bonds occupied by a-bonds, = N=(L + 1). It has been recently established that, in the thermodynamic limit, the Ruelle pressure of random walkers on disordered lattices becomes independent of the density at 6= 1, and it is completely controlled by a rare con guration of the disorder. So, it does not carry any information on the average structure of this disorder. This has been rigorously proved in the case of a Lorentz lattice gas with identical point-like scatterers [14]. The rare con gurations which control the Ruelle pressure on a lattice Lorentz gas are those
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with a large void for ¿ 1 or a large cluster of scatterers for ¡ 1, which are the “most deterministic” and “most chaotic”, respectively. This phenomenon is also present in the random bond model, and probably in any di usion model with static disorder, but with a richer and more complex dynamical phase diagram. In other words, the trajectories may localize in three di erent kinds of clusters: sequences of a-bonds (aa-clusters), sequences of b-bonds (bb-clusters) or sequences of pairs formed by an a-bond and a b-bond (ab-clusters). We will follow the arguments in Ref. [14] in order to sketch the corresponding derivation of these results for the random bond model. For a closed system it follows from the de nition of the generalized transition matrix W (r 0 |r) and Eq. (2) that X W (r 0 |r) = [w(r)] ˆ + [w(r ˆ − 1)] + [1 − w(r) ˆ − w(r ˆ − 1)] ; (13) r0
where w(r ˆ − 1) and w(r) ˆ are the hopping rates of the bond whose right end is site r, and the bond whose left end is site r, respectively. In our case, Eq. (13) can also be written in the following form: X W (r 0 |r) = Waa ( )n(r ˆ − 1)n(r) ˆ r0
ˆ − 1))(1 − n(r)) ˆ +Wbb ( )(1 − n(r ˆ − 1)(1 − n(r)) ˆ + n(r)(1 ˆ − n(r ˆ − 1))] ; +Wab ( )[n(r
(14)
with Waa ( ) = 2wa + (1 − 2wa ) ;
(15)
Wbb ( ) = 2wb + (1 − 2wb ) ;
(16)
Wab ( ) = wa + wb + (1 − wa − wb ) :
(17)
For open systems the equality sign in Eq. (15) is replaced by a ‘less than’ sign in case r is a site adjacent to the borders, that is r = {1; L}. So, in general, we can write X
W (r 0 |r)6Waa ( )
if (r − 1; r) and (r; r + 1) are a-bonds ;
W (r 0 |r)6Wbb ( )
if (r − 1; r) and (r; r + 1) are b-bonds ;
W (r 0 |r)6Wab ( )
if (r − 1; r) is an a-bond and
r0
X r0
X r0
(r; r + 1) is a b-bond; or vice versa :
(18)
We now de ne W ( ) = max{Waa ( ); Wbb ( ); Wab ( )} as the maximum of the three functions given in Eqs. (15) for a given set of parameter values { ; wa ; wb }. According to this de nition and Eq. (18) it follows that the sum of all terms of any column of
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the matrix W satis es the inequality with Eqs. (6) and (7) yields
P r0
135
W (r 0 |r)6W ( ). This inequality combined
Z L ( ; t|x0 )6W ( )Z L ( ; t − 1|x0 )6(W ( ))t :
(19)
Then, an upper bound of the Ruelle pressure (8) is given by ˆ W ( ) L ( |w)6ln
:
(20)
A lower bound to Z L ( ; t|x0 ) is constructed by considering the largest cluster of a-bonds, b-bonds or that in which the a- and b-bonds appear alternatively, if W ( ) = Waa ( ); W ( ) = Wbb ( ) or W ( ) = Wab ( ), respectively. We suppose that this cluster has length M , so it contains exactly M bonds. A partition function ZM ( ; t|x0 ) corresponding to the subset of trajectories that remain con ned for t time steps to this cluster is de ned. As all terms in the sum (7) are nonnegative, any sum over a subset of trajectories will give an exact lower bound for Z L and, consequently, we can write Z L ( ; t|x0 )¿ZM ( ; t|x0 ) :
(21)
This partition function ZM ( ; t|x0 ) is that of an open system of length M in a rare con guration {a; a; : : : ; a}; {b; b; : : : ; b} or, {a; b; a; b; : : : ; a; b}. The Ruelle pressure of such a system is then given by the logarithm of the largest eigenvalue of the matrix W [19]: M ( ) = W ( ) − ( )q02 + O(q03 ) ;
(22)
where q0 = =M is the wave number of the slowest decaying mode in the largest dynamics dominant cluster. By using the mean eld theory for the dynamical properties of Ref. [19], it can be easily found that the function ( ) is approximately given by pq ( ) = (wp + wq )=2, with p; q = {a; b}. Starting from Eqs. (8), (21) and (22) the following lower bound of the Ruelle pressure follows: 2 : (23) L ( )¿hln M ( )i = ln W ( ) − ( ) M2 It can be shown that the average h1=M 2 i tends to zero in the thermodynamic limit, L → ∞, so the upper and the lower bound of L ( ) coincide. This means that, for a given set of parameter values { 6= 1; wa ; wb } the con gurational average of the Ruelle pressure is given by L ( )
= ln W ( );
L→∞;
(24)
with W ( ) ≡ Wpq ( ) = max{Waa ( ); Wbb ( ); Wab ( )}. Consequently, ( ) = limL→∞ L ( ) is independent of the proportion between a- and b-bonds. Nevertheless, this result is only approximately valid for very large systems as the moment h1=M 2 i tends to zero very slowly with the system length. It has been shown that h1=M 2 i ∼ (ln L)− , where is a positive exponent dependent on the model [14]. The scheme of ( ) values in the parameter space { ; wa ; wb } is called dynamical phase diagram. For the sake of simplicity, we take wb = 12 , so this diagram is reduced to a plane. In Fig. 1 the physical region ¿ 0; 06wa 6 12 of this plane is represented and
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Fig. 1. Dynamical phase diagram in a bond model with wb = 12 . The Ruelle pressure is the logarithm of the largest value W ( )=max{Waa ( ); Wbb ( ); Wab ( )} in every region separated from the others by coexistence lines.
divided in six di erent portions according to the numerical order of the three quantities Waa ( ); Wbb ( ), and Wab ( ) in Eq. (15). The Ruelle pressure in every region of the -wa plane is given by the logarithm of the largest of these functions (24). These regions are separated by three coexistence curves: Waa ( ) = Wbb ( ) = Wab ( ) ( = 1); Wab ( ) = Waa ( ) (wa = 14 ) and Waa ( ) = Wbb ( ). The last one intersects the = 1 line at wa = wac = 0:113(1), which plays the role of a triple point, and approaches asymptotically to wa = 14 following the law = ln 2=(1 − 4wa ). The Ruelle pressure is a continuous function at these coexistence curves but its rst derivative has a nite jump. This means that L ( ) becomes nonanalytic in the thermodynamic limit.
4. Numerical results on rare conÿgurations In the last section we have concluded that in a bond model at 6= 1, the trajectories localize on rare con gurations of the disorder which depends upon the set of parameter values { ; wa ; wb }. Here, we intend to perform a numerical analysis of this phenomenon by means of the chaotic dynamical quantities distribution over the con gurational space. In particular, we are interested in the extreme values of these quantities and the corresponding con gurations, especially that of the Lyapunov exponent. The reasons for this choice are clear. The Ruelle pressure, L ( ), is an analytic function of { ; wa ; wb } in any nite system but it becomes nonanalytic in the thermodynamic limit, L → ∞, and its rst derivative changes abruptly at the coexistence curves of the dynamical phase diagram (Fig. 1). For example at the curve = 1, this phenomenon occurs as a consequence of the localization of trajectories on the “most chaotic” and the “most deterministic” clusters at the ¡ 1 and ¿ 1 branches of the diagram, respectively. From this result and Eq. (9), we can argue that the maximum, max (L), and minimum, min (L), Lyapunov exponent of a nite system relate with the left, 0 ( − ), and right
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derivative, 0 ( + ), of the Ruelle pressure at = 1 in the thermodynamic limit through the following expressions: 1 0 ; (25) ( − ) = −max (L) + O ln L
0
( + ) = −min (L) + O
1 ln L
:
(26)
These equations are the base to the numerical inspection of the neighbourhood of = 1 in the dynamical phase diagram. According to them, the extreme values of the Lyapunov exponent in an open bond model with length L and a xed set of parameters {N; wa ; wb } are given by the derivatives of the Ruelle pressure of an in nite system at ± =1±, with 1, save logarithmical corrections in the system length. The presence of these corrections was already established in Section 3. Nevertheless, we can expect that the localization phenomena may already be visible in the small systems attainable in simulations (L ' 100), at least in an incipient way. In the following subsections it will be shown that this assumption is true. 4.1. The Monte Carlo method The chaotic properties depend strongly on the bond con gurations, wˆ = {w(r) ˆ |r = 0; : : : ; L}, in the case of the open system (ABC), so the Lyapunov exponent and the escape rate assumes a number of order 2L of di erent values in a relatively broad interval. The situation is very di erent in the closed system (PBC). Here, the escape rate is null, = 0, and the Lyapunov exponent depends only on the number of clusters of a-bonds, or clusters of b-bonds (these numbers coincide). So, if the number of a-bonds is Na 6Nb , the Lyapunov exponent takes only Na di erent values over the con gurational space [19]. This degeneracy precludes the use of the closed system in a numerical analysis of the dynamical phase transition. In the rest of the paper we consider absorbing boundary conditions (ABC). For a given quenched con guration, the Ruelle pressure, and the chaotic dynamical quantities linked with it through the relations in Eq. (9), can be calculated from a standard method based on the numerical evaluation of the largest eigenvalue of the random sparse matrix W (see appendix). This method has already been used in the calculation of the con gurational average of the chaotic properties [12,19], but we are now interested in the extreme values of these quantities. The Monte Carlo method we have used with these purpose consists of three steps performed successively in a loop: (i) A con guration wˆ = {w(r) ˆ | r = 0; 1; : : : ; L}, with a given N number of a-bonds, is chosen at random and the dynamical quantity of interest is calculated numerically by the method discussed in the appendix. (ii) The positions of an a-bond and a b-bond selected at random are interchanged and the mentioned dynamical quantity is calculated again.
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L. Acedo, M.H. Ernst / Physica A 262 (1999) 129–143 Table 1 The average, h (w)i, ˆ minimum, min , and maximum, max , escape rate values in a system of length L = 99 with hopping rates wa = 0:1; 0:2; 0:3; 0:4; wb = 12 and a density =N=(L+1) of a-bonds. These values have been scaled with the factor 104 = 0:3
= 0:7
wa
h (w)i ˆ
min
max
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
2.2431 3.4033 4.1123 4.5905 1.2986 2.4072 3.3646 4.1998
1.5144 2.6627 3.5963 4.3424 1.033 2.0453 3.0265 3.9849
3.8735 4.5728 4.7596 4.8729 1.6823 2.8966 3.7543 4.4208
(iii) If the dynamical quantity is increased (or decreased if we are seeking the smallest values), we take the new con guration as the starting point of the algorithm, otherwise the previous one is reset. This method provides a fast convergence to the extreme regions of con guration space. 4.2. Extreme values of the dynamical properties. Rare conÿgurations As a test of the method explained in the last section we have applied it to the numerical calculation of the extreme values of the escape rate, and the con gurations associated to them, in an open system with length L = 99. The results for two typical a-bond densities and hopping rates, = 0:3; 0:7; wa = 0:1; 0:2; 0:3, and 0.4 are displayed in Table 1. The corresponding con gurations are best detailed in terms of the random variable n(r) ˆ de ned by Eq. (4). As these con gurations are very regular, it is convenient to abridge them by grouping the patterns {a; b; c; : : : ; a; b; c; : : :} which repeats n times by n(a; b; c; : : :). The following results are found for the rare con guration corresponding to the largest escape rates: = 0:3
{35(0); 30(1); 35(0)} ;
= 0:7
{15(0); 70(1); 15(0)} :
(27)
These results are valid for arbitrary values of the hopping rates in the interval (0; 12 ) whenever the condition wa ¡ wb is satis ed. Similarly, the largest escape rates are attained at the con gurations = 0:3
{15(1); 70(0); 15(1)} ;
= 0:7
{35(1); 30(0); 35(1)} :
(28)
The generalization of these results to any system of length L with N a-bonds is obvious. The largest (smallest) escape rate is attained at a con guration with only one aa-cluster
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Table 2 The maximum Lyapunov exponent and the corresponding rare con guration in the random bond model with L = 99 and wb = 12 at three di erent a-bond densities, , and for several a-bond hopping rates, wa . In the last column the theoretical size independent prediction of Eq. (25) is shown wa
Con guration
max (L)
max (L → ∞)
{25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {25(0); 50(1); 25(0)} {20(0); 20(1); 10(0; 1); 20(1); 20(0)} {16(0); 16(1); 18(0; 1); 16(1); 16(0)} {15(0); 14(1); 22(0; 1); 14(1); 13(0)} {7(0); 13(1); 24(0; 1); 13(1); 19(0)}
0.9184 1.0244 1.0781 1.0958 1.0796 1.0250 0.9803 0.8985 0.7877
0.9489 1.0549 1.0961 1.0889 1.0397 1.0296 0.9986 0.9433 0.8557
{32(0); 3(0; 1); 23(1); 4(0; 1); 31(0)} {28(0); 7(0; 1); 17(1); 6(0; 1); 29(0)} {26(0); 6(1); 18(0; 1); 6(1); 26(0)} {26(0); 5(1); 20(0; 1); 5(1); 24(0)}
0.9242 0.9914 0.9778 0.8958
1.0549 1.0889 1.0296 0.9433
{15(0); 70(1); 15(0)} {15(0); 70(1); 15(0)} {15(0); 70(1); 15(0)} {21(1); 2(0); 28(0; 1); 21(1)}
1.0666 1.1290 1.0300 0.8512
1.0549 1.0889 1.0296 0.9433
= 0:5 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 = 0:3 0.4 0.3 0.2 0.1 = 0:7 0.4 0.3 0.2 0.1
(bb-cluster) whose central site coincides with the middle of the system: nˆ = {(L − N )=2(0); N (1); (L − N )=2(0)}
for the maximum escape rate ;
(29)
nˆ = {(L − N )=2(1); N (0); (L − N )=2(1)} for the minimum escape rate :
(30)
As the escape rate depends only on the value of the Ruelle pressure strictly at =1 (9), these con gurations have no relation with the localization phenomena discussed in Section 3. Its interpretation is simple: the trajectories are trapped on those con gurations with clusters of a-bonds adjacent to the absorbing boundaries as we suppose that the hopping rate associated to them is smaller than the one corresponding to b-bonds. The aa-clusters adjacent to the boundaries are identical in order to provide the most e ective trapping of the moving particle. The con gurations corresponding to the largest escape rate can be given similar interpretation. The maximum Lyapunov exponent values, max (L), and the rare con gurations associated to them, calculated by the same numerical method as those of the escape rate, are shown in Table 2. The same results for the minimum Lyapunov exponent are displayed in Table 3. In the last column of these tables the prediction given by Eq. (25) is shown as a comparison. The relative di erence between the numerical value and the theoretical estimation based on the in nite system limit of the Ruelle pressure
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L. Acedo, M.H. Ernst / Physica A 262 (1999) 129–143 Table 3 The minimum Lyapunov exponent and the corresponding rare con guration in the random bond model with L = 99 and wb = 12 at three di erent a-bond densities, , and for several a-bond hopping rates wa . In the last column the theoretical size independent prediction of Eq. (25) is shown wa
Con guration
min (L)
min (L → ∞)
{25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {25(1); 50(0); 25(1)} {2(1); 4(0; 1); 20(1); 42(0); 20(1); 4(0; 1)} {2(1); 8(0; 1); 16(1); 34(0); 16(1); 8(0; 1)} {2(1); 11(0; 1); 13(1); 28(0); 13(1); 11(0; 1)} {13(1; 0); 12(1); 25(0); 11(1); 12(0; 1); 2(1)} {22(0); 5(0; 1); 39(1); 6(0; 1); 17(0)}
0.7627 0.8037 0.8337 0.8481 0.8597 0.8376 0.7925 0.7247 0.5333
0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.6931 0.3944
{11(1); 4(0; 1); 62(0); 4(0; 1); 11(1)} {1; 7(0; 1); 7(1); 55(0); 7(1); 8(0; 1)} {1; 10(0; 1); 4(1); 50(0); 4(1); 10(0; 1); 1} {34(0); 29(1); 0; 1; 35(0)}
0.7442 0.7668 0.7675 0.5738
0.6931 0.6931 0.6931 0.3944
{35(1); 30(0); 35(1)} {35(1); 30(1); 35(1)} {1; 4(0; 1); 30(1); 22(0); 30(1); 4(0; 1); 1} {3(0); 11(1; 0); 47(1); 11(0; 1); 5(0); 1}
0.8908 0.9395 0.8990 0.4645
0.6931 0.6931 1.6931 0.3944
= 0:5 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 = 0:3 0.4 0.3 0.2 0.1 = 0:7 0.4 0.3 0.2 0.1
is usually large, specially for the minimum Lyapunov exponents, but in some cases it can be less than 5%. These large di erences were expected as a consequence of the logarithmical corrections in the system length. By direct inspection of the rare con gurations and Fig. 1, we can infer that the highest Ruelle pressure phase is always at the center of the system surrounded by clusters with lower Ruelle pressure. In summary, we have (i) The con gurations associated to the smallest Lyapunov exponent ( ¿ 1 branch of the dynamical phase diagram) are characterized by a centered bb-cluster in the range of hopping probabilities wa ∈ (wac ; 12 ), and an aa-cluster in the range wa ∈ (0; wac ). (ii) The con gurations corresponding to the largest Lyapunov exponent ( ¿ 1 branch) and characterized by an aa-cluster for systems with hopping rates wa ∈ ( 14 ; 12 ), and an ab-cluster in the wa ∈ (0; 14 ) range. The dominant cluster is always in the center of the system in order to prevent the particle from escaping for as long a time as possible. That way the fractal set of nonescaping trajectories is bigger and, consequently, the sum in Eq. (7) and the Ruelle pressure is maximized. This central cluster is an incipient version of the one which controls the dynamics of the in nite system, so the numerical results con rm, within the nite system errors, the localization phenomena in the bond model discussed in Section 3.
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5. Conclusions (1) In this paper we have studied the dynamical phase space of a random bond model with static disorder by means of the thermodynamic formalism [15,16]. The recent results which indicate that localization phenomenon occurs in lattice Lorentz gases and other di usive models in the in nite system limit have been extended to the random bond model. In particular, we have found that the Ruelle pressure, ( ), becomes a non-analytic function of the temperature-like parameter and the bond hopping rates in the thermodynamic limit. (2) The case with two species of bonds, a- and b-bonds, with associated hopping rates, wa and wb , has been explicitly considered. It has been shown that the thermodynamic limit of the Ruelle pressure, ( ), is a function independent of density (the fraction of a-bonds in the system) as a consequence of the localization of trajectories in rare con guration of the disorder. These rare con gurations have three di erent forms: clusters of a-bonds (aa-clusters), clusters of b-bonds (bb-clusters) or sequences {a; b; a; b; : : : ; a; b} (ab-clusters) depending on the set of parameter values { ; wa ; wb }. This is in contrast with the simpler dynamic behaviour of the lattice Lorentz gas where the localization occurs only in two rare con gurations: the largest void for ¿ 1 and the largest cluster of scatterers for ¡ 1 [13]. (3) The dynamical phase diagram of ( ) in the -wa plane (with wb = 12 ) has been deduced. This diagram consists of six di erent regions separated by three coexistence curves. (4) The extreme values of the chaotic dynamical properties have been calculated with a Monte Carlo method based on the thermodynamic formalism. The con gurations corresponding to the largest and smallest values of the Lyapunov exponent in a system with L = 99 have been related with the ¡ 1 and ¿ 1 branches of the diagram. These con gurations are characterized by its regularity and a conspicuous central cluster corresponding to the rare con guration which dominates the dynamics for a given set of parameter values { ; wa }. In spite of the small length of the system studied, transitions at = 1 − ; wa = 14 and = 1 + ; wa = w ac (with 1), predicted theoretically, are observed. Acknowledgements L.A. gratefully acknowledges support from the Ministerio de Educacion y Ciencia (Spain) and the hospitality of the University of Utrecht during a visit in the autumn of 1994 where this work was started. Appendix A. Recursion relations for the determinant of Wÿ In the ABC case, the con guration wˆ = {w(r) ˆ | r = 0; 1; 2; : : : ; L} contains (L + 1) independent random variables: in the PBC case, wˆ = {w(r) ˆ | r = 1; 2; : : : ; L} contains
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L such variables as w(0) ˆ = w(L). ˆ The random matrix, de ned below Eq. (6), has the form W (r|r 0 ) = br−1 ;
(r 0 = r − 1) ;
W (r|r 0 ) = ar ;
(r 0 = r) ;
W (r|r 0 ) = br ;
(r 0 = r + 1) ;
(A1)
where r = 1; 2; : : : ; L and ˆ ; br = (w(r)) ˆ − w(r ˆ − 1)) : ar = (1 − w(r)
(A2)
Moreover, the determinant, det |W |, in the ABC and PBC case is denoted by D L (0; 1; 2; : : : ; L) and E L (1; 2; : : : ; L) respectively, where the arguments refer to the set of random variables w. ˆ To derive the recursion relation we write E L explicitly a1 b1 bL b a b 1 2 2 b2 a3 b3 (A3) EL (1; 2; : : : ; L) = b3 a4 b4 ··· b a L−1 L−1 bL bL−1 aL and D L (0; 1; 2; : : : ; L) follows from Eq. (A3) by setting b L = 0. We calculate D L by developing Eq. (A3) (with b L = 0) with respect to the last column. This yields at once the recursion relation for the ABC case: D L (0; 1; 2; : : : ; L) = a L DL−1 (0; 1; 2; : : : ; L − 1) − b2L−1 DL−2 (0; 1; 2; : : : ; L − 2) (A4) with initial conditions D1 (1) = a1 ;
D0 = 1 :
(A5)
The determinant E L is slightly more complicated. Developing E L with respect to the last column yields E L (1; 2; : : : ; L) = a L DL−1 (L; 1; 2; : : : ; L − 1) − bL−1 ML−1; L + (−1)L−1 b L M1; L : (A6) The two minors are developed with respect to the last row and yield ML−1; L = bL−1 DL−2 (L; 1; 2; : : : ; L − 2) + (−1)L b L b1 b2 · · · bL−2 ; M1; L = b1 b2 · · · bL−1 + (−1)L b L DL−2 (2; : : : ; L − 1) :
(A7)
Combination of Eqs. (A6), (A7) and (A4) yields the relation E L (1; 2; : : : ; L) = D L (L; 1; 2; : : : ; L) − b2L DL−2 (1; 2; : : : ; L − 1) −2(−1)L b1 b2 · · · b L :
(A8)
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The recursion relation (A4) provides a very ecient numerical algorithm for calculating the determinants D L and E L for a given set of quenched variables. The secular determinant, det |W −|=0, which is a polynomial in , of degree L, can be calculated by replacing ar by ar − . The largest zero L ( ) 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] [22]
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