On the assumption of initial factorization in the master equation for weakly coupled systems II: Solvable models

Annals of Physics 322 (2007) 657–676 www.elsevier.com/locate/aop

On the assumption of initial factorization in the master equation for weakly coupled systems II: Solvable models K. Yuasa a,*, S. Tasaki b, P. Facchi c,d, G. Kimura H. Nakazato a, I. Ohba a, S. Pascazio e,d b

a,1

,

a Department of Physics, Waseda University, Tokyo 169-8555, Japan Department of Applied Physics and Advanced Institute for Complex Systems, Waseda University, Tokyo 169-8555, Japan c Dipartimento di Matematica, Universita` di Bari, I-70125 Bari, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy e Dipartimento di Fisica, Universita` di Bari, I-70126 Bari, Italy

Received 23 February 2006; accepted 28 June 2006 Available online 9 August 2006

Abstract We analyze some solvable models of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. We apply Nakajima–Zwanzig’s projection method by choosing a reference state of the reservoir endowed with the mixing property. In van Hove’s limit, the dynamics is described in terms of a master equation. We observe that Markovianity becomes a valid approximation for timescales that depend both on the form factors of the interaction and on the observables of the reservoir that can be measured.  2006 Elsevier Inc. All rights reserved. PACS: 03.65.Yz; 05.30.d Keywords: Master equation; van Hove’s limit; Dissipation; Nakajima–Zwanzig’s projection method; Correlations

* Corresponding author. Present address: Research Center for Information Security, National Institute of Advanced Industrial Science and Technology (AIST), 1-18-13 Sotokanda, Chiyoda-ku, Tokyo 101-0021, Japan. Fax: +81352984522. E-mail address: [email protected] (K. Yuasa). 1 Present address: Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.

0003-4916/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2006.06.003

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1. Introduction The dissipative dynamics of a small quantum system weakly coupled to a large reservoir is described in terms of a master equation [1–4]. In the standard approach to this problem, one usually takes for granted that there are no initial correlations between the system and the reservoir. In the preceding article, hereafter referred to as Article I, we reconsidered this hypothesis in the framework of Nakajima–Zwanzig’s projection method [2,4–6] and proved that, in order to get a consistent description, the reference state of the reservoir should be endowed with the mixing property. In such a case, the initial correlations disappear in the Markovian (van Hove) limit and the system behaves as if it started from a factorized initial condition. Interestingly, one arrives at the same conclusions also for uncorrelated initial conditions. The mixing property is therefore crucial, and a ‘‘wrong’’ choice of the reservoir state provokes the appearance of secular terms. In this article, we shall focus on the hypotheses that are necessary for the derivation of the theorem proved in Article I. These will be scrutinized in terms of two exactly solvable models, in which an oscillator is coupled to a bosonic reservoir. This will enable us to describe the onset to Markovianity and the timescales at which Markovianity becomes a valid approximation. This article is organized as follows. We introduce notation and summarize previous results in Section 2. The first exactly solvable model is introduced in Section 3 and solved in Sections 4–6. The second model is briefly discussed in Section 7. Section 8 is devoted to a discussion and some concluding remarks. Two Appendices contain the details of the derivations. 2. Summary of previous results 2.1. Notation We start by briefly summarizing the main ideas of Article I and introduce notation. Let the total system consist of a ‘‘large’’ reservoir B and a ‘‘small’’ (sub)system S, so that the total Hilbert space can be expressed as the tensor product of the Hilbert spaces of the reservoir HB and of the system HS , Htot ¼ HS  HB :

ð2:1Þ

The Hamiltonian and the corresponding Liouvillian of the total system read H ¼ H 0 þ kH SB ¼ H S þ H B þ kH SB ;

ð2:2Þ

L ¼ L0 þ kLSB ¼ LS þ LB þ kLSB ;

ð2:3Þ

respectively, where k is the coupling constant. Clearly, ½H S ; H B  ¼ 0;

½LS ; LB  ¼ 0:

ð2:4Þ

We assume that the system Hamiltonian HS admits a pure point spectrum, and the system ~ m, Liouvillian LS is resolved in terms of its eigenprojections Q X X ~ mQ ~ m; ~ m ¼ 1; Q ~ n ¼ dmn Q ~ m: LS ¼ i xm Q ð2:5Þ Q m

m

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2.2. Nakajima–Zwanzig’s projection method Let q(t) be the density operator of the total system at time t, which has evolved from the initial state q0 qðtÞ ¼ eLt q0

ð2:6Þ

and is the solution of the von Neumann equation d qðtÞ ¼ LqðtÞ; dt

qð0Þ ¼ q0 :

ð2:7Þ

We are interested in the reduced dynamics of system S, which is described by the density operator of S, qS ðtÞ ¼ trB qðtÞ:

ð2:8Þ

In order to derive a master equation for qS(t), Nakajima–Zwanzig’s procedure makes use of the projection operators [2,4–6] Pq ¼ trB fqg  XB ¼ r  XB ;

Q ¼ 1  P;

ð2:9Þ

where XB is a certain reference state of the reservoir. Due to normalization trBXB = 1, it follows that P 2 ¼ P and Q2 ¼ Q. In particular, PqðtÞ ¼ qS ðtÞ  XB ;

QqðtÞ ¼ qðtÞ  qS ðtÞ  XB ;

ð2:10Þ

where we used the definition (2.8). In the standard derivation of a master equation, the initial state of the total system, q0, is taken to be the tensor product of a system initial state qS and a reservoir state qB, q0 ¼ qS  qB :

ð2:11Þ

This is an uncorrelated initial state. The reservoir is assumed to be at equilibrium (with respect to the reservoir free evolution LB ) LB qB ¼ 0;

ð2:12Þ

bH B and in most applications qB ¼ Z 1 is a thermal state at the inverse temperature b e 1 b = (kBT) with the normalization constant Zb. Then, the reservoir state qB in the uncorrelated initial state (2.11) is usually taken as the reference state XB. When the assumption of a factorized initial state is not justified, however, an ambiguity arises regarding the choice of the reference state XB. Indeed, if

q0 ¼ qS  qB þ dq0 ;

ð2:13Þ

where qS ¼ trB q0 ;

qB ¼ trS q0 ;

ð2:14Þ

and the term dq0 represents the correlation between system S and reservoir B, the relation between qB and XB is by no means obvious. We discussed this point in Article I and proved the following theorem.

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2.3. Theorem Given a correlated initial state q0, if (i) 0 is the unique simple eigenvalue of the reservoir Liouvillian LB corresponding to the eigenvector XB and the remaining part of the spectrum of LB is absolutely continuous (strictly speaking, the spectrum of LB can be defined only once the sector has been specified: in our case, the relevant sector is that containing the state XB); (ii) the initial (correlated) state of the total system is given in the form X q0 ¼ Kð1S  XB Þ ¼ Li ð1S  XB ÞLyi ; ð2:15Þ i

where K is a bounded superoperator (i.e., Li’s are bounded operators) satisfying the normalization condition trq0 = 1, namely, the initial state q0 belongs to the sector specified by 1S  XB, then van Hove’s ‘‘k2t’’ limit [1,7,8] of the P-projected density operator in the interaction picture, 2

ðkÞ

qI ðsÞ ¼ lim qI ðsÞ ¼ lim eLS s=k Pqðs=k2 Þ; k!0

is the solution of qI ðsÞ ¼ Pq0 þ

k!0

Z

ð2:16Þ

s

ds0 KqI ðs0 Þ

ð2:17Þ

0

with K¼

X

~ m LSB PQ

m

Q ~ m P; LSB Q L0 þ ixm  0þ

ð2:18Þ

or equivalently, d q ðsÞ ¼ KqI ðsÞ; ds I

qI ð0Þ ¼ Pq0 ¼ trB fq0 g  XB :

ð2:19Þ

That is, even if the initial state q0 is not in a factorized form, all correlations disappear in van Hove’s limit and system S behaves as if the total system started from the factorized initial state in (2.19) with the reservoir state XB. In addition, we showed that lim Qqðs=k2 Þ ¼ 0; k!0

ð2:20Þ

which makes the dynamics consistent, for no spurious term will develop in the master equation and no correlations can appear at later times: not only the initial state, but also the state at any moment t is factorized in van Hove’s limit. This supports the validity of the assumption of the factorized state, that is frequently applied in literature in order to derive a master equation [2–4]. The state of system S evolves according to the master equation (2.19), while the reservoir B remains in the state XB. It is important to note that, in van Hove’s limit, the reservoir state immediately relaxes into XB, which is the eigenstate of the reservoir Liouvillian LB belonging to its unique

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simple eigenvalue 0, and the spectral properties required in hypothesis (i) imply that it is a mixing state. The right choice for the reference state of the projection (2.10) is this mixing state XB, and such a projection is nothing but the eigenprojection of the reservoir Liouvillian LB belonging to the simple eigenvalue 0. This is the criterion for the reference state, that covers both equilibrium states and nonequilibrium steady states. Furthermore, as clarified in Article I, the interaction between system S and reservoir B is not essential to the factorization or the mixing; the total system is factorized and the reservoir relaxes into the mixing state through its own free evolution. The purpose of the present article is to scrutinize these issues in some explicit examples. In particular, we shall focus on: (a) the disappearance of the initial correlation, (b) the factorization of the total system, and (c) the relaxation of the reservoir into the mixing state, in van Hove’s limit. This will also enable us to discuss the relevant timescales for the factorization and the mixing. 3. An exactly solvable model Let us corroborate the above general arguments by scrutinizing an exactly solvable model. We consider an oscillator a coupled to a reservoir bx, whose Hamiltonian is given by (2.2) with Z 1 Z 1 y y H S ¼ xS a a; H B ¼ dx xbx bx ; H SB ¼ i dxðgx ay bx  gx abyx Þ; ð3:1Þ 0

0

where a(a ) and bxðbyx Þ are annihilation (creation) operators satisfying the canonical commutation relations ½a; ay  ¼ 1;

½bx ; byx0  ¼ dðx  x0 Þ;

ð3:2Þ

and gx is the form factor of the interaction. Even though system S has an infinite number of levels, and does not fulfill the conditions of the main theorem proved in Article I, the following explicit calculation will show that all the conclusions are still valid and therefore the theorem has a wider applicability. The above model is exactly solvable [9–11]. Indeed, the Heisenberg equations of motion for a(t) = eiHtaeiHt and bx ðtÞ ¼ eiHt bx eiHt read Z 1 _ ¼ ixS aðtÞ þ k aðtÞ dx gx bx ðtÞ; ð3:3aÞ 0

b_ x ðtÞ ¼ ixbx ðtÞ  kgx aðtÞ;

ð3:3bÞ

and by integrating the second equation and inserting it into the first, one obtains an integro-differential equation for a(t), Z t 2 _ ¼ ixS aðtÞ  k aðtÞ dt0 Kðt  t0 Þaðt0 Þ þ kBðtÞ; ð3:4Þ 0

with KðtÞ ¼

Z 0

1

2

dx jgx j eixt ;

BðtÞ ¼

Z

1

dx gx eixt bx ;

0

which is solved via Laplace transform to yield

ð3:5Þ

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K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

aðtÞ ¼ GðtÞa þ k

Z 0

bx ðtÞ ¼ eixt bx  k

t

dt0 Gðt  t0 ÞBðt0 Þ; Z

t

ð3:6aÞ

0

dt0 eixðtt Þ gx aðt0 Þ;

ð3:6bÞ

0

where GðtÞ ¼

Z CB

ds est ; ^ 2pi s þ ixS þ k2 KðsÞ

^ KðsÞ ¼

Z

2

1

dx 0

jgx j ; s þ ix

ð3:7Þ

_ þÞ ¼ CB being the Bromwich path on the complex s-plane. Note that G(0+) = 1 and Gð0 ixS . 4. A correlated initial state Any physical preparation of a quantum state is based on concrete physical procedures that cannot be controlled with complete accuracy. The real initial state is therefore unknown to some extent and in general has certainly some correlations built in. As an example of a correlated initial state, that has the advantage of being solvable, we take q0 ¼

1 ay ny b y e ðrS  qW Þeb na ; Z0

ð4:1Þ

with any positive operator rS of system S and a reservoir state qW ¼

1 by Wb e ; ZW

ð4:2Þ

where the summations over the reservoir modes x are implicit (and so henceforth as long as no confusion can arise) Z 1 Z 1 Z 1 ny b ¼ dx nx bx ; by Wb ¼ dx dx0 byx W xx0 bx0 : ð4:3Þ 0

0

0 ð0Þ

W xx0 is Hermitian ðW xx0 ¼ and consists of W xx0 , that is proportional to d(x  x 0 ), ~ xx0 , and the remaining square integrable part W W x0 x Þ

ð0Þ

~ xx0 ; W xx0 ¼ W xx0 þ W

ð0Þ

W xx0 ¼ W ðxÞdðx  x0 Þ:

ð4:4Þ

The states q0 and qW are normalized with the normalization constants Z0 and Z W , and nx is the relevant parameter to the initial correlation between system S and reservoir B. For nx = 0, the state (4.1) is obviously factorized, while it becomes a tightly correlated state for any nx „ 0, with correlations proportional to nx, as will be shown later in (5.7). y y Actually, the operator ea n b appearing in the initial state (4.1) generates a correlation between S and B: it changes the n-particle states b g1    b gn| vacæ of the reservoir into y y ea n b by g1    by gn jvaci ¼ ðby g1 þ ay ny g1 Þ    ðby gn þ ay ny gn Þjvaci, so that S and B are entangled for any nonvanishing value of nx. It is also possible to explicitly compute the correlation functions in the initial state (4.1): see the generating functional (5.4) and the correlation function (5.7) below. The choice of this particular form for the initial state q0 is mainly due to the fact that it allows us to solve the dynamics of the total system exactly and to discuss the correlation between system S and reservoir B. One can think of the

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663

correlations in (4.1) as engendered by a linear interaction of the form Hprep / a n b + h.c. in a rotating-wave-like approximation. As shown in Appendix A of Article I, the (normalized) reservoir state XB ¼

1 by W ð0Þ b e Z W0

ð4:5Þ

is mixing with respect to the reservoir dynamics driven by the Hamiltonian HB in (3.1), and the initial state q0 in (4.1) belongs to the sector specified by 1S  XB in the sense of (2.15). Indeed, q0 is the state perturbed from 1S  XB by a local operator L, q0 ¼ Lð1S  XB ÞLy ;

pffiffiffiffiffi 1 y y L ¼ pffiffiffiffiffi ea n b ð rS  LB Þ; Z0

ð4:6Þ

where LB ¼

1=2 1=2 qW XB

rffiffiffiffiffiffiffiffi  Z 1=2  Z W0  y bW ð0Þ ~ bW ð0Þ ¼ db b e b We T exp  ZW 0

ð4:7Þ

is a local perturbation such that qW ¼ LB XB LyB ;

ð4:8Þ

 denoting the anti-chronologically ordered product and T Z 1 Z 1 ð0Þ ~ bW ð0Þ b ¼ ~ xx0 ebW ðx0 Þ bx0 : dx dx0 byx ebW ðxÞ W by ebW We 0

ð4:9Þ

0

Even though the initial state q0 does not satisfy the hypotheses of the theorem proved in Article I, the following analysis extends the general results valid for a bounded perturbation. Note that the reservoir Gaussian state qW in (4.2) is fully characterized by the two-point function N xx0 ¼ hbyx0 bx iqW ¼ trB fbyx0 bx qW g

ð4:10Þ

and, as shown in Appendix A in Article I, it is also composed of two parts like W xx0 in (4.4), ð0Þ ~ xx0 : N xx0 ¼ N xx0 þ N

ð4:11Þ

The first term is the two-point function in the mixing state XB, ð0Þ

N xx0 ¼ hbyx0 bx iXB ¼ N ðxÞdðx  x0 Þ;

N ðxÞ ¼

1 eW ðxÞ

1

;

ð4:12Þ

which is the Bose distribution function when W(x) = bx, while the second one is a local function representing the effect of the local perturbation LB in (4.8). 5. Dynamics of the total system Since we are interested in the correlation between system S and reservoir B, we need to look at the state of the total system, q(t). In order to treat the reservoir degrees of freedom rigorously, we should restrict ourselves to reservoir observables whose expectation values are finite and discuss the state of the total system through a set of such expectation values.

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The relevant quantity for our discussion is therefore a characteristic functional of the state q(t), e.g., 

y

y

y

G½J a ; J a ; J b ; J yb ; t ¼ trfeJ a a eJ b b eb J b ea J a qðtÞg; ð5:1Þ R1 y  where J b b ¼ 0 dx J b;x bx , which is the generating functional of the expectation values of any anti-normally ordered products of a, a , bx, and byx and characterizes the state of the total system, q(t). It is important to note that we are not interested in infinitely extended objects, such as the Hamiltonian of the reservoir HB, since their expectation values are infinite: our targets are locally distributed objects. Such a formalization of the problem is reasonable, since we cannot observe infinitely extended objects in practice, and this is nothing but the starting point of the C*-algebraic approach to quantum statistical mechanics [12]. In the characteristic functional (5.1), the bandwidth of Jb,x represents the locality of the observables. Let us begin with the characteristic functional of the initial state q0 in (4.1), 

y

y

y

G 0 ½J a ; J a ; J b ; J yb  ¼ G½J a ; J a ; J b ; J yb ; 0 ¼ trfeJ a a eJ b b eb J b ea J a q0 g; which, in the coherent-state representation (Q-representation [4]) Z 2 2 da 0 2  0 jaihaj ¼ 1S ; ajai ¼ ajai; haja0 i ¼ ejaj =2ja j =2þa a ; p

ð5:2Þ

ð5:3Þ

is evaluated as G 0 ½J a ; J a ; J b ; J yb  Z 2 y y 1 da y    y hajrS jaieJ a aa J a heb ðnaJ b Þ eða n þJ b Þb iW ¼ eJ b J b Z0 p Z y 1 J y ð1þN ÞJ b d2 a y  y   hajrS jaiea n N na eðJ a þJ b N nÞa ea ðJ a þn N J b Þ ¼ e b Z0 p

ð5:4Þ

y

¼ eJ b ð1þN ÞJ b G S ðJ a þ ny N J b ; J a þ J yb N nÞ; where 

y

G S ðJ a ; J a Þ ¼ G 0 ½J a ; J a ; 0; 0 ¼ trS feJ a a ea J a qS g

ð5:5Þ

is the characteristic function of the initial state of system S and qS ¼ trB q0 :

ð5:6Þ

One can see from this characteristic functional how the parameter nx embodies the initial correlation. For example,   o d y  y  habx iq0 ¼   G 0 ½J a ; J a ; J b ; J b  oJ a dJ b;x J a ;J a ;J b ;J yb ¼0  Z 1 o2 G S ðJ a ; J a Þ 0  ð5:7Þ ¼ dx nx0 N x0 x oJ a oJ a J a ;J a ¼0 0 Z 1 ¼ dx0 nx0 N x0 x haay iqS : 0

K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

665

Let us discuss the evolution of the state of the total system qðtÞ ¼ eiHt q0 eiHt :

ð5:8Þ

The characteristic functional (5.1) of the state q(t) is easily computed in the Heisenberg picture 

y

y

y



y

y

y

G½J a ; J a ; J b ; J yb ; t ¼ trfeJ a aðtÞ eJ b bðtÞ eb ðtÞJ b ea ðtÞJ a q0 g ð5:9Þ

¼ trfeJ a ðtÞa eJ b ðtÞb eb J b ðtÞ ea J a ðtÞ q0 g G 0 ½J a ðtÞ; J a ðtÞ; J b ðtÞ; J yb ðtÞ;

¼

where Ja(t) and Jb(t) are functionals of Ja and Jb, defined via a (t)Ja + b (t)Jb = a  Ja(t) + b Jb(t). Note that the solutions (3.6) for a(t) and bx(t) are linear in a and bx, but do not contain a  or byx . The characteristic functional of the initial state q0 is given in (5.4) and Eq. (5.9) is further reduced to y

G½J a ; J a ; J b ; J yb ; t ¼ eJ b ðtÞð1þN ÞJ b ðtÞ G S ðJ a ðtÞ þ ny N J b ðtÞ; J a ðtÞ þ J yb ðtÞN nÞ:

ð5:10Þ

We thus obtain the exact characteristic functional of the state of the total system, q(t), G½J a ; J a ; J b ; J yb ; t ¼ eJ

y AðtÞJ

G S ðhy ðtÞJ ; J y hðtÞÞ;

ð5:11Þ

where J AðtÞJ ¼ y



J a

J yb

  Aaa ðtÞ Aba ðtÞ

Aab ðtÞ Abb ðtÞ



Ja Jb

 ;

y

h ðtÞJ ¼



ha ðtÞ

hyb ðtÞ

   Ja Jb ð5:12Þ

with Aaa ðtÞ ¼ k

2

Z

t

dt

0

Z

0

t

dt00 Gðt  t0 ÞUgg ðt0 ; t00 ÞG ðt  t00 Þ;

ð5:13aÞ

0

Z t dt0 ðK J b g  GÞðt  t0 ÞUgJ b ðt0 ; tÞ J yb Abb ðtÞJ b ¼ UJ b J b ðt; tÞ  2k2 Re 0 Z t Z t þ k4 dt0 dt00 ðK J b g  GÞðt  t0 ÞUgg ðt0 ; t00 ÞðG  K J b g Þðt  t00 Þ; 0

0

ð5:13bÞ J yb Aba ðtÞ ¼ k

Z

t

dt0 UJ b g ðt; t0 ÞG ðt  t0 Þ 0 Z t Z t  k3 dt0 dt00 ðK J b g  GÞðt  t0 ÞUgg ðt0 ; t00 ÞG ðt  t00 Þ 0

ð5:13cÞ

0

and ha ðtÞ ¼ GðtÞ þ kðG  K gðN nÞ ÞðtÞ;

ð5:13dÞ

J yb hb ðtÞ ¼ K J b ðN nÞ ðtÞ  kðK J b g  GÞðtÞ  k2 ðK J b g  G  K gðN nÞ ÞðtÞ:

ð5:13eÞ

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K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

We have introduced Z 1 dx fx eixt gx ; K fg ðtÞ ¼ 0 Z 1 Z 1 0 0 Ufg ðt; t0 Þ ¼ dx dx0 fx eixt ð1 þ N Þxx0 eix t gx0 ; 0

ð5:14Þ

0

where 1xx0 ¼ dðx  x0 Þ, and the convolution Z t dt0 F ðt  t0 ÞGðt0 Þ: ðF  GÞðtÞ ¼

ð5:15Þ

0

The characteristic functional of the total system (5.11) is exact and valid for any time t. The functions kðG  K gðN nÞ ÞðtÞ in ha(t) and K J b ðN nÞ ðtÞ in hb(t) describe how the initial correlation propagates, while Aba ðtÞ and kðK J bg  GÞðtÞ in hb(t) describe the correlation established through the interaction between system S and reservoir B. System S forgets its initial state through the decay of G(t) and approaches an equilibrium state via the action of Aaa ðtÞ, while Abb ðtÞ governs the relaxation of reservoir B into its equilibrium, i.e., the mixing state XB, as explained in the following. 6. The van Hove limit of the characteristic functional and discussion We are now in a position to discuss the van Hove limit of the evolution of the total system and demonstrate the validity of the general theorem proved in Article I: (a) the disappearance of the initial correlation, (b) the factorization of the total system, and (c) the relaxation into the mixing state, in van Hove’s limit. In order to discuss van Hove’s limit, let us remove the (rapid) oscillation of system S. That is, let us look at the characteristic functional of the density operator eiH S t qðtÞeiH S t in the scaled time s = k2t, 2

ðkÞ

2

G I ½J a ; J a ; J b ; J yb ; s ¼ G½J a eixS s=k ; J a eixS s=k ; J b ; J yb ; s=k2 :

ð6:1Þ

Then, the van Hove limits of the constituent functions (5.13) (Appendix A), lim Aaa ðs=k2 Þ ¼ ½1 þ N ðxS Þð1  eCðxS Þs Þ; k!0 Z 1 y 2 dx J b;x ½1 þ N ðxÞJ b;x ; lim J b Abb ðs=k ÞJ b ¼ k!0

ð6:2aÞ ð6:2bÞ

0 2

2

lim eixS s=k Aab ðs=k2 ÞJ b ¼ lim J yb Aba ðs=k2 ÞeixS s=k ¼ 0; k!0

k!0

2

lim ha ðs=k2 ÞeixS s=k ¼ eCðxS Þs=2 eiDðxS Þs ; k!0

lim J yb hb ðs=k2 Þ ¼ 0; k!0

ð6:2cÞ ð6:2dÞ

lead us to the van Hove limit of the characteristic functional (5.11) ðkÞ

G I ½J a ; J a ; J b ; J yb ; s ¼ lim G I ½J a ; J a ; J b ; J yb ; s k!0



¼ eJ a J a ½1þN ðxS Þð1e

CðxS Þs Þ

e



R1 0

dx J b;x ½1þN ðxÞJ b;x

 G S ðJ a eCðxS Þs=2 eiDðxS Þs ; J a eCðxS Þs=2 eiDðxS Þs Þ;

ð6:3Þ

K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

where 2

CðxÞ ¼ 2pjgx j ;

DðxÞ ¼ P

Z

1 0

dx0 Cðx0 Þ : 2p x  x0

667

ð6:4Þ

It is clear from (6.3) that (a) the initial correlation (or, equivalently, nx) disappears and (b) the state of the total system is factorized at all times in van Hove’s limit. Furthermore, (c) the local perturbation in the initial state q0, i.e., L in (4.6) (especially, the contribution of ~ xx0 ), decays out and the res~ xx0 , which appears in the characteristic functional through N W ervoir relaxes into the mixing state XB given in (4.5). The dynamics of the system in van Hove’s limit is exactly the same as that derived from the uncorrelated initial state tr{q0}  XB with the mixing state XB and it is actually possible to show that the density operator qI(s) characterized by the characteristic functional (6.3) obeys the master equation d q ðsÞ ¼  i½DðxS Þay a; qI ðsÞ ds I  12½1 þ N ðxS ÞCðxS Þ½ay aqI ðsÞ þ qI ðsÞay a  2aqI ðsÞay 

ð6:5Þ

 12N ðxS ÞCðxS Þ½aay qI ðsÞ þ qI ðsÞaay  2ay qI ðsÞa: This is nothing but the familiar master equation derived from the factorized initial condition with the reservoir in the thermal equilibrium state at a finite temperature, q0  qS  ebH B [3,4]. These points corroborate the theorem in Article I, suggesting that the mixing state XB, which is contained in the initial state q0, should be selected as the reference state of Nakajima–Zwanzig’s projection P. Note that the characteristic functional in van Hove’s limit, Eq. (6.3), approaches R1   s!1  dxJ b;x ½1þN ðxÞJ b;x G I ½J a ; J a ; J b ; J yb ; s ! eJ a J a ½1þN ðxS Þ e 0 ; ð6:6Þ which means that the equilibrium state (in van Hove’s limit) is qeq ¼

1 W ðxS Þay a e  XB ; Z eq

W ðxS Þ Z 1 ; eq ¼ 1  e

ð6:7Þ

i.e., system S relaxes into the equilibrium state with the same structure as that of the mixing state XB. As discussed in Article I, the state of the total system is factorized through its free evolution and the interaction between system S and reservoir B is not essential, which is also confirmed by the present exact solution. In the absence of the interaction, the exact characteristic functional (5.11) reads G½J a ; J a ; J b ; J yb ; t ¼ eUJ b J b ðt;tÞ G S ðJ a eixS t þ K J b ðN nÞ ðtÞ; J a eixS t þ K J b ðN nÞ ðtÞÞ and approaches t!1 

G½J a ; J a ; J b ; J yb ; t ! e

R1 0

dx J b;x ½1þN ðxÞJ b;x

G S ðJ a eixS t ; J a eixS t Þ

ð6:8Þ

ð6:9Þ

by Riemann–Lebesgue’s lemma [see the discussion below and Eq. (A.1b)]. The state is thus factorized into t!1

qðtÞ ! qS ðtÞ  XB ;

qS ðtÞ ¼ eLS t trB q0

ð6:10Þ

through the free evolution, which confirms the second part of the theorem in Article I.

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K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

The timescales of the factorization and the relaxation into the mixing state are clear from (6.8): the former is governed by the function  Z 1 Z 1 0  K J b ðN nÞ ðtÞ ¼ dx dx J b;x N xx0 nx0 eixt ð6:11Þ 0

0

J yb hb ðtÞ

in (5.13e), and the latter by the leading term of Abb ðtÞ in (5.13b), ! Z 1 Z 1 ð0Þ  ~ ðxþx=2Þð  J b;xþx=2 N UJ b J b ðt; tÞ ¼ UJ b J b ð0Þ þ 2Re dx dx J b;xx=2 eixt ;    xx=2Þ 

contained in

0

x=2

ð6:12Þ where ð0Þ Ufg ðtÞ

¼

Z

1

dx fx ½1 þ N ðxÞgx eixt :

ð6:13Þ

0

The timescales of the decay of these functions are determined by the bandwidths of their Fourier transforms, Z 1 ~ J ðN nÞ ðxÞ ¼ 2p K dx0 J b;x N xx0 nx0 ð6:14Þ b 0

for the former, and ~ J J ðxÞ ¼ 2p U b b

Z

1

~ ðxþx=2Þð  J b;xþx=2 N dx J b;xx=2    xx=2Þ 

ð6:15Þ

x=2

for the latter. Therefore, besides the spread of the initial correlation (nx) and of the per~ xx0 Þ, the size of the relevant reservoir observables turbation from the mixing state ðN (Jb, x) influences the timescales of the factorization and the mixing. In the weak-coupling regime they are very short compared with the timescale of the dissipative dynamics of system S, which is of order 1/k2 in the original time t. It is interesting to discuss what happens from a physical point of view. The initial correlations and the local perturbations propagate outwards from the region of interest (defined by the ‘‘size’’ of the relevant local observables) and never come back. What remains is the ‘‘unperturbed’’ state, that is the mixing state XB and is the stable ‘‘ground state’’ within the sector it specifies. The relaxation time of such a process is the time necessary for the disturbance to pass through the range of the interaction, that of the initial correlation, and the extension of the observable. It should be noted that, when we work in the interaction picture eiH 0 t qðtÞeiH 0 t , instead of eiH S t qðtÞeiH S t considered above in (6.1), we should duly take into account the time dependence of the observables in such a picture, X ðtÞ ¼ eiH 0 t X eiH 0 t , otherwise mixing is not observed. 7. A solvable model with counter-rotating interaction Let us look at another solvable example: the same model as the previous one (3.1) but with a different interaction Hamiltonian Z 1 H SB ¼ iða þ ay Þ dxðgx bx  gx byx Þ; ð7:1Þ 0

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669

containing counter-rotating terms. This model is also exactly solvable [9,11,13–15]. Let us only briefly sketch the main results. More details are given in Appendix B. The exact characteristic functional of the state of the total system, q(t), reads G½J a ; J a ; J b ; J yb ; t ¼ eJ

y

   y ðtÞJ AðtÞJ J y AðtÞJ J T A

e

e  G S ðh ðtÞJ þ J y  hðtÞ; J y hðtÞ þ hy ðtÞJ Þ y

ð7:2Þ

for the same correlated initial state as before, q0 in (4.1), where T denotes the transpose matrix, and !    ab ðtÞ  J a   aa ðtÞ A   y A    Ja y y  y    J AðtÞJ ¼ J a J b ; h ðtÞJ ¼ ha ðtÞ hb ðtÞ :  ba ðtÞ A  bb ðtÞ J b Jb A ð7:3Þ The details of these functions are given in Appendix B. This characteristic functional contains different types of terms from those in the previous example (5.11): the counter-rotating interaction provokes ‘‘squeezing.’’ In van Hove’s limit, however, these contributions disappear. Indeed, the van Hove limits of the constituent functions (B.4)–(B.6) of the characteristic functional (7.2) are lim Aaa ðs=k2 Þ ¼ ½1 þ N ðxS Þð1  eCðxS Þs Þ; k!0 Z 1 dx J b;x ½1 þ N ðxÞJ b;x ; lim J yb Abb ðs=k2 ÞJ b ¼ k!0

ð7:4bÞ

0 2



lim ha ðs=k2 ÞeixS s=k ¼ eCðxS Þs=2 ei½DðxS ÞDðxS Þs ; k!0

ð7:4aÞ

ð7:4cÞ

while all other limits vanish (see Appendix B.3), and one ends up with the same dynamics as the previous one (6.3) except for the frequency shift; D(xS) must be substituted with  S Þ, where DðxÞ  DðxS Þ  Dðx is defined in (B.10). The present example again supports the validity of the theorem proved in Article I: (a) the initial correlation disappears, (b) the state of the total system is factorized at all times, and (c) the reservoir remains in the mixing state, in van Hove’s limit. The effect of the counter-rotating interaction manifests itself only in the frequency shift; no other differences in the resultant dynamics from the previous example with the rotating-wave interaction [8]. Furthermore, the timescales of the factorization and of the mixing are governed by the functions K J b ðN nÞðtÞ and UJ b J b ðt; tÞ, respectively (see Appendix B.2); they are the same as those in the previous example [Eqs. (6.11) and (6.12)]. This also supports the general conclusion that the free evolution of the reservoir plays an essential role for the factorization and the mixing, but the interaction does not. The counter-rotating interaction gives rise to no significant effect on the factorization or the mixing. 8. Concluding remarks We have investigated two solvable models in the light of the general theorem proved in Article I. In both cases, we confirmed that when the initial state of the quantum system and the reservoir is not factorized, a correct application of Nakajima–Zwanzig’s projection method requires that the reference state of the latter be mixing. In addition, close scrutiny of the solvable models enabled us to focus on the relevant timescales. It turns

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out that an effective factorization of the state of the total system depends on the free dynamics of the reservoir (responsible for mixing) as well as on the interaction. Indeed, the free dynamics itself is sufficient to drive a complete factorization. Moreover, the timescales for mixing (that in turn govern the very applicability of the projection method in terms of the ‘‘reference’’ state of the reservoir) depend on the ‘‘size’’ of local observables of the reservoir: clearly, if one has access to information that is distributed over larger portion of the reservoir, one can in general detect finer deviations from mixing. The timescales at which Markovianity can be considered a good approximation depend on the structure of the local observables that one can measure, that is on the dimension of the (sub)system whose evolution one wants to describe. This conclusion, physically sound, is in some sense a strict consequence of the philosophy at the basis of the C*-algebraic approach to infinite systems (in the case at hand, the reservoir, whose observables one can measure). There are other very interesting problems that we have not analyzed and that are related to the general features of the evolutions when it is not permissible to consider a factorized initial state [11,13,15–19]. Among others, the problems related to the (complete) positivity of the evolution requires additional investigations [20–27]. Another interesting issue would be to discuss the applicability of this method to more articulated (and intriguing) thermodynamical situations, such as those of nonequilibrium steady states [28], shortly discussed in Article I (see Fig. 1 in Article I). It is indeed possible to apply the method we propose to discuss the relaxation of a system driven by a reservoir at a nonequilibrium steady state and this aspect will be discussed elsewhere. Acknowledgments This work is partly supported by the bilateral Italian–Japanese Projects II04C1AF4E on ‘‘Quantum Information, Computation and Communication’’ of the Italian Ministry of Instruction, University and Research, and 15C1 on ‘‘Quantum Information and Computation’’ of the Italian Ministry for Foreign Affairs, by the European Community through the Integrated Project EuroSQIP, by the Grant for The 21st Century COE Program ‘‘Holistic Research and Education Center for Physics of Self-Organization Systems’’ at Waseda University, the Grant-in-Aid for the COE Research ‘‘Establishment of Molecular Nano-Engineering by Utilizing Nanostructure Arrays and Its Development into Micro-Systems’’ at Waseda University (No. 13CE2003), and the Grants-in-Aid for Scientific Research on Priority Areas ‘‘Control of Molecules in Intense Laser Fields’’ (No. 14077219), ‘‘Dynamics of Strings and Fields’’ (No. 13135221), and for Young Scientists (B) (No. 18740250) from the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by Grants-in-Aid for Scientific Research (C) (Nos. 14540280, 17540365, and 18540292) from the Japan Society for the Promotion of Science. Appendix A. Prototypes of the van Hove limits The characteristic functional G½J a ; J a ; J b ; J yb ; t in (5.11) is expressed in terms of the functions given in (5.13). The van Hove limits of these functions fall into the following types: by taking the weak-coupling limit k fi 0 keeping s = k2t finite, one obtains ðiÞ GðtÞeixS t ! eCðxS Þs=2 eiDðxS Þs ;

ðA:1aÞ

K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

ðiiÞ K fg ðtÞ ! 0;

Ufg ðt; tÞ !

Z

671

1

dxfx ½1 þ N ðxÞgx ;

ðA:1bÞ

0

^ fg ðixS þ 0þ ÞK ^ gf 0 ðixS þ 0þ ÞeCðxS Þs=2 eiDðxS Þs ; ðiiiÞ ðK fg  G  K gf 0 ÞðtÞeixS t ! K

ðivÞ k2

Z

t

dt0 0

Z

ðA:1cÞ t

dt00 ðK fg  GÞðt  t0 ÞUgg ðt0 ; t00 ÞðG  K f 0 g Þðt  t00 Þ

0

^ fg ðixS þ 0þ Þ½K ^ f 0 g ðixS þ 0þ Þ ½1 þ N ðxS Þð1  eCðxS Þs Þ; !K ðA:1dÞ Z 1 Z t þ ^ fg ðix þ 0 Þ ½K dt0 UJ b g ðt; t0 ÞðG  K fg Þðt  t0 Þ ! dxJ b;x ½1 þ N ðxÞgx ; ðvÞ iðx  xS Þ þ 0þ 0 0 ðA:1eÞ ^ fg ðsÞ is the Laplace transform of Kfg(t) in where C(x) and D(x) are defined in (6.4), and K (5.14). Let us prove these results. (i) The van Hove limit of G(t), which is defined in (3.7), is the ordinary one [8]: Z Z d~s e~ss d~s e~ss k!0 ixS t GðtÞe ¼ ! ; ðA:2Þ þ ^ 2~s  ixS Þ ^ s þ Kðk s þ Kðix C B 2pi ~ C B 2pi ~ S þ0 Þ ^ which results in (A.1a), by noting the formula for KðsÞ in (3.7), þ 1 ^ Kðix S þ 0 Þ ¼ 2CðxS Þ þ iDðxS Þ;

ðA:3Þ

with C(x) and D(x) defined in (6.4). (ii) The van Hove limits in (A.1b) are just the long-time limits and are due to Riemann– Lebesgue’s lemma. The timescales of the decays are determined by the band widths of their Fourier transforms. See Eqs. (5.14) and (6.12). (iii) In terms of the inverse Laplace transform, the convolution in (A.1c) is written as Z ^ fg ðsÞK ^ gf 0 ðsÞ ds K ðK fg  G  K gf 0 ÞðtÞeixS t ¼ eðsþixS Þt ; ðA:4Þ 2^ C B 2pi s þ ixS þ k KðsÞ whose van Hove limit proceeds like in (A.2). ~ xx0 to Eq. (A.1d) through the function (iv) Notice first that the contribution of N 0 Ufg(t,t ), which represents the effect of the local perturbation LB for qW in (4.8), decays out in van Hove’s limit, since the van Hove limit of this contribution is a generalization of (iii) but with a vanishing prefactor k2 in (A.1d). Therefore, the main ð0Þ contribution comes from the mixing state through Ufg ðt  t0 Þ defined in (6.13): by noting that 1 ð0Þ 0 U ðt  t0 ÞeixS ðtt Þ 2 fg k Z 1 1 0 ¼ 2 dx fx gx ½1 þ N ðxÞeiðxxS Þðtt Þ k 0 Z 1 0Þ ~ ~ fk2 xþx ~ þ xS Þeixðss ¼ dx g 2 ~ S ½1 þ N ðk2 x ~ S k xþx xS =k2

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K. Yuasa et al. / Annals of Physics 322 (2007) 657–676 k!0

!

Z

1

0

~ Þ ~ fxS gxS ½1 þ N ðxS Þeixðss dx

1

¼ 2pfxS gxS ½1 þ N ðxS Þdðs  s0 Þ;

ðA:5Þ

Eq. (A.1d) is deduced via 1 k2

Z

s

ds

0

Z

0

s

0

2

0

ð0Þ ds00 ðK fg  GÞððs  s0 Þ=k2 ÞeixS ðss Þ=k Ugg ððs0  s00 Þ=k2 ÞeixS ðs s

00 Þ=k2

0

 ðG  K f 0 g Þððs  s00 Þ=k2 ÞeixS ðss 

00 Þ=k2

k!0

^ fg ðixS þ 0þ Þ½K ^ f 0 g ðixS þ 0þ Þ CðxS Þ½1 þ N ðxS Þ !K

Z

s

0

ds0 eCðxS Þs :

ðA:6Þ

0

~ xx0 decays out in van Hove’s limit, which is shown by (v) While the contribution of N ð0Þ generalizing (ii) and (iii), that of N xx0 , ! ðsþixÞs=k2 ^ K ds ðsÞ e fg dx J b;x ½1 þ N ðxÞgx 2^ s þ ix C B 2pi s þ ixS þ k KðsÞ 0 ! Z Z 1 ^ fg ðk2~s  ixÞ K d~s e~ss  ¼ dx J b;x ½1 þ N ðxÞgx ; 2 ^ 2~s  ixÞ ~s s  iðx  xS Þ þ k2 Kðk C B 2pi k ~ 0

Z

Z

1

ðA:7Þ yields (A.1e). The prototypes (i)–(v) lead to the van Hove limits of the components (6.2). Appendix B. Solution to the model with the counter-rotating interaction We summarize the exact solution to the model with the counter-rotating interaction (7.2). B.1. Heisenberg operators The exact solution to the Heisenberg equations of motion for a(t) = eiHtaeiHt and bx ðtÞ ¼ eiHt bx eiHt reads Z t dt0 F ðt  t0 Þ½Bðt0 Þ  By ðt0 Þ; ðB:1aÞ aðtÞ ¼ ½F ðtÞ þ k2 F ðtÞa þ k2 F ðtÞay þ k bx ðtÞ ¼ eixt bx  k

0

Z

t

dt0 e

ixðtt0 Þ

gx ½aðt0 Þ þ ay ðt0 Þ;

ðB:1bÞ

0

where B(t) is defined in (3.5) and F ðtÞ ¼

Z CB

ds s  ixS est ; ^ 2pi s2 þ x2S þ 2k2 xS LðsÞ

ðB:2aÞ

K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

F ðtÞ ¼ 

Z CB

with ^ ¼ LðsÞ

Z

673

^ ds iLðsÞ est 2 2 ^ 2pi s þ xS þ 2k2 xS LðsÞ

ðB:2bÞ

2x : þ x2

ðB:3Þ

1

2

dxjgx j

0

s2

Note that F(0+) = 1, F_ ð0þ Þ ¼ ixS and F ð0þ Þ ¼ 0, F_ ð0þ Þ ¼ 0. B.2. Characteristic functional The characteristic functional of the state of the total system, q(t), is given by (7.2), which is composed of the functions 1 2 2 Aaa ðtÞ ¼ ½1  jF ðtÞ þ k2 F ðtÞj  k4 jF ðtÞj  2 Z t Z t 2 þk dt0 dt00 F ðt  t0 ÞReUbgg ðt0 ; t00 ÞF  ðt  t00 Þ; 0

ðB:4aÞ

0

1 2 2 J yb Abb ðtÞJ b ¼UJ b J b ðt; tÞ  k2 ½jðK J b g  F ÞðtÞj þ jðK J b g  F ÞðtÞj  2 Z t þ 2k2 Im dt0 ðK J b g  ImF Þðt  t0 ÞUbgJ b ðt0 ; tÞ 0 Z t Z t þ 4k4 dt0 dt00 ðK J b g  ImF Þðt  t0 Þ 0

ðB:4bÞ

0

 ReUbgg ðt0 ; t00 ÞðImF  K J b g Þðt  t00 Þ; 1 J yb Aba ðtÞ ¼ kðK J b g  F ÞðtÞF  ðtÞ þ ik3 ðK J b g  ImF ÞðtÞF  ðtÞ 2 Z t 1 þ k dt0 UbJ b g ðt; t0 ÞF  ðt  t0 Þ 2 0 Z t Z t 3 þ 2k dt0 dt00 ðK J b g  ImF Þðt  t0 ÞImUbgg ðt0 ; t00 ÞF  ðt  t00 Þ; 0

0

 aa ðtÞ ¼ 1 k2 ½F ðtÞ þ k2 F ðtÞF  ðtÞ  1 k2 A 2 2

Z

t

dt0

0

Z

ðB:4cÞ

t

dt00 F  ðt  t0 ÞReUbgg ðt0 ; t00 ÞF  ðt  t00 Þ;

0

Z

ðB:5aÞ t

 bb ðtÞJ  ¼  1 k2 ðK J g  F ÞðtÞðK J g  F  ÞðtÞ þ ik2 J yb A dt0 UbJ b g ðt;t0 ÞðImF  K J b g Þðt  t0 Þ b b b 2 0 Z t Z t þ 2k4 dt0 dt00 ðK J b g  ImF Þðt  t0 ÞReUbgg ðt0 ; t00 ÞðImF  K J b g Þðt  t00 Þ; 0

0

ðB:5bÞ Z

t  ba ðtÞ ¼  1 k3 ðK J g  F ÞðtÞF  ðtÞ  1 k J yb A dt0 UbJ b g ðt; t0 ÞF ðt  t0 Þ b 2 2 0 Z t Z t þ ik3 dt0 dt00 ðK J b g  ImF Þðt  t0 ÞUbgg ðt0 ; t00 ÞF ðt  t00 Þ; 0

0

ðB:5cÞ

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K. Yuasa et al. / Annals of Physics 322 (2007) 657–676

 ab ðtÞJ  ¼ 1 k½F ðtÞ þ k2 F ðtÞðK J g  F  ÞðtÞ A b b 2 Z t Z t þ ik3 dt0 dt00 F ðt  t0 ÞUbgg ðt0 ; t00 ÞðK J b g  ImF Þðt  t00 Þ; 0

ðB:5dÞ

0

and ha ðtÞ ¼ F ðtÞ þ kðF  K gðN nÞ ÞðtÞ þ k2 F ðtÞ;

ðB:6aÞ

J yb hb ðtÞ ¼ K J b ðN nÞ ðtÞ  kðK J b g  F ÞðtÞ  2ik2 ðK J b g  ImF  K gðN nÞ ÞðtÞ;

ðB:6bÞ

 ha ðtÞ ¼ kðF 

ðB:6cÞ

K gðN nÞ ÞðtÞ

2 

 k F ðtÞ;

J yb  hb ðtÞ ¼ kðK J b g  F  ÞðtÞ þ 2ik2 ðK J b g  ImF  K gðN nÞ ÞðtÞ; where Ubfg ðt; t0 Þ ¼

Z

Z

1

1

0

0 0

dx0 fx eixt ð1 þ 2N Þxx0 eix t gx0 :

dx

ðB:6dÞ

ðB:7Þ

0

B.3. van Hove’s limit In addition to the prototypes (A.1), the following limits are necessary for the van Hove limit of the characteristic functional (7.2): by taking the weak-coupling limit k fi 0 keeping s = k2t finite, we have Z d~s k2~s  2ixS ixS t F ðtÞe ¼ e~ss 2 2 ^ 2~s  ixS Þ s  2ixS~s þ 2xS Lðk C B 2pi k ~ ðB:8aÞ Z d~s 1 k!0  S Þs ~ss CðxS Þs=2 i½DðxS ÞDðx e ¼e ! e ; þ ^ s þ iLðix C B 2pi ~ S þ0 Þ Z d~s k2~s k!0 F ðtÞeixS t ¼ e~ss ! 0; ðB:8bÞ 2 2 2 ^ s þ 2ixS~s þ 2xS Lðk ~s þ ixS Þ C B 2pi k ~ Z ^ 2~s ixS Þ d~s iLðk k!0 e~ss ! 0; k2 F ðtÞe ixS t ¼ k2 ðB:8cÞ 2 2 2 ^ s 2ixS~s þ 2xS Lðk ~s ixS Þ C B 2pi k ~ where þ 1  ^ iLð ix S þ 0 Þ ¼ 2CðxS Þ i½DðxS Þ  DðxS Þ

with C(x) and D(x) in (6.4), and Z 1 dx0 Cðx0 Þ  DðxÞ ¼ : 2p x þ x0 0

ðB:9Þ

ðB:10Þ

Then, the van Hove limits of the components (B.4)–(B.6) of the characteristic functional (7.2) yield (7.4). References [1] H. Spohn, Rev. Mod. Phys. 52 (1980) 569. [2] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, second ed., Springer, Berlin, 1995.

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