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Master equation, Anderson localization and statistical mechanics
2 February 1998 PHYSICS
LETTERS
A
Physics Letters A 238 ( 1998) 169- 178
ELSECVIER
Master equation, Anderson localization and statistical mechanics Alessandro
Mazza a, Paolo Grigolini a*b*c
a Dipartimento di Fisica dell’(lniversitci di Piss, Piazza Torricelli 2, 56126 Pisa, Italy h Department
ofPhysics
of the University of North Texas, PO. Box 5368, Denton, 7X 76203, USA ’ Istituto di Biojsica de1 CNR. Via San Lorenzo 26. 56127 Piss. Italy
Received 28 April 1997; accepted for publication 5 September 1997 Communicated by A.R. Bishop
Abstract A theoretical
picture is established
which sheds light on the apparent conflict between the arguments used by Zwanzig
to derive the Pauli master equation, and the localization process generated by the random distribution of site energies on a one-dimensional
tight-binding
Hamiltonian.
The possible theoretical consequences are discussed. @ 1998 Elsevier Science
B.V. PACS: 72.10.Bg; 72.15.Rn; 74.40.+k Keywords: Master equation; Quantum localization; Genuine randomness
The foundation of the master equation in terms of microscopic dynamics has been and remains one of the most challenging problems of modern physics. To make it possible for the reader to appreciate this aspect, we refer to the fundamental work by Zwanzig [ 1,2]. Zwanzig adopts a projection method to derive from the Liouville equation of the whole system of interest a reduced equation of motion only concerning the diagonal elements of the density matrix. Then he proves that using the Markov approximation it is possible to express the resulting reduced equation in a form distinguishable from the Pauli master equation, thereby deriving with the elegant formalism of projection operators the same conclusion as that of the earlier work by Pauli [ 31 and Van Hove [ 41. We apply the same method to studying the transport properties of an electron moving in a one-dimensional random lattice described by a tight-binding Hamiltonian with a random distribution of site energies. The
explicit form of the Hamiltonian
H=Ho+AW,
is
(1)
where
(2) and AW=V~(~m)(m+l~+~m+l)(m~~. m
(3)
We have adopted a notation identical to that used by Zwanzig [2] to make it easier for the reader to relate the result of this Letter to the theoretical problem discussed by Zwanzig 121. Thus, we are considering the Hamiltonian terms which generate a quantum transition from a state ]m) to another state jm’) as a sort of weak perturbation compared to the diagonal part of the Hamiltonian. For Zwanzig these are transitions
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between different eigenstates of the unperturbed part of the Hamiltonian. If we look at the same Hamiltonian from the perspective adopted by Anderson [ 5,6] to predict the phenomenon now known as Anderson localization, the states Im) are distinct sites of a lattice, in this case a one-dimensional lattice, and the transport is made to deviate from the ideal ballistic condition of a perfect crystal by means of a random distribution of the site energies, E,, = E + %I .
(4)
Here we are assuming that with changing site there is a fluctuation pnt around the common value E. We assume no correlation among different sites, namely (cp,,,~,,,~)= AS,,,,,,/
(5)
Notice that to define the correlation function (5) we are using the Gibbs picture, in spite of the fact that according to Anderson [ 61 the study of a disordered crystal should imply only a single realization of the random distribution of energies. However, since the approach to the master equation illustrated by Zwanzig rests on the quantum Liouville equation, and so on the adoption of the Gibbs picture, we adopt for convenience the same perspective, too. Of course there is a subtle difference between the following two ideal cases. The former is the study of single realizations of the random distribution of energies by means of the density matrix. Although done so as to emphasize the statistical nature of quantum mechanics, it would not conflict with the recommendation made by Anderson [ 61 to study a single sample with a single realization of the energy distribution. The latter case, providing the theoretical perspective adopted in this Letter, is the study of a statistical density matrix which is the average over infinitely many single statistical density matrices, each one corresponding to a given distribution of site energies. This perspective bypasses, in a given sense, the need of adopting an extremely complex Hamiltonian, with extremely high degeneracy, or with a high density of states, imagined by Zwanzig [ I ,2 1. We shall see that this perspective supplemented by the adoption of a generalized projection operator /7, makes it possible, in principle, to realize the coarse graining condition which according to Zwanzig is necessary to derive the Pauli master equation.
Letters A 238 (1998)
169-178
Furthermore, we make the assumption that the random distribution of the site energies follows the Cauchy prescription p(q)
= 1r lr y2 + p= ’
(6)
where p(q) denotes the probability that the energy of a given site fluctuates by the quantity p about the common value E. Note that the Cauchy distribution is one of the cases explicitly considered to study the tight-binding electron model ( 1) (see, for instance, Ref. [7]). According to the earlier arguments we consider the statistical density matrix ps defined by ps(r)
=
J
d-MW({cD})p({&r).
(7)
This means an average over the single realizations , pi,. . . each characterized by {+‘} = $‘l,p2~$‘3,... the density matrix p( {(p}, t). The weighing function corresponding to the random distribution of energy fluctuations, w({qo}), in accordance to the lack of memory (5), is given by W({P))
=
wn,-1~{40”,-I}~~n,~{SPm}~~nt+l~{~~~+l}~~~~
1 (8)
with the weighing distribution
of each site identical to
(6). We adopt the following projection IIp=
operator,
d{4 c J w({wl)
l~)(~lP~sP~l~)(4~
m
(9) We have adopted the symbol n rather than the more conventional P, since it is the result of two distinct procedures, the average over the realizations of energy fluctuations and the cancellation of the diagonal matrix elements prescribed by Zwanzig [ 1,2]. Note that this is proved to be a projection operator as a consequence of the fact that the weighing p(p) is properly normalized. We note that the projection operator I7 is essentially the same as that adopted by Kenkre [ 81 to study exciton dynamics. This is not fortuitous, since the motivation for the generalization of the projection operator is simiIar in both cases: it should help
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the process of irreversibility creation. In a paper by Kenkre and Knox [9] an illuminating discussion can be found of why the picture of Zwanzig must be supplemented with a coarse graining ingredient. This essential ingredient generates an element of irreversibility and makes it possible to obtain a generalized version of the master equation from which to derive, in the due limiting condition, the standard Pauli master equation. Our purpose here is to show that this irreversibility ingredient must be genuine and that much attention must be devoted to distinguishing real from apparent seeds of “irreversibility”. As we shall see, the random distribution of energies, here referred to as Anderson noise, is a false ingredient, though it has the key role of making easier and more effective the action of a genuine seed of irreversibility. We shall see that even a relatively weak but genuine seed of irreversibility can produce the Pauli master equation, and, hence, according to the perspective established in Ref. [ IO], statistical mechanics from mechanics. However, without the action of the genuine seed of irreversibility after a temporary condition of ordinary diffusion, the system would make always a transition to localization. Thus we can say that the purpose of the generalization of the projection operator is similar but not identical to that of Kenkre [ 81 and Kenkre and Knox [ 91. We introduce the operator II for the purpose of giving birth to an intermediate rather than a steady regime of irreversibility and statistical mechanics. This temporary condition of statistical mechanics is killed by the same physical condition from which it is generated. Using the generalized projection operator Z7 but essentially the same calculations as those illustrated by Zwanzig [ 1,2] and adapted to the case here under discussion, we find the following equation of motion,
;pn =- nt+tI c
dt’Bnnl (t - t’) [P,([‘)
- Prn(Ol *
(10) Note that
P,,(f) = (mbslm)
(11)
and E,,,(t)
= (n({ZZLexp[
x (1 - fl>Wz)(4}ln),
(1 - n)L(t
- t’)] (12)
171
Letters A 238 (1998) 169-I 78
where L denotes the commutator superoperator associated to the Hamiltonian ( 1) . As pointed out earlier, the disordered distribution of site energies has seemingly, according to Zwanzig [ 1,2], the key role of realizing a condition favorable to Fermi’s golden rule. One can render this condition as favorable as possible increasing the intensity of Anderson noise, namely the width y of the distribution (6). In other words, we make the important assumption YBV,
(13)
which has the beautiful consequence of rendering the interaction term of ( 1) much smaller than the unperturbed part, and consequently the superoperator associated to the interaction part, Lt , negligible compared to the unperturbed part of it, LO. Note that L = LO+ Lt and that, on the basis of ( 13) we make the assumption of replacing L with ,!.c in the argument of the exponential of ( 12). As an effect of this approximation, and only as an effect of it, we derive for the memory kernel the following simple expression, ZILexp[(l
-n)L(t--t’)](l
-Z7>Llm)(ml
= 2( V*/fi*) exp[ -2(y/fi)
It - t’ j 1
x (]m+
l)(rn - 11 -21m)(m]).
l)(~+
I]+ Im-
(14) The exponential nature of the resulting relaxation process is due to the choice of (6). According to Lee [ 111, in full agreement with the view expressed by Fonda, Ghirardi and Rimini [ 121, there is no room in quantum mechanics for a decay process which is exactly exponential. However, their warnings refer to a dynamical derivation from a well-defined Hamiltonian. It refers to a problem discussed at length by Zwanzig himself in his fundamental work [ 1,2] aiming at deriving from a rigorous microscopic approach the Pauli master equation: he established that the exponential behavior is only admitted in an intermediate region of times. Here, we are already making with Anderson a statistical assumption about the random distribution of site energies. Thus, under this assumption the exponential decay of (14) is an exact property. The counterpart of Zwanzig’s discussion within this context would be a condition where the fluctuations of the site energies do not result in a continuous
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spectrum, so that the memory kernel would not be exponential. The case of dichotomous fluctuations is an example of this kind, resulting in fact in oscillations rather than exponential behavior. An imperfect realization of the Cauchy distributions, with finite rather than infinite moments, would result in the breakdown of the exponential behavior at long times, as stressed by many authors [ 1,2,11.12]. However, this breakdown, if it ever exists, does not have anything to do with Anderson’s localization, as will be made clear by the results of this Letter. For this reason, the assumption of Cauchy distribution of the site energies can be safely made. In other words, we plan to explore the intermediate region of times that according to Zwanzig should not conflict with the realization of the Pauli master equation, and we plan to show that, in spite of traditional wisdom, a breakdown of ordinary statistical mechanics occurs much earlier than the breakdown of the coarse graining condition. This is due to the fact that Anderson noise is not sufficient to realize the second-order approximation behind ordinary statistical mechanics. The memory kernel of ( 14) leads at long times to the Pauli master equation
which, in turn, implies Brownian with the second moment given by (x’(t))
= (2V2/riy)L%.
motion
diffusion
(16)
Note that
(X2(f))= L2-yP”,mn2
9
(17)
“I
where L is the distance between two nearest-neighbor sites. The standard result of (16) is obtained adopting the assumption of intermediate asymptotics [ 131, namely, as well known [ 141, it seems to imply that many sites are occupied. However, from the earlier derivation it is clear that Eq. (15) is already valid at times t B h/y. As we shall see, it is broken at the much larger time scale t x h/V and throughout the whole time interval ranging from t = 0 to f M h/V the system population of a distribution originally located at the site m = 0 remains virtually confined in
Letters A 238 (1998)
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the lattice region spanned by the sites m = 0, m = 1 and m= -1. Apparently the result of Eq. ( 15) is a correct consequence of ( 13) and of the fact that the coarse graining over the infinitely many energy levels corresponding to a transition from the site Im)to the site Jm+ 1)implies the realization of the conditions for the validity of the Pauli master equation. Why should the random distribution of site energies, supplemented by the suitable Gibbs picture of Eq. (7), not produce the same coarse graining effect as those directly or indirectly assumed by Zwanzig and Kenkre? It is difficult to answer this question only on the basis of the mathematical properties of the memory kernel E. This is so because, in general, to derive its exact expression is beyond the range of an analytical treatment. Furthermore, according to Zwanzig [ 1,2] the Pauli master equation rests on the second-order approximation to B and on the assumption that the breakdown of its irreversible behavior takes place at times much larger than those of experimental interest. There is no hint, however, on how to establish the physical conditions necessary to make legitimate the adoption of the second-order approximation. Yet, if a numerical study is made of the problem in the correspondence of the condition ( 13), with the wave function initially localized in the site m = 0, the result found is that illustrated by Fig. 1. This means that, after an earlier transient process, not visible in the long-time scale of these figures, and which might or might not correspond to the prediction of ordinary statistical mechanics, see Eq. (16), the second moment ( 17) of the quantum mechanical distribution shows fluctuations about a time-independent value. In the light of the well-known arguments used by Anderson [5,6] this result is not surprising, and it is a consequence of the fact that disorder, in addition to realizing the coarse graining helping the validity of the Pauli master equation, also establishes the condition for the occurrence of Anderson localization. We are not aware of any discussion in the literature of the reason for the failure of the Pauli master equation, in spite of the existence of conditions ensuring the necessary coarse graining, but the subtly related discussion of Kubo et al. [ 151 on the molecular origin of diffusion, recovering the result of an earlier work by Kirkwood [ 161. These authors proved that the process of Brownian diffusion implies an ineluctable form of
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Lerters A 238 (I 998) 169-178
0.016
0.012 A C “x V
0.008
0.004
3
I
I
40
80
I
I
160
120
I
+
200
t
Fig.
1. The
second moment (x2(f)),
Eq.
( 17),
as a function of time in the long-time region. The dashed line is the result of the numerical
calculation. The full line denotes the theoretical prediction, based on the numerical integration of ( 18). The pammeters used are y = 100 and V = I.
T 0.016
1
0.012 A =;‘
V t
0.008
0 o”----I-
I
I
40
120 8O
Fig. 2. The second moment
(x*(r)),
calculation
lattice. The full line denotes the theoretical
with a three-site
The parameters
Eq. ( 17). as a function
used are y = 100 and V = I.
160
t
of time in the long-time prediction
region. The dashed line conesponds
( 18). The corresponding
integral
to the numerical
is evaluated numerically.
174
A. Mazza. I? Grigolini/Physics
localization at long times. In a quantum mechanical context, we are only aware of the fact that Zwanzig pointed out [ I ] that there exists a time, tt, beyond which the continuum approximation, responsible for relaxation properties, is broken and the discrete nature of quantum mechanics is perceived again. This time can be rendered so large as to become infinite when compared to experimental times. This important time, however, must not be confused with the time at which localization occurs. The purpose of this Letter is to prove that the Anderson localization has to do with a more crucial condition, which can be realized at finite times even when the time tl is virtually infinite, and this condition implies that the second-order approximation behind the master equation approximation is overcome. To shed light on this important aspect we note first of all that when condition ( 13) is adopted, the amount of population entering states different from IO), 1I) and 1- I) is negligible. This can be assessed by comparing the many-site result of Fig. 1 to the three-site result of Fig. 2. Thus, rather than using a computer we might limit ourselves to diagonalizing third-order matrices. We decided to make our discussion still simpler by exploring the case of only two sites, the dimer. This can be justified by assuming that, as an effect of a strong Anderson noise, the transition from the site lm) to the site lm + I) is statistically independent of that from the site (m) to the site Irn - 1). Under this assumption we can derive an expression for the time evolution of p,,,(r), P,~,+, (r) andp,,,_t (t) and from this wederive, assuming that p,,(O) = I and P,,,+I (0) = p,,,-I (0) = 0. 00
(.G( t)) =
16V*L”y 7r
1
dx I
I
x* + 4y* x* + 4v2
-m
x sin*[ (x7 + 4V’) ‘/*t/2fi]
.
(18)
This formula was obtained by taking into account that the distribution of relative energies, pn, -q,,,*, , is still a Cauchy distribution, with a doubled width. Our numerical calculation, shown in Fig. 2, results in very good agreement with the theoretical prediction (18) and thus it fully supports the assumption that the backward transition is statistically independent of the forward transition. The expression of Eq. ( 18) is not yet a useful ana-
Letters A 238 (1998)
169-178
lytical formula. It can be made so as follows. We can split it into the sum of two terms. The first is obtained by neglecting the dependence of the harmonic function on V, as a consequence of ( 13). The latter aims at correcting this too strong approximation by noticing that in the space region close to the maximum of the Cauchy distribution, we can neglect the dependence of the harmonic function on n. Thus, we get
(x2(O)= -V[l
-$${Y[1 - exp(-2U/fi)
I
2L2V -exp(-2yr/fi)l}+----+-sm*(Vt/li). (19)
In Fig. 3 we compare the prediction (19) with the numerical treatment of the tight-binding model with infinitely many sites (virtually the exact solution of the problem) and we find that ( 19) reproduces satisfactorily the phenomenon of Anderson localization. In conclusion, Eq. (19), though approximate, is an acceptable analytical formula which makes it possible to shed light on the problem here under discussion. In Fig. 4 we zoom on the early-time region, with a standard diffusion-like character, and we find that the agreement between the theory represented by Eq. ( 18) and the numerical results is very good. Furthermore, we note that the slope of the diffusion process predicted by the Pauli master equation agrees very well with the early slope of the numerical result. Let us now reverse the standard perspective. Rather than using the memory kernel fl( t) to find, using the generalized master equation of ( lo), the time evolution of the second moment (x2(t)), let us do the reverse: from the time evolution of (x*(t)), let us derive an expression for the memory kernel s”(t) . Of course ( 18) would be more convenient from a quantitative point of view. However, the conceptual clarification that we have in mind is made more transparent if we adopt ( 19). Therefore let us use ( 19) to derive the expression for the memory kernel 9(t) . Setting
I”n,n’ =
we find
2K(t)(&,~+, + a,,,/_,) ,
(20)
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L&ten
A 238 (1998)
169-178
17s
O.OlE
0.011 A z 5 0.008
0.004
0
I
40
I
8o
t
I
I
120
160
I >
200
Fig. 3. The second moment (am), Eq. (17), as a function of time in the long-time region. The dashed line is the result of the numerical calculation. The full line denotes the theoretical analytical prediction, based on the approximation to ( 18) given by ( 19). The parameters usedarey=lOOandV=l.
Fig. 4. The second moment (r’(t)), IQ. (17). as a function of time in the short-time region. The diamonds denote the result of the numerical calculation. The crosses denote the prediction of the numerical integration of ( 18). The dashed line denotes the slope of the diffusion process predicted by the Pauli theory (Eqs. ( 15) and ( 16)).
A. Maua, I? Grigolini/Physics Letters A 238 (1998) 169-178
176
K(f)
_Vexp( -2Vr/Fi)]
+ 2
cos( 2Vr/Fi) .
(21)
By inspection of (2 1) we notice that a second-order approximation would yield a result agreeing with ( 14). This means that the Anderson perspective goes beyond the conventional statistical mechanics of transport processes. The analytical result of Eq. (2 1) establishes a link between the Pauli and the Anderson perspective. The early-time regime is indistinguishable from that which would be produced by the Pauli master equation, and represents the predictions of ordinary statistical mechanics. At times of the order of /i/V, however, the influence of the negative tail of (21) becomes significant and the system makes a transition to the localized state. This picture, simple as it is, has the remarkable effect of making it very easy to establish the influence of an uncorrelated fluctuation. If the energy of the sites fluctuates in time as an effect of the interaction between the system and an external environment, it is easy to prove that the new form of the memory kernel is K(t)
=
v2y 1 ~y= - P @
V3 _ V exp( -2vt//%) 1+ cos(2Vr/~) ‘TTY@ x exp( -~t/fi)
.
> (22)
y > CJ > V ,
(24)
then, for all practical purposes diffusion takes place according to the predictions of the Pauli master equation and the resulting diffusion coefficient is distinguishable from the prediction of Fermi’s golden rule. This is reminiscent of the conclusions recently reached by Giovannetti et al. [ 171 who pointed out that the Markovian approximation is often equivalent to tacitly assuming the presence of an extra source of stochasticity, which has the role of making the approximation legitimate. In this specific case the extra source of stochasticity obeying (22) has the key effect of recovering the second-order structure therefore adhering to the general prescription of Bianucci et al. [ lo] : these authors have shown that the microscopic foundation of the linear response would be equivalent to the microscopic foundation of statistical mechanics. Similar results have been found using numerical calculation by Evensky, Scalettar and Wolynes [ 181. The nice aspect of (22) is that it results in the analytical prediction for the diffusion coefficient Y -_~ 2y+a
where g=akgT,
Eq. (21) is an important result which makes it possible for us to draw a conclusion on the intriguing problem of the conflict between the Anderson and the Pauli perspective. The Pauli perspective requires that the condition (13) is fulfilled. On the other hand, if the condition (13) is fulfilled, and an extra source of stochasticity of intensity (7 is used, and the following condition is realized,
(23)
ka is the Boltzmann constant and T is the temperature of the environment. This means that we establish a subtle connection with thermodynamics without discussing its microscopic derivation. The remarkable aspect of this seemingly trivial result is that in the case of very strong Anderson noise, an external fluctuation resulting in a rate c comparable to v/i, and thus extremely weak compared to the fast decorrelation process caused by the random distribution of energies, destroys localization and results again in a process of transport indistinguishable from that predicted by the Pauli master equation.
V 2v+a
(25) which at least qualitatively accounts very well for the numerical result of Ref. [ 181. As shown in Fig. 5, with increasing CTthe diffusion coefficient makes a fast transition from the vanishing value to a finite value and then, upon a further increase beyond the value 2fi, regresses to zero very slowly. Upon increase of the ratio y/V the regression to zero becomes so slow as to be virtually independent of the noise intensity, and the constant value kept, 2V2L2/y, coincides with the Pauli prediction ( 16). We think that the important results of this Letter are as follows.
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F! Grigolini/Physics
0 02
/ ,’
0.01
cl
:
1’
!i
----.___
-.
.-.-
o.oosl
ool_._i__
169-178
20
-‘----% 4o 0
Fig. 5. The diffusion coefficient noise intensity V, for different top to the bottom curve, vy is 1000,500.300,200, 150, 120,
~
80
100
D of FZq. (25) as a function of the values of the ratio V/y. From the given by the following sequence: 100.
(a) A simple theoretical picture is developed establishing the connection between two conflicting predictions: Anderson localization and ordinary diffusion generated by the Pauli master equation. (b) The breakdown of the Pauli condition is independent of the discrete nature of the systems studied. The time at which this kind of breakdown should occur, in principle, as pointed out by Zwanzig [ 1,2], could be rendered as large as we like by increasing the intensity of the coarse graining processes. In a sense, increasing the intensity of the Anderson noise, y, should produce the same effect. This is so much so because, as a result of a statistical average over infinitely many realizations of the distribution of the site energies (Pi,, see Eq. (4), the memory kernel becomes an exponential. An imperfect realization of this condition would result in the recurrence processes and in the violation of the conditions necessary for relaxation processes to be realized. The breakdown of the Pauli master equation resulting in quantum localization is provoked by the fact that the second-order approximation is not legitimate, in general. On the other hand, it can be rendered such by the addition of a genuine source of stochasticity. (c) The condition of adopting a strong Anderson noise is not enough for the Pauli master equation to be valid. It is a necessary but not sufficient condition. For the system to adhere to the prescriptions of Pauli, and to those of ordinary statistical mechanics [ lo], the whole system must be perturbed by a genuine
177
source of stochastic fluctuations. If this perspective is adopted, the problem of a rigorous foundation of the Pauli master equation from within ordinary quantum mechanics becomes equivalent to that of deriving randomness from quantum mechanics. This is a delicate problem still generating vivacious debates and controversies. For a recent discussion, see Ref. [ 171 and references quoted therein. A much more extended discussion would be required. Let us limit ourselves to remarking that if attention is focused on the macroscopic process and a strong condition of time scale separation between macroscopic diffusion and microscopic dynamics is set by making the coupling V smaller and smaller, then the intensity of the genuinely stochastic process necessary to recover the Pauli master equation becomes smaller and smaller, too. This makes the settlement of the problem more difficult rather than easier. It is so because a source of genuinely stochastic fluctuations, be it compatible or not with ordinary quantum mechanics, produces effects identical to those of a Markovian approximation arbitrarily made, and so theoretical predictions indistinguishable from those generated by conventional wisdom. In addition to shedding light on the apparent conflict between Anderson localization and the Pauli master equation, the result of this Letter affords further benefits. The equivalence between the quantum kicked rotator (QKR) and the Anderson localization is attracting the attention of an ever increasing number of authors [ 20-231. For a very recent example, see Ref. [ 241. On the other hand, since the QKR has a classical counterpart, the study of its dynamics is also done having in mind the problem of the correspondence principle. The classical kicked rotator does not result in any process of localization. The most widely accepted view is that a complete correspondence between quantum and classical physics is recovered if the QKR is made to interact with an external source of fluctuations [ 201. The numerical results show that an external fluctuation of even extremely small intensity can break localization and allow the system to diffuse with a diffusion coefficient almost indistinguishable from the classical diffusion coefficient in the absence of any perturbation. This way of realizing the Pauli master equation could be questioned if irreversibility has to be regarded as being intrinsic [ 2.51 rather than extrinsic to quantum mechanics. However, if some modifications are needed to make irreversibility really
.. --%___. -------___.__ --x
------.
: 0.015
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intrinsic to quantum mechanics, then the results of this paper support the view that they might be extremely weak [17]. In a recent paper [26] it has been pointed out that the conviction that a weak fluctuation produced by the environment can always be used to account for the transition from quantum to classical mechanics can be challenged by the phenomena of anomalous diffusion. The authors of Ref. [ 261 noticed that the kicked rotor in the presence of the so called accelerator islands according to the correspondence principle should exhibit an initial condition of superdiffusion. This happens to be the case. However, before localization the system shows a sort of transition to a quantum mechanically induced state of normal diffusion, and this physical condition is found to be surprisingly robust against the influence of environmental fluctuations. We think that this regime of normal diffusion is essentially the same as the Pauli regime discussed in this paper. Consequently, the time region where the correspondence principle was found to apply [ 261 is nothing but a sort of ballistic regime made more extended in time by the adoption of the special condition of acceleration state [26]. If the correspondence between the kicked rotor and the Anderson model is taken for granted, this time enhancement of the ballistic regime corresponds to a tight-binding Hamiltonian with long-range correlations among the site energy fluctuations. From a qualitative point of view the role of space fluctuation is equivalent to resealing the site distance L to a much larger, even macroscopically larger [ 271, value A. As a consequence the ballistic regime is enhanced, and the extension of the regime of anomalous diffusion is dilated. All this requires further investigation. However, we think that the proper theoretical tools to do that are already available: the methods introduced more than twenty years ago by Kenkre and co-workers [ 8,9] are still a very convenient way to approach this intriguing problem, and probably this can be done by properly extending the results of this paper. It has to be remarked that a unified model for the study of diffusion localization and dissipation has been very recently proposed by Cohen [ 281 by means of a Feynman-Vernon type of path integral, and it would be interesting to compare his theoretical approach to the generalized version of the Zwanzig projection method proposed by Kenkre and co-workers [ 8,9].
III R. Zwanzig, in: Quantum Statistical Mechanics,
ed. PH.E Meijer (Gordon and Breach, London, 1966) p. 139. 121R. Zwanzig, in: Lectures in Theoretical Physics, Vol. 3, eds. W.E. Brittin et al. (Interscience, New York, 1961). 131 W. Pauli. in: Probleme der Modemen Physik, Festschrift zum 60. Geburtstag A. Sommerfeld, ed. P Debye (Hirzel, Leipzig, 1928). I41 L. Van Hove, Physica 21 (1955) 517: 23 (1957) 441. ]5l PW. Anderson, Phys. Rev. 109 (1958) 1492. 161 P.W. Anderson, Rev. Mod. Phys. 50 (1978) 191. 171 K. Ishii, Suppl. Prog. Theor. Phys. 53 (1973) 77. 181 V.M. Kenkre, in: Springer Tracts in Modem Physics. Vol. 94, Excitation Dynamics in Molecular Crystal and Agregates (Springer, Berlin, 1982) p. 1. 191 V.M. Kenkre, R.S. Knox, Phys. Rev. B 9 (1974) 5279. IlO1 M. Bianucci, R. Mannella, B.J. West, P Grigolini, Phys. Rev. E 51 (1995) 3002. (Ill M.H. Lee, Phys. Rev. Lett. 51 (1983) 1227. II21 L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Prog. Phys. 41 ( 1977) 587. Lectures on Phases Transitions and the 1131 N. Goldenfeld, Renormalization Group (Addison-Wesley. Reading, 1992). I 141 N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 198 I ). 151 R. Kubo, M. Toda, N. Hashitsume, Statistical Mechanics, Vol. 2. Nonequilibrium Statistical Mechanics (Springer, Berlin, 1985) p. 105. 161 J. Kirkwood, J. Chem. Phys. I4 (1946) 180. 171 V. Giovanetti, P Grigolini, G. Tesi, D. Vitali, Phys. Lett. A 224 (1996) 31. [IS] D.A. Evensky, R.T. Scalettar, PG. Wolynes, J. Phys. Chem. 94 (1990) 1149. 1191 A. Bohm, I. Antoniou, P. Kielanowski, Phys. Lett. A 189 ( 1994) 442. 1201 T. Dittrich, R. Graham, Phys. Rev. A 42 ( 1990) 4647; D. Cohen, S. Fishman, Phys. Rev. A 39 ( 1989) 6478; D. Cohen, Phys. Rev. A 43 ( 1991) 639; Phys. Rev. Lett. 67 (1991) 1945. [2l] G. Casati, B.V. Chirikov, J. Ford, EM. Izrailev, Lect. Notes Phys. 93 (1979) 334. [22] S. Fishman, D.R. Grempel, R.E. Pranges, Phys. Rev. Lett. 49 (1982) 509. [23] A. Altland, Phys. Rev. Lett. 71 (1993) 69. [ 241 A. Altland, M.R. Zimbauer, Phys. Rev. Lett. 77 ( 1996) 4536. 1251 1. Prigogine, T. Petrosky, Physica A 147 (1987) 33. [26] L. Bonci, P. Grigolini, A. Laux, R. Roncaglia, Phys. Rev. A 54 (1996) 112. [ 271 P. Allegrini, L. Bonci, P Grigolini, B.J. West, Phys. Rev. B 54 ( 1996) 11899. [28] D. Cohen, Phys. Rev. L&t. 78 (1997) 2878; Phys. Rev. E 55 (1997) 1422.
LETTERS
A
Physics Letters A 238 ( 1998) 169- 178
ELSECVIER
Master equation, Anderson localization and statistical mechanics Alessandro
Mazza a, Paolo Grigolini a*b*c
a Dipartimento di Fisica dell’(lniversitci di Piss, Piazza Torricelli 2, 56126 Pisa, Italy h Department
ofPhysics
of the University of North Texas, PO. Box 5368, Denton, 7X 76203, USA ’ Istituto di Biojsica de1 CNR. Via San Lorenzo 26. 56127 Piss. Italy
Received 28 April 1997; accepted for publication 5 September 1997 Communicated by A.R. Bishop
Abstract A theoretical
picture is established
which sheds light on the apparent conflict between the arguments used by Zwanzig
to derive the Pauli master equation, and the localization process generated by the random distribution of site energies on a one-dimensional
tight-binding
Hamiltonian.
The possible theoretical consequences are discussed. @ 1998 Elsevier Science
B.V. PACS: 72.10.Bg; 72.15.Rn; 74.40.+k Keywords: Master equation; Quantum localization; Genuine randomness
The foundation of the master equation in terms of microscopic dynamics has been and remains one of the most challenging problems of modern physics. To make it possible for the reader to appreciate this aspect, we refer to the fundamental work by Zwanzig [ 1,2]. Zwanzig adopts a projection method to derive from the Liouville equation of the whole system of interest a reduced equation of motion only concerning the diagonal elements of the density matrix. Then he proves that using the Markov approximation it is possible to express the resulting reduced equation in a form distinguishable from the Pauli master equation, thereby deriving with the elegant formalism of projection operators the same conclusion as that of the earlier work by Pauli [ 31 and Van Hove [ 41. We apply the same method to studying the transport properties of an electron moving in a one-dimensional random lattice described by a tight-binding Hamiltonian with a random distribution of site energies. The
explicit form of the Hamiltonian
H=Ho+AW,
is
(1)
where
(2) and AW=V~(~m)(m+l~+~m+l)(m~~. m
(3)
We have adopted a notation identical to that used by Zwanzig [2] to make it easier for the reader to relate the result of this Letter to the theoretical problem discussed by Zwanzig 121. Thus, we are considering the Hamiltonian terms which generate a quantum transition from a state ]m) to another state jm’) as a sort of weak perturbation compared to the diagonal part of the Hamiltonian. For Zwanzig these are transitions
0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO375-9601(97)00754-S
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between different eigenstates of the unperturbed part of the Hamiltonian. If we look at the same Hamiltonian from the perspective adopted by Anderson [ 5,6] to predict the phenomenon now known as Anderson localization, the states Im) are distinct sites of a lattice, in this case a one-dimensional lattice, and the transport is made to deviate from the ideal ballistic condition of a perfect crystal by means of a random distribution of the site energies, E,, = E + %I .
(4)
Here we are assuming that with changing site there is a fluctuation pnt around the common value E. We assume no correlation among different sites, namely (cp,,,~,,,~)= AS,,,,,,/
(5)
Notice that to define the correlation function (5) we are using the Gibbs picture, in spite of the fact that according to Anderson [ 61 the study of a disordered crystal should imply only a single realization of the random distribution of energies. However, since the approach to the master equation illustrated by Zwanzig rests on the quantum Liouville equation, and so on the adoption of the Gibbs picture, we adopt for convenience the same perspective, too. Of course there is a subtle difference between the following two ideal cases. The former is the study of single realizations of the random distribution of energies by means of the density matrix. Although done so as to emphasize the statistical nature of quantum mechanics, it would not conflict with the recommendation made by Anderson [ 61 to study a single sample with a single realization of the energy distribution. The latter case, providing the theoretical perspective adopted in this Letter, is the study of a statistical density matrix which is the average over infinitely many single statistical density matrices, each one corresponding to a given distribution of site energies. This perspective bypasses, in a given sense, the need of adopting an extremely complex Hamiltonian, with extremely high degeneracy, or with a high density of states, imagined by Zwanzig [ I ,2 1. We shall see that this perspective supplemented by the adoption of a generalized projection operator /7, makes it possible, in principle, to realize the coarse graining condition which according to Zwanzig is necessary to derive the Pauli master equation.
Letters A 238 (1998)
169-178
Furthermore, we make the assumption that the random distribution of the site energies follows the Cauchy prescription p(q)
= 1r lr y2 + p= ’
(6)
where p(q) denotes the probability that the energy of a given site fluctuates by the quantity p about the common value E. Note that the Cauchy distribution is one of the cases explicitly considered to study the tight-binding electron model ( 1) (see, for instance, Ref. [7]). According to the earlier arguments we consider the statistical density matrix ps defined by ps(r)
=
J
d-MW({cD})p({&r).
(7)
This means an average over the single realizations , pi,. . . each characterized by {+‘} = $‘l,p2~$‘3,... the density matrix p( {(p}, t). The weighing function corresponding to the random distribution of energy fluctuations, w({qo}), in accordance to the lack of memory (5), is given by W({P))
=
wn,-1~{40”,-I}~~n,~{SPm}~~nt+l~{~~~+l}~~~~
1 (8)
with the weighing distribution
of each site identical to
(6). We adopt the following projection IIp=
operator,
d{4 c J w({wl)
l~)(~lP~sP~l~)(4~
m
(9) We have adopted the symbol n rather than the more conventional P, since it is the result of two distinct procedures, the average over the realizations of energy fluctuations and the cancellation of the diagonal matrix elements prescribed by Zwanzig [ 1,2]. Note that this is proved to be a projection operator as a consequence of the fact that the weighing p(p) is properly normalized. We note that the projection operator I7 is essentially the same as that adopted by Kenkre [ 81 to study exciton dynamics. This is not fortuitous, since the motivation for the generalization of the projection operator is simiIar in both cases: it should help
A. Maua,
P: Grigolini/Physics
the process of irreversibility creation. In a paper by Kenkre and Knox [9] an illuminating discussion can be found of why the picture of Zwanzig must be supplemented with a coarse graining ingredient. This essential ingredient generates an element of irreversibility and makes it possible to obtain a generalized version of the master equation from which to derive, in the due limiting condition, the standard Pauli master equation. Our purpose here is to show that this irreversibility ingredient must be genuine and that much attention must be devoted to distinguishing real from apparent seeds of “irreversibility”. As we shall see, the random distribution of energies, here referred to as Anderson noise, is a false ingredient, though it has the key role of making easier and more effective the action of a genuine seed of irreversibility. We shall see that even a relatively weak but genuine seed of irreversibility can produce the Pauli master equation, and, hence, according to the perspective established in Ref. [ IO], statistical mechanics from mechanics. However, without the action of the genuine seed of irreversibility after a temporary condition of ordinary diffusion, the system would make always a transition to localization. Thus we can say that the purpose of the generalization of the projection operator is similar but not identical to that of Kenkre [ 81 and Kenkre and Knox [ 91. We introduce the operator II for the purpose of giving birth to an intermediate rather than a steady regime of irreversibility and statistical mechanics. This temporary condition of statistical mechanics is killed by the same physical condition from which it is generated. Using the generalized projection operator Z7 but essentially the same calculations as those illustrated by Zwanzig [ 1,2] and adapted to the case here under discussion, we find the following equation of motion,
;pn =- nt+tI c
dt’Bnnl (t - t’) [P,([‘)
- Prn(Ol *
(10) Note that
P,,(f) = (mbslm)
(11)
and E,,,(t)
= (n({ZZLexp[
x (1 - fl>Wz)(4}ln),
(1 - n)L(t
- t’)] (12)
171
Letters A 238 (1998) 169-I 78
where L denotes the commutator superoperator associated to the Hamiltonian ( 1) . As pointed out earlier, the disordered distribution of site energies has seemingly, according to Zwanzig [ 1,2], the key role of realizing a condition favorable to Fermi’s golden rule. One can render this condition as favorable as possible increasing the intensity of Anderson noise, namely the width y of the distribution (6). In other words, we make the important assumption YBV,
(13)
which has the beautiful consequence of rendering the interaction term of ( 1) much smaller than the unperturbed part, and consequently the superoperator associated to the interaction part, Lt , negligible compared to the unperturbed part of it, LO. Note that L = LO+ Lt and that, on the basis of ( 13) we make the assumption of replacing L with ,!.c in the argument of the exponential of ( 12). As an effect of this approximation, and only as an effect of it, we derive for the memory kernel the following simple expression, ZILexp[(l
-n)L(t--t’)](l
-Z7>Llm)(ml
= 2( V*/fi*) exp[ -2(y/fi)
It - t’ j 1
x (]m+
l)(rn - 11 -21m)(m]).
l)(~+
I]+ Im-
(14) The exponential nature of the resulting relaxation process is due to the choice of (6). According to Lee [ 111, in full agreement with the view expressed by Fonda, Ghirardi and Rimini [ 121, there is no room in quantum mechanics for a decay process which is exactly exponential. However, their warnings refer to a dynamical derivation from a well-defined Hamiltonian. It refers to a problem discussed at length by Zwanzig himself in his fundamental work [ 1,2] aiming at deriving from a rigorous microscopic approach the Pauli master equation: he established that the exponential behavior is only admitted in an intermediate region of times. Here, we are already making with Anderson a statistical assumption about the random distribution of site energies. Thus, under this assumption the exponential decay of (14) is an exact property. The counterpart of Zwanzig’s discussion within this context would be a condition where the fluctuations of the site energies do not result in a continuous
172
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P Grigolini/Plzysics
spectrum, so that the memory kernel would not be exponential. The case of dichotomous fluctuations is an example of this kind, resulting in fact in oscillations rather than exponential behavior. An imperfect realization of the Cauchy distributions, with finite rather than infinite moments, would result in the breakdown of the exponential behavior at long times, as stressed by many authors [ 1,2,11.12]. However, this breakdown, if it ever exists, does not have anything to do with Anderson’s localization, as will be made clear by the results of this Letter. For this reason, the assumption of Cauchy distribution of the site energies can be safely made. In other words, we plan to explore the intermediate region of times that according to Zwanzig should not conflict with the realization of the Pauli master equation, and we plan to show that, in spite of traditional wisdom, a breakdown of ordinary statistical mechanics occurs much earlier than the breakdown of the coarse graining condition. This is due to the fact that Anderson noise is not sufficient to realize the second-order approximation behind ordinary statistical mechanics. The memory kernel of ( 14) leads at long times to the Pauli master equation
which, in turn, implies Brownian with the second moment given by (x’(t))
= (2V2/riy)L%.
motion
diffusion
(16)
Note that
(X2(f))= L2-yP”,mn2
9
(17)
“I
where L is the distance between two nearest-neighbor sites. The standard result of (16) is obtained adopting the assumption of intermediate asymptotics [ 131, namely, as well known [ 141, it seems to imply that many sites are occupied. However, from the earlier derivation it is clear that Eq. (15) is already valid at times t B h/y. As we shall see, it is broken at the much larger time scale t x h/V and throughout the whole time interval ranging from t = 0 to f M h/V the system population of a distribution originally located at the site m = 0 remains virtually confined in
Letters A 238 (1998)
169-178
the lattice region spanned by the sites m = 0, m = 1 and m= -1. Apparently the result of Eq. ( 15) is a correct consequence of ( 13) and of the fact that the coarse graining over the infinitely many energy levels corresponding to a transition from the site Im)to the site Jm+ 1)implies the realization of the conditions for the validity of the Pauli master equation. Why should the random distribution of site energies, supplemented by the suitable Gibbs picture of Eq. (7), not produce the same coarse graining effect as those directly or indirectly assumed by Zwanzig and Kenkre? It is difficult to answer this question only on the basis of the mathematical properties of the memory kernel E. This is so because, in general, to derive its exact expression is beyond the range of an analytical treatment. Furthermore, according to Zwanzig [ 1,2] the Pauli master equation rests on the second-order approximation to B and on the assumption that the breakdown of its irreversible behavior takes place at times much larger than those of experimental interest. There is no hint, however, on how to establish the physical conditions necessary to make legitimate the adoption of the second-order approximation. Yet, if a numerical study is made of the problem in the correspondence of the condition ( 13), with the wave function initially localized in the site m = 0, the result found is that illustrated by Fig. 1. This means that, after an earlier transient process, not visible in the long-time scale of these figures, and which might or might not correspond to the prediction of ordinary statistical mechanics, see Eq. (16), the second moment ( 17) of the quantum mechanical distribution shows fluctuations about a time-independent value. In the light of the well-known arguments used by Anderson [5,6] this result is not surprising, and it is a consequence of the fact that disorder, in addition to realizing the coarse graining helping the validity of the Pauli master equation, also establishes the condition for the occurrence of Anderson localization. We are not aware of any discussion in the literature of the reason for the failure of the Pauli master equation, in spite of the existence of conditions ensuring the necessary coarse graining, but the subtly related discussion of Kubo et al. [ 151 on the molecular origin of diffusion, recovering the result of an earlier work by Kirkwood [ 161. These authors proved that the process of Brownian diffusion implies an ineluctable form of
A. Moz,m. P: Gri@ini/Physics
173
Lerters A 238 (I 998) 169-178
0.016
0.012 A C “x V
0.008
0.004
3
I
I
40
80
I
I
160
120
I
+
200
t
Fig.
1. The
second moment (x2(f)),
Eq.
( 17),
as a function of time in the long-time region. The dashed line is the result of the numerical
calculation. The full line denotes the theoretical prediction, based on the numerical integration of ( 18). The pammeters used are y = 100 and V = I.
T 0.016
1
0.012 A =;‘
V t
0.008
0 o”----I-
I
I
40
120 8O
Fig. 2. The second moment
(x*(r)),
calculation
lattice. The full line denotes the theoretical
with a three-site
The parameters
Eq. ( 17). as a function
used are y = 100 and V = I.
160
t
of time in the long-time prediction
region. The dashed line conesponds
( 18). The corresponding
integral
to the numerical
is evaluated numerically.
174
A. Mazza. I? Grigolini/Physics
localization at long times. In a quantum mechanical context, we are only aware of the fact that Zwanzig pointed out [ I ] that there exists a time, tt, beyond which the continuum approximation, responsible for relaxation properties, is broken and the discrete nature of quantum mechanics is perceived again. This time can be rendered so large as to become infinite when compared to experimental times. This important time, however, must not be confused with the time at which localization occurs. The purpose of this Letter is to prove that the Anderson localization has to do with a more crucial condition, which can be realized at finite times even when the time tl is virtually infinite, and this condition implies that the second-order approximation behind the master equation approximation is overcome. To shed light on this important aspect we note first of all that when condition ( 13) is adopted, the amount of population entering states different from IO), 1I) and 1- I) is negligible. This can be assessed by comparing the many-site result of Fig. 1 to the three-site result of Fig. 2. Thus, rather than using a computer we might limit ourselves to diagonalizing third-order matrices. We decided to make our discussion still simpler by exploring the case of only two sites, the dimer. This can be justified by assuming that, as an effect of a strong Anderson noise, the transition from the site lm) to the site lm + I) is statistically independent of that from the site (m) to the site Irn - 1). Under this assumption we can derive an expression for the time evolution of p,,,(r), P,~,+, (r) andp,,,_t (t) and from this wederive, assuming that p,,(O) = I and P,,,+I (0) = p,,,-I (0) = 0. 00
(.G( t)) =
16V*L”y 7r
1
dx I
I
x* + 4y* x* + 4v2
-m
x sin*[ (x7 + 4V’) ‘/*t/2fi]
.
(18)
This formula was obtained by taking into account that the distribution of relative energies, pn, -q,,,*, , is still a Cauchy distribution, with a doubled width. Our numerical calculation, shown in Fig. 2, results in very good agreement with the theoretical prediction (18) and thus it fully supports the assumption that the backward transition is statistically independent of the forward transition. The expression of Eq. ( 18) is not yet a useful ana-
Letters A 238 (1998)
169-178
lytical formula. It can be made so as follows. We can split it into the sum of two terms. The first is obtained by neglecting the dependence of the harmonic function on V, as a consequence of ( 13). The latter aims at correcting this too strong approximation by noticing that in the space region close to the maximum of the Cauchy distribution, we can neglect the dependence of the harmonic function on n. Thus, we get
(x2(O)= -V[l
-$${Y[1 - exp(-2U/fi)
I
2L2V -exp(-2yr/fi)l}+----+-sm*(Vt/li). (19)
In Fig. 3 we compare the prediction (19) with the numerical treatment of the tight-binding model with infinitely many sites (virtually the exact solution of the problem) and we find that ( 19) reproduces satisfactorily the phenomenon of Anderson localization. In conclusion, Eq. (19), though approximate, is an acceptable analytical formula which makes it possible to shed light on the problem here under discussion. In Fig. 4 we zoom on the early-time region, with a standard diffusion-like character, and we find that the agreement between the theory represented by Eq. ( 18) and the numerical results is very good. Furthermore, we note that the slope of the diffusion process predicted by the Pauli master equation agrees very well with the early slope of the numerical result. Let us now reverse the standard perspective. Rather than using the memory kernel fl( t) to find, using the generalized master equation of ( lo), the time evolution of the second moment (x2(t)), let us do the reverse: from the time evolution of (x*(t)), let us derive an expression for the memory kernel s”(t) . Of course ( 18) would be more convenient from a quantitative point of view. However, the conceptual clarification that we have in mind is made more transparent if we adopt ( 19). Therefore let us use ( 19) to derive the expression for the memory kernel 9(t) . Setting
I”n,n’ =
we find
2K(t)(&,~+, + a,,,/_,) ,
(20)
A. Mazza.
f? Grigolini/Physics
L&ten
A 238 (1998)
169-178
17s
O.OlE
0.011 A z 5 0.008
0.004
0
I
40
I
8o
t
I
I
120
160
I >
200
Fig. 3. The second moment (am), Eq. (17), as a function of time in the long-time region. The dashed line is the result of the numerical calculation. The full line denotes the theoretical analytical prediction, based on the approximation to ( 18) given by ( 19). The parameters usedarey=lOOandV=l.
Fig. 4. The second moment (r’(t)), IQ. (17). as a function of time in the short-time region. The diamonds denote the result of the numerical calculation. The crosses denote the prediction of the numerical integration of ( 18). The dashed line denotes the slope of the diffusion process predicted by the Pauli theory (Eqs. ( 15) and ( 16)).
A. Maua, I? Grigolini/Physics Letters A 238 (1998) 169-178
176
K(f)
_Vexp( -2Vr/Fi)]
+ 2
cos( 2Vr/Fi) .
(21)
By inspection of (2 1) we notice that a second-order approximation would yield a result agreeing with ( 14). This means that the Anderson perspective goes beyond the conventional statistical mechanics of transport processes. The analytical result of Eq. (2 1) establishes a link between the Pauli and the Anderson perspective. The early-time regime is indistinguishable from that which would be produced by the Pauli master equation, and represents the predictions of ordinary statistical mechanics. At times of the order of /i/V, however, the influence of the negative tail of (21) becomes significant and the system makes a transition to the localized state. This picture, simple as it is, has the remarkable effect of making it very easy to establish the influence of an uncorrelated fluctuation. If the energy of the sites fluctuates in time as an effect of the interaction between the system and an external environment, it is easy to prove that the new form of the memory kernel is K(t)
=
v2y 1 ~y= - P @
V3 _ V exp( -2vt//%) 1+ cos(2Vr/~) ‘TTY@ x exp( -~t/fi)
.
> (22)
y > CJ > V ,
(24)
then, for all practical purposes diffusion takes place according to the predictions of the Pauli master equation and the resulting diffusion coefficient is distinguishable from the prediction of Fermi’s golden rule. This is reminiscent of the conclusions recently reached by Giovannetti et al. [ 171 who pointed out that the Markovian approximation is often equivalent to tacitly assuming the presence of an extra source of stochasticity, which has the role of making the approximation legitimate. In this specific case the extra source of stochasticity obeying (22) has the key effect of recovering the second-order structure therefore adhering to the general prescription of Bianucci et al. [ lo] : these authors have shown that the microscopic foundation of the linear response would be equivalent to the microscopic foundation of statistical mechanics. Similar results have been found using numerical calculation by Evensky, Scalettar and Wolynes [ 181. The nice aspect of (22) is that it results in the analytical prediction for the diffusion coefficient Y -_~ 2y+a
where g=akgT,
Eq. (21) is an important result which makes it possible for us to draw a conclusion on the intriguing problem of the conflict between the Anderson and the Pauli perspective. The Pauli perspective requires that the condition (13) is fulfilled. On the other hand, if the condition (13) is fulfilled, and an extra source of stochasticity of intensity (7 is used, and the following condition is realized,
(23)
ka is the Boltzmann constant and T is the temperature of the environment. This means that we establish a subtle connection with thermodynamics without discussing its microscopic derivation. The remarkable aspect of this seemingly trivial result is that in the case of very strong Anderson noise, an external fluctuation resulting in a rate c comparable to v/i, and thus extremely weak compared to the fast decorrelation process caused by the random distribution of energies, destroys localization and results again in a process of transport indistinguishable from that predicted by the Pauli master equation.
V 2v+a
(25) which at least qualitatively accounts very well for the numerical result of Ref. [ 181. As shown in Fig. 5, with increasing CTthe diffusion coefficient makes a fast transition from the vanishing value to a finite value and then, upon a further increase beyond the value 2fi, regresses to zero very slowly. Upon increase of the ratio y/V the regression to zero becomes so slow as to be virtually independent of the noise intensity, and the constant value kept, 2V2L2/y, coincides with the Pauli prediction ( 16). We think that the important results of this Letter are as follows.
A. Maua,
F! Grigolini/Physics
0 02
/ ,’
0.01
cl
:
1’
!i
----.___
-.
.-.-
o.oosl
ool_._i__
169-178
20
-‘----% 4o 0
Fig. 5. The diffusion coefficient noise intensity V, for different top to the bottom curve, vy is 1000,500.300,200, 150, 120,
~
80
100
D of FZq. (25) as a function of the values of the ratio V/y. From the given by the following sequence: 100.
(a) A simple theoretical picture is developed establishing the connection between two conflicting predictions: Anderson localization and ordinary diffusion generated by the Pauli master equation. (b) The breakdown of the Pauli condition is independent of the discrete nature of the systems studied. The time at which this kind of breakdown should occur, in principle, as pointed out by Zwanzig [ 1,2], could be rendered as large as we like by increasing the intensity of the coarse graining processes. In a sense, increasing the intensity of the Anderson noise, y, should produce the same effect. This is so much so because, as a result of a statistical average over infinitely many realizations of the distribution of the site energies (Pi,, see Eq. (4), the memory kernel becomes an exponential. An imperfect realization of this condition would result in the recurrence processes and in the violation of the conditions necessary for relaxation processes to be realized. The breakdown of the Pauli master equation resulting in quantum localization is provoked by the fact that the second-order approximation is not legitimate, in general. On the other hand, it can be rendered such by the addition of a genuine source of stochasticity. (c) The condition of adopting a strong Anderson noise is not enough for the Pauli master equation to be valid. It is a necessary but not sufficient condition. For the system to adhere to the prescriptions of Pauli, and to those of ordinary statistical mechanics [ lo], the whole system must be perturbed by a genuine
177
source of stochastic fluctuations. If this perspective is adopted, the problem of a rigorous foundation of the Pauli master equation from within ordinary quantum mechanics becomes equivalent to that of deriving randomness from quantum mechanics. This is a delicate problem still generating vivacious debates and controversies. For a recent discussion, see Ref. [ 171 and references quoted therein. A much more extended discussion would be required. Let us limit ourselves to remarking that if attention is focused on the macroscopic process and a strong condition of time scale separation between macroscopic diffusion and microscopic dynamics is set by making the coupling V smaller and smaller, then the intensity of the genuinely stochastic process necessary to recover the Pauli master equation becomes smaller and smaller, too. This makes the settlement of the problem more difficult rather than easier. It is so because a source of genuinely stochastic fluctuations, be it compatible or not with ordinary quantum mechanics, produces effects identical to those of a Markovian approximation arbitrarily made, and so theoretical predictions indistinguishable from those generated by conventional wisdom. In addition to shedding light on the apparent conflict between Anderson localization and the Pauli master equation, the result of this Letter affords further benefits. The equivalence between the quantum kicked rotator (QKR) and the Anderson localization is attracting the attention of an ever increasing number of authors [ 20-231. For a very recent example, see Ref. [ 241. On the other hand, since the QKR has a classical counterpart, the study of its dynamics is also done having in mind the problem of the correspondence principle. The classical kicked rotator does not result in any process of localization. The most widely accepted view is that a complete correspondence between quantum and classical physics is recovered if the QKR is made to interact with an external source of fluctuations [ 201. The numerical results show that an external fluctuation of even extremely small intensity can break localization and allow the system to diffuse with a diffusion coefficient almost indistinguishable from the classical diffusion coefficient in the absence of any perturbation. This way of realizing the Pauli master equation could be questioned if irreversibility has to be regarded as being intrinsic [ 2.51 rather than extrinsic to quantum mechanics. However, if some modifications are needed to make irreversibility really
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intrinsic to quantum mechanics, then the results of this paper support the view that they might be extremely weak [17]. In a recent paper [26] it has been pointed out that the conviction that a weak fluctuation produced by the environment can always be used to account for the transition from quantum to classical mechanics can be challenged by the phenomena of anomalous diffusion. The authors of Ref. [ 261 noticed that the kicked rotor in the presence of the so called accelerator islands according to the correspondence principle should exhibit an initial condition of superdiffusion. This happens to be the case. However, before localization the system shows a sort of transition to a quantum mechanically induced state of normal diffusion, and this physical condition is found to be surprisingly robust against the influence of environmental fluctuations. We think that this regime of normal diffusion is essentially the same as the Pauli regime discussed in this paper. Consequently, the time region where the correspondence principle was found to apply [ 261 is nothing but a sort of ballistic regime made more extended in time by the adoption of the special condition of acceleration state [26]. If the correspondence between the kicked rotor and the Anderson model is taken for granted, this time enhancement of the ballistic regime corresponds to a tight-binding Hamiltonian with long-range correlations among the site energy fluctuations. From a qualitative point of view the role of space fluctuation is equivalent to resealing the site distance L to a much larger, even macroscopically larger [ 271, value A. As a consequence the ballistic regime is enhanced, and the extension of the regime of anomalous diffusion is dilated. All this requires further investigation. However, we think that the proper theoretical tools to do that are already available: the methods introduced more than twenty years ago by Kenkre and co-workers [ 8,9] are still a very convenient way to approach this intriguing problem, and probably this can be done by properly extending the results of this paper. It has to be remarked that a unified model for the study of diffusion localization and dissipation has been very recently proposed by Cohen [ 281 by means of a Feynman-Vernon type of path integral, and it would be interesting to compare his theoretical approach to the generalized version of the Zwanzig projection method proposed by Kenkre and co-workers [ 8,9].
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