A remark on ergodicity, dissipativity, return to equilibrium

Physica 68 (1973) 587-594 0 North-Holland Publishing Co.

A REMARK

ON ERGODICITY,

RETURN P. J. M. BONGAARTS*

DISSIPATIVIT’Y,

TO EQUILIBRIUM #, M. FANNES’

and A. VERBEURE

Instituut voor Theoretische Natuurkunde, Katholieke UniversiteitLeuven, Leuven, Belgi? b

Received 11 April 1973

Synopsis Some notions of ergodicity, dissipativity and return to equilibrium for quantum systems are exhibited and compared in a simple situation of a pure state. As a by-product we develop a socalled “Liouville-space formalism” for systems of infinitely many degrees of freedom.

1. Introduction. The problems on quantum ergodicity’*2*3), dissipativity4s5Jj) and return to equilibrium’-lo) seem all to have something to do with the problem of a stable state becoming unstable under the influence of a perturbation and its asymptotic behaviour. However, the intuitive point of view and the interpretation of the different schools of thought, represented in refs. l-11 seem to be different. If one looks closer at the definitions and notions developed one gets the impression that one is talking about different things. However, it is very difficult to compare the different results directly because of the different formalisms used by the different schools. In this note we try to clarify a little this puzzling situation by treating in a mathematically rigorous scheme some of the essential ingredients of the various theories. We are looking at the asymptotic behaviour of the expectation value of a constant of the motion under the influence of a local perturbation. We consider the case of an ergodic system in the sense of ref. 1, rigorously formulated in ref. 11, and

* On leave of absence from the Instituut Lorentz voor Theoretische Natuurkunde, Leiden, The Netherlands. * Partially supported by the Foundation for Fundamental Research of Matter (F.O.M.), The Netherlands. ’ Aspirant N.F.W.O., Belgium. * Postal address: Instituut voor Theoretische Natuurkunde, Departement Natuurkunde, Celestijnenlaan 200 D, B-3030 Heverlee, Belgium. 587

588

P. J. M. BONGAARTS,

M. FANNES

AND A. VERBEURE

we are able to make a clear distinction between dissipativity and return to equilibrium under the influence of a local perturbation (theorem 3.2). As a by-product we show, the sense in which nonequilibrium statistical mechanics,can be seen as a scattering phenomenon (theorem 2.5) and by theorem 3.2 we also give the rigorous framework in order to study the dissipative character of the Friedrichs model as was done in ref. 5, together with a so-called “Liouville space” adapted to infinite systems. 2. Ergodicity-return to equilibrium. Let d be a C*-algebra of observables with unit element; an evolution is defined by the strongly continuous map t: R + at, a homomorphism of the additive group of the real line (R)into. the *-automorphisms of &‘; denote by d,, the C*-subalgebra of invariant elements of d, i.e., do is the set {A E d 1a,(A) = A for all t E R).Let co be a time-invariant state on &, i.e., o (+(A)) = co(A) for all A E d and for all t E R. 2.1. Definition. above.

We call a dynamical system any triplet (&‘, 01,, w) as defined

2.2. Definition (see ref. 11). A dynamical system (d, LYE, CO)is called ergodic if for all elements A E .& and at least one element BE ,zZOone has the following equality : At (0 (A*&, (4))

= b W*W/~

(B*B),

where Jz’, means the time average in the sense of ref. 12. Let zdo, &‘,, Sz, be, respectively, the GNS representation, representation space and cyclic vector induced by the state o on d (see ref. 13, proposition 2.4.4), i.e., for all A E d: o(A) = (Q,, q,,(A) Q,). The state u being invariant, there exists a unitary representation t --f UY, of the real line on &‘, such that %I (%(A)) = K%O (A) u_“,

u$2, = f-2,

for all

for all

AEJZI,

tER.

Let F be the orthogonal projection on the invariant vectors of X,, i.e., the projection on Za, with range {YE %,I UplY = Y for all t E R} then we have the following proposition. 2.3. Proposition. A dynamical system (d, mt, CO)is ergodic if and only if F is a one-dimensional projection operator. If this condition is satisfied then COis an extremal invariant state (o is extremal invariant if it cannot be written as a convex combination of other invariant states). Proof. See ref. 11, theorem 2. Let (d, a:, me) be a dynamical system, from now on referred to as the unperturbed

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system. Let V = V* E d be a perturbation, then the total evolution, or the perturbed evolution is described by the automorphism group (QtE a : for all A E &‘.

Because VE d, the sum converges absolutely, making it possible to check the group property of (& ER . By a calculation as one finds in scattering theory one deduces from (1) and (2):

Now we prove a first property on nonequilibrium statistical mechanics linking the time-asymptotic expectation value of a constant of the unperturbed motion under the total evolution, with some form of the Y-operators of scattering theory14). This property enhances the viewpoint of the Brussels school in so far as they look upon nonequilibrium phenomena as a scattering problem4). 2.4. Theorem. Let (~2, a:, mo) be a dynamical system and V = V* E& a perturbation; if Em,,, coo (&,(A)) exists for all A E do then

lim w. MA)) = oo(A) + (QoyG(n0

f-rrn

UK Al))-Qo),

(4)

where

PO, G+(~o(4) Qo) = lim w. f’rn

Proof.

(iids n.(A))

Insert (3) into w. (&,(A))

m. (qa~2 (A>) = o. (d’,+t,(A)) + w.

i ? ds G[K 0 .

As A E do this becomes: m. (c&O) = we(A) +

~0

i ids a,W, 0

4

.

a0,,,,,-sG41))

.

590

P. J. M. BONGAARTS,

M. FANNES

AND A. VERBEURE

Now the map t + w0 (i Ji ds a,(A)) being continuous and using the fact that the limit t + co exists, we get q.e.d.

Note that, that lim,,, exists, and was linked

as A E do, we also have o. (or,(A)) = w. (~~01:f (A)) ; hence requiring moo(a,(A)) exists is equivalent to requiring that lim,,, coo(a&, (A)) this is clearly a weaker condition than the one of ref. 9 which in turn to the asymptotic integrability.

3. Dissipativity. In this section we link the asymptotic behaviour, under the influence of a perturbation, of the expectation value of a constant of the motion with the so-called collision operator (see ref. 4). Consider again the unperturbed dynamical system (d, oly, u)~). As mentioned before (GNS representation) the automorphism group ap is represented by unitary operators U:: no (c&A)) = Upno (A) U!, . As the map t + &A) for all A E d is continuous, also the map t -+ coo(B*&’ (A)) = (no(B) Go, @no (A) Q,) is continuous for all A, B E ~4, hence t -+ Up is weakly continuous. Because the Up are unitary, the map is strongly continuous. By Stone’s theorem Up = e”F” where Ho is a self-adjoint operator on so ; Ho is called the hamiltonian of the unperturbed system. In order to avoid domain questions, we shall assume in this section that Ho is bounded. 3.1. Theorem. Let (&, a:, coo) be a dynamical system with bounded hamiltonian. With the notation of theorem 2.4 we get: (i) 7co(a:(A)) = eitLono (A) for all A E d, where Lo is an operator on the elements X E g (so) : LOX = HoX - XH, = [Ho, X] such that the spectrum of Lo is contained in the real line; (ii) no (a,(A)) =e irLzo (A) for A E & and again spectrum L c R and LX = [Ho + no(V), X]. Proof.

Starting with what is said above, no ($(A))

= enHo no (A) eeitHo. AS calculation,

Ho is bounded, Lo maps g(&‘,) into &?(Zo); by a straightforward

eitLo no(A) = f 0

(l/n!) (it)” LEno (A) = eitHono(A) emitHo = no (&4)

*

It remains to prove the spectrum Lo c R. It is also clear that eLoX = eHoXeeHo for all XE g(Xo). Define x = eLo, then x = x1x2 where x1X = eHoX and xzX = XeeHo. Clearly spectrum x1 = spectrum eHo c (0, co) where eHois now a multiplication operator on 8(X0); also spectrum xz = spectrum eeHo c (0, co). Because x1 and xz commute spectrum x c spectrum x1 x spectrum xz ; hence spectrum x c (0, co). By operator calculus on Banach spaces log x is well defined and the spectrum of log x c R. But log x = Lo. This proves (i).

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To prove (ii) note that the perturbation V E d, hence the representation zq, is also covariant for the evolution 01~(see ref. 9, theorem 1); in particular it is implemented by unitary operators U, = eitH where H = Ho + no(V): n,, (a,(A)) = eitHzo (A) emirH. As H,, and q,(V) are bounded, also His a bounded operator on Z0 and repeating the same arguments as in the proof of(i) we prove (ii), q.e.d. 3.2. Lemma. With the notation and assumptions of theorem 3:1, we have the following: (i) define for y > 0 I’,‘(v) n,(A) = if emyt eitL no(A) dt; then F,‘(J) n,(A) = -(iy

+ L)-l n&t).

(ii) denote by P the operator on ?8(Z0) defined by PX = FXF; F is the orthogonal projection operator on the subspace of &‘,, invariant under Up (range of F=(~E~~,U:~~=~ f or all t E R));then P2 = P, (1 - P)” = (1 - P) and the spectrum of PLP is real as an operator on the Banach space SI(SO). Proof. I’:(y)

For any XE W(S,,), because of the boundedness of L, X = i r ewyt f [(iL>“/n!] Xt” dt 0

0

m i(iL)

=L!

X&“dt=fyX&

This proves (i). Furthermore

.

_-

0.

0

Y?I+1

iy +

L

it is clear that PzX = F2XF2 = FXF = PX and

(1 - P)” X = (1 - P) [X - FXF] =X-FXF-FXF+F2XF2=X-FXF=(l-P)X

for all X E a(%,). Because multiplication is norm-continuous, as (1 - P) (~29(%?~)are Banach spaces, and

WZo) = lJ PWo)l + (1 - PI WWo)l * For all X E (1 - P) [W(&‘,)], PLPX = 0 and for X E P [W(%‘,)] PLPX

= [FHF, FXF]

or ePLP_y

=

eFIIF

xe-F”F

= [FHF, X],

P (&3(.%fo)as well

P. J. M. BONGAARTS,

592

M. FANNES

AND A. VERJ3EURE

Repeating the same argument as in the proof of theorem 3.1 yields that the spectrum of PLP is real, q.e.d. 3.3. Theorem. Let (9X,alp, oO) be a dynamical system and V = I’* E % a perturbation ; if lim,, m CC)~ (a,(A)) exists for all A E 5!&, , then (i) for A E alo lim w0 (&,(A)) = w,(A) - lim f-too

y-+0+

1

Dn,, PC

iy + L

[

%(&0

;

J

>

(ii) If furthermore ojo is a pure state, A = PA E no(!J) and the imaginary axis of the complex plane does not belong to the spectrum of (1 - P) L (1 - P) then lim(.!Z,, UtAU_t4,)

= + lim

f-trn

y-+0+

Proof.

ly (Qo 3 AQo). iy + !P(iy)

Let I’,$z, (A) = i ji ds no (a,(A)) for all A E ‘X It is clear that

lim (Go, I’: (no(A))Do) = lim PO, y+o+

t-al

U’,‘(Y)n0W1

DO).

When the left-hand side exists. Hence from lemma 3.2 lim (Qo, r,’ (n,(A)) Qo) = lim

’

-

y-to+

t-+m

iy

+

L

no(A)

7

If A E ‘2X0,then Lono (A) = 0 and Lno (A) = ‘x0 ([V, A]). From theorem 2.4 lim w. (a,(A)) = w,(A) t-c.2

-

lim

P 1

y+o+

iy + L

no(A)

this proves (i). Now we prove (ii): as A = PA, (i) gives P -

’

Pn, (A)

iy + L Now we exhibit the operator !P(iy); use the following expansion formula (well known in the Liouville-space formalism of the Hilbert-Schmidt operators5)) for the resolvent of the operators L, for y > 0 -=

1

L + iy

PJ + g

WI

’

Y (iy) + iy

[P + 9 WI + 9 (iA,

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where Y(iy)=PLP-PL(1

9(iy)

= -PL(l

-P)

- P)

(1 - P)L(l

(1 - P)LP, -P)

+ iy

1 (l-P)L(l-P)+iy’ 1

% (iy) = -

(1 -P)L(l Y (iy) =

1

(1 - P) LP, -P)+iy

1

(1 - P)

(l-P)L(l-P)+iy

and also PLP + $2 (iy) LP = Y(iy). One gets easily 1

P-

iy + L

y (iv)

LP =

iy + Y(iy) ’

hence lim (Q, , U,A U_,QJ

= (ii?,, A&),,) -

t-+00

lim

y+o+

!iy iy + K (iy) Although the manipulations and results,of theorems 3.1, 3.2 and lemma 3.2 bear much resemblance to what one meets in Liouville-space formalism, we stress that we are not working in this Liouville space but in the GNS representation induced by the state w. which in general describes an infinite system (thermodynamic limit) at least if one wants to satisfy the condition of theorem 2.5, 3.1 and 3.3. The operator Y(z) is called the collision operator; it is clear that it determines the asymptotic behaviour of the expectation value of a constant of the motion under the perturbed evolution. Finally we shall analyze the asymptotic limit for t + co of (Go, U,AU_ t, Q,) in terms of the collision operator for ergodic pure systems in the sense of definition 2.2. We formulate the result in the following theorem. 3.4. Theorem. With the notations and assumptions of theorem y(z) is analytic around z = 0 then (i) if Y(0) # 0 it follows that lim (Q,, U,AU_.,, 8,)

f-ra,

= 0;

3.3 (ii) if

594

P. J. M. BONGAARTS,

M. FANNES

AND A. VERBEURE

(ii) if Y(0) = 0 then lim (Q,, U,A U_,sZ,) = [I + u/l(O)]-’ (Q,, AQ,

t-+m

where Y’(0) = $

Y(z)

z=o

Proof. This follows in a straightforward manner from theorem 3.3, q.e.d. If Y(0) # 0, we found that the expectation value of a constant of the motion vanishes and we say that it behaves dissipativily. This is in analogy with the results of the Brussels schoo14,5*6). This school is very much interested in the nonvanishing of the collision operator. Their treatment of the Friedrichs models fits in theorem 3.4 with Y(0) # 0 if the di$crete part of the spectrum lies in the continuum, and with Y(0) = 0 if it lies outside of the continuum. It is also interesting to note that lim,,, (Q,, U,AU_,Qo) = (L$, ADO), if Y(0) = 0, Y’(O) = 0. This is the situation of return to equilibrium in which many other investigators are interested1p7*g*‘0). H ence the difference between dissipative systems4) and systems returning to equilibrium under the influence of a bounded (local) perturbation becomes clear. Acknowledgements. One of use (P.J.M.B.) thanks the members of the Instituut voor Theoretische Natuurkunde, University of Leuven, Belgium, for their kind hospitality. REFERENCES

1) Mazur, P., Physica 43 (1969) 533. 2) Siskens, Th. J., Physica 59 (1972) 639. 3) Verbeure, A., The Notion of Ergodicity in Solvable Models-Statistical Mechanics and Field Theory, Halsted Press (Jerusalem, London, 1972). 4) Prigogine, I., Non Equilibrium Statistical Mechanics, Wiley-Interscience (New York, London, 1962). 5) GrCcos, A. and Prigogine, I., Physica 59 (1972) 77; Proc. Nat. Acad. Sci. USA 69 (1972) 1629. 6) Rae, J. and Davidson, R., J. stat. Phys. 3 (1971) 135. 7) Lima, R. and Verbeure, A., Local Perturbations and Approach to Equilibrium, Marseille preprint. 8) Lanford, 0. and Robinson, D.W., Commun. math. Phys. (1972). 9) Robinson, D. W., Return to Equilibrium, Marseille preprint (1972). 10) Abraham, D.B., Barouch, E., Gallavotti, G. and Martin-LGf, A., Phys. Rev. Letters 25 (1970) 1449. 11) Sirugue, M. and Verbeure, A., Physica 65 (1973) 181. 12) Godement, R., Trans. Amer. Math. Sot. 63 (1948) 1. Gauthier-Villars et Cie (Paris, 1964). 13) Dixmier, J., Les C*-Algkbres et leurs Repr&entations, 14) Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag (Berlin, Heidelberg, New York, 1966), ch. X, sec. 5.