the master equation and the arrow of time

IL NUOVO CIMENTO VoL. 3 B, 1~. 2 11 Giugno 1971

The Master Equation and the Arrow of Time.

H. KI~IPS

Mathematical Physics Department, University o/ Adelaide - Adelaide

(rieevuto il 15 Settembre 1970; manoscritto revisionato rieevuto il 28 Settembre 1970)

S u m m a r y . - - The aim of this article is to reformulate the derivation of the master equation made by Fulinski and Kramarezyk, by introducing a boundary condition. This move facilitates physical interpretation. Comments will be included on the topic of the arrow of time.

1 . - I n t r o d u c t i o n .

Consider a sys tem M whose s ta te kets are the elements of a separable Hi lber t space =*F. The coarse-grained mas te r equat ion for M is

(1) pt(A)l . Pt(,J'> G(LI,,~)

A e A '

The t e rms in (1) are defined as follows. Pt<~ is the p robabi l i ty of measur ing the s ta te ke t of M a t t ime t in a subspace A of gg. A is a ssumed to be u closed l inear subspace, wi th finite dimension Gc~>: TrE(~>, where E(~) is the pro-

jection opera tor onto /1, and Tr is the opera t ion of tak ing the t race in gg. The set (Ec~)} is assumed or thogonal and complete in gg, so t ha t we can assume

the existence of a complete o r thonormal set of kets (}m)} in gg for which

m~A

where ~ denotes s u m m a t i o n over the subset of {Ira)} which spans A. The meA

operat ion Tr can then be defined b y T r A = ~ (mlA[m). The Born stat is t ical

11 - I I N u o v o C i m e n ~ o :B. 1 5 3

1 5 4 H. K m e s

in te rpre ta t ion asserts t ha t

(2) Pt(a) = Tr E~a)~(,~ ,

where ~ct) equat ion

is the densi ty opera tor for M at t ime t, which obeys the yon H e u m a n n

(3) ~,) = i[q,~, H] .

(For a discussion of the Born in te rpre ta t ion , and its re lat ion to m e a s u r e m e n t

see (~.~).) The G(aa, ) in (1) are s imply scalar coefficients, which m~y be t ime- dependent , and have the p r o p e r t y t ha t

G(a~, ) = 0 . .4

The der iva t ion of (1) f rom (3) will be made via the theory of t e t rads (a).

I n order to es tabl ish notat ion, a brief s u m m a r y of t e t r ad theory will now be

given. Consider the set of l inear operators A on o%f for which T r A * A ~ ~ . These operators are called Hi lber t -Sehmidt operators . The l inear i ty of A en-

sures t h a t A is uniquely defined b y the set of scalar ma t r i x e lements Am~----[A]m~= (m]A]n}. The set of A const i tutes the e lements of a Hi lbe r t

space ~ (Liouville space), wi th scalar p roduc t ( A , B ) = T r A * B and cor-

responding no rm HAll = TrA*A. A t e t r ad ~ ' is then defined as being an opera tor on ~ which l inear ly maps e lements of ~f onto e lements of ~q~ (*).

The t e t r ad norm [Idll will be defined in the usual way (4) as the 1.u.b.

(least upper bound) of the C for which HdAll < CIIAH, for any A e .~e. The l inear i ty of d ensures t ha t ~ is uniquely defined b y the set of scalar ma t r i x

elements---- dmn~,~,---- [J~Jmnm,n, ~-" (m[(s/lm'} (n'[)[n}. I t is easi ly p roved f rom the definitions t h a t

[ ~ ] . . . . ,~, = ~ d . . . . ~,,~,~,, r . $

T,8

�9 he te t rads to be used here are denned as fonows:

[Tr E.~)A] E.~, for any A e ~e .@A Y

~ , ) A U. ~,)A U* = . ,.~0) for ~ny A ~ s

(1) H. KRIPS: Nuovo Cimento, 60B, 278 (1969). (2) H. KRIPS: ~uovo Cimento, 61B, 12 (1969). (a) G. E~c~: Helv. Phys. Acta, 37, 533 (1964). (4) M. NAI~ARK: 2~ormed Rings (Groningen, 1964). (*) We use script Latin characters for both Hilber~ spaces and tetrads, but the context will make clear whether a space or an operator is intended.

T H E M A S T E R E Q U A T I O N A N D T H E A R R O W O F T I M E 1 5 5

where U,.,.) is the un i t a ry propagator in M defined by

J A = A

~ A = [H, A]

for any A e s

for any A e Lf,

where H is the Hamil tonian for M. The condition g'A e s required b y the last definition does of course const i tute a restr ict ion on H. This restr ict ion is simply met however by assuming H bounded, because the product of a t t i lber t - Schmidt operator with a bounded operator is a Hilber t -Sehmidt operator. (This is easily shown by choosing a representa t ion for which the bounded operator is diagonal.) The boundedness of H, its assumed l ineari ty and self- adjoint na ture entai l tha t the energy spectrum for H has finite upper and lower eut-offs (physically not an unreasonable requirement) . This in no

way prevents the energy spectrum from being continuous of course, and, in fact, cont inui ty of the energy spectrum will be referred to later. The boundedness of H also allows one to define U,.~~ exp [--ill(t--to)I, in the case of t ime- independent H (5) as a consequence of (3).

2 . - D e r i v a t i o n .

Our derivat ion of (1) f rom (3) follows the moves made in (6), except tha t

we make use of a boundary condition corresponding to a postulate of initial r andom phases. This boundary condition has the advantage of not only al- lowing a generalization of certain features of the derivation, bu t also facilitates the physical in terpre ta t ion. We will incorporate coarse-grMning into the derivation.

The derivat ion depends on two lemmata , which we now prove.

I Zemma 1. For each t there exists a bounded t e t r ad ~ , ) such that , for a given set of {P(A)},

!

A .t

where p(~) > 0, ~p(~) : 1, and A

~f,, = J + 2 ( ~ . ) - - J ) ,

where the form of Jff,) m a y depend on the choice of (P(a)}.

(5) T. KATO: Perturbation Theory o] ~inear Operators (Berlin, 1966). (6) A. FULINSXI and W. KRAMARCZYK: Physica, 39, 575 (1968).

156 H. ~mPs

Obviously Lemma 1 holds if and only if, for all t, ./V((~) ~, p(~)E(.4)V= 0 for .4

!

the given {p(,~)}, because in tha t case we can define .A'~ by specifying

.4

The ~dzi~ ) so defined, for a part icular {p(.4)} and a particular t, is not, of course, uniquely defined, since it is simply one of the bounded and linear extensions of ~n operator (qua tetrad) which transforms the given vector g~,)~p<,~)E<.4~ into ~ T<.4~ E(.4). .4

.4

~qow let us examine just what the condition ~C(~> ~ p(.4)E(~): 0 amounts to. A

Subst i tut ing from the definitions gives us tha t it is equivalent to

[Tr E<.4., U(,.,,, ~ p(~)E(~) U~,.,o>] E<~,) = 0,

~-" G(~,~ A '

i.e. Zp,.4, Z Z 0 for an A' . A m ~ A m'~ .4 t

Since at least one p(~)~= 0, this is equivalent to U<~.~o~mm'= 0 for all m', and for all m such tha t ]m} e A. This obviously contradicts the uni ta r i ty of U<t.t.> , which requires tha t

~. ]U(~.~o)mm'[ ~ - 1 for any m . m ~

Thus we see tha t Lemma 1 is proved in so far as we have shown tha t �9 A~t~ ~p~.4)E(.4)----0 leads to contradiction for any {P<d)}.

Note tha t if...V~ exists then it is a possible solution for .A~t) which is moreover independent of the choice of {p<~)}. The reason why, in general, we must al-

!

low .V m to depend on {p(~)} is that , although, in general, ..,f<~>~p<~)E<~>~O, .4

it is not necessarily the ease tha t ,I~+ ~ C(.4~E(,~)= 0 for any {C(~>} sueh tha t .4

~ C(.4, = O. (This reason is adequate because if there did exist a {C(~)} for which .4

~df~) ~ C(~)E<A , = O, and 2 C(~, = O, and if ~ > was independent of {p(~,}, then A .4

!

~'~(~ could not exist as a single-valued tetrad, since it would have to map the single veetor

into each of the ~(p<~)-i-2Cc.4))E<z~, for arbi t rary 2 small enough so tha t .4

p~.4)§ 2C(~)~0 for all A.)

T H E M A S T E R E Q U A T I O N A N D T H E A R R O W O F T I M E 157

Z e m m a 2.

where

= _ i~r ) .

Now from the definitions above,

~,~,m, = [ ~ ] m ' > <m'l ]~m = [ ~ A ] ~

Therefore,

~ r = ~ [-@~fA]mm = : [ T r E ( , 4 , ( ~ d A ) ] = [ T r ( ~ E,,4,)(C~A)] . m m ,4

Therefore, since the set {E(,4)} is complete and orthogonal,

= T r ~ A = T r [ H , A ] = 0 .

Now we use these lemmata to derive (1) from (3). Equat ion (3) is obviously equivalent to

(4) ~(t) = - - i < r

We now assume, as boundary condition at to,

(5) ~(~.) = ~p(,t)E~,4). A

This boundary condition is simply a postulate of initial random phases. Note tha t it does not include a postulate of equal a p r i o r i probabilities, nor does it apply continuously over un interval of t imes like the stosszahlansatz of classical statistical mechanics. Substi tut ing (4) into (3), and using Lemma 1, we get

Operating on both sides with ~ , and expanding out r gives

Equat ing coefficients of E(,4) on both sides, and using (2), gives

~t{,4} __ P t ( , 4 ' ) Z 2, ,4' ~t(A ) m'~,4

where m E A.

for some A .

158 n. xmPs

Summing both sides over all m eA , and lett ing

we get

~a~z~ m ~ A e

ibt(A) = ~ pt(4,~ G.~.~,>. A ' ~ ( A ' )

From Lemma 2 however we get ~G~A,~ = 0, and therefore get (1).

I t is worth not ing here tha t the G(~,> m a y depend on {p(~>}, which are the occupation probabilities of the various E(~> at an initial t ime to. Hence what we have derived is actually a family of master equations, of which the coefficients depend on initial conditions. This lat ter property does seem to be undesirable; but in defence of it two facts can be pointed out. First , when we approximate the terms in the master equation in order to get out a practical solution (see Sect. 3) the undesirable dependence on {p(~)} disappears. Second, it is t radi t ional ly postulated tha t the {p(~>} do have a fixed form at an initial t i m e - - t h a t given by the (~ postulate of equal a priori probabilities >>. These two

!

facts indicate t ha t the dependence of ~4z(t> on the {P(z>} is not worrisome in practice, even though it may offend aesthetically.

We also note tha t the above proof may be facilitated by taking TrE(,~> of both sides of (6), and redefining fr as --i~eJV'it~ (Lemma 2 still holds). We have not done this, in order to retain the notat ion of (6).

3. - Markoff and Pauli equations.

We now consider the relation between (1) and the Markoff equation. We will define the Markoff equation as being tha t version of (1) for which G~,~ is identifiable with the transit ion probabili ty per unit t ime from A to A' at t ime t, for any A, A'. The Markoff equation is physically significant because it can be taken (see later) as governing the diffusion process by which a system attains thermodynamic equilibrium.

Now whatever the relation between (1) and the Markoff equation m a y be, it is certainly not one of identi ty, because (1) is equivalent to (3), together with a boundary condition. And E~cE has shown (7) tha t the equations of quantum mechanics (qua (3)) are incompatible with a nontrivial (i.e. nonsta- t ionary) Markoff equation; and therefore, not only is (1) not Markovian, but if (1) is true for some system then the Markoff equation is false for tha t system.

(~) G. EMcg" Helv. Phys. Acta, 38, 164 (1965).

THE I~IASTER :EQUATION AND TIIE ARROW OF TIME 159

(E~[cH uses a different definition of the Markoff equat ion f rom the one used here, bu t the difference does not affect the proof.)

W h a t we do therefore is to introduce a new rela t ion (which is weaker t h a n t h a t of ident i ty) be tween two equations isomorphic wi th (1), and then show tha t this new rela t ion does indeed hold be tween the Markoff equat ion

and (1). We define <~ equat ion e reduces to equat ion e', to order A )> as meaning

t h a t e and e' are ident ical wi th (1) except tha t in each of t h e m the G<zz, ~ m a y

be replaced b y t e rms which differ f rom the G<zz, ~ only b y order A or less. The

point of in t roducing this new relat ion is tha t , even though e' m a y be false for M, and e is t rue , i t still m a y be possible to use e' to predic t s tates of M,

to wi thin a good approx imat ion , if e reduces to e' to order A for small A. I f this possibi l i ty is realized, we say t ha t e is well condit ioned for M, in so far as a small pe r tu rba t ion (to order A) in the coefficients of e only results in

small pe r tu rba t ions in the solutions of e. The assumpt ion t ha t there are equa-

t ions which are well condit ioned for any ac tual ly occurring sys tem is of course

not a physical ly implausible assumpt ion , as indicated b y the success of ap- p rox ima t ion techniques, pe r tu rba t ion theory, etc., in the physical sciences.

Assume tha t we can choose {Ira>} for which, not only do we have ~ct0>----

= ~ P<z~ E(~) for some to, bu t also we can split up H = Ho-4- ,~V, <m]'VIm) ---- O,

H o l m ) =- E ~ [ m } , Tr V* V = 1, and ~ is a scalar such t ha t 0 < ~ << 1. H, Ho, V are assumed t ime- independent . The point of this spli t-up is t ha t i t can be shown, for 2 small enough (r t ha t ] ] ~ ( : Y - - J ) ] ] < I and hence t ha t the yon N e u m a n n series (')

))=0

is convergent to ~-~. Therefore for A small enough, we can evaluate JV'-'. FuLI~S~ and KRA~AI~-

CZu (~) do this to order ~, obtaining, for m # m',

(7) Gm~,m, =

sin (t--to)(E,~-- Em.) Em # F~, ,

2,~=]V~m,l ~ ( t - - t o ) , E. , = B~ , .

6t . . . . is t hen given b y ~ Gm~:,m, = 0.

F r o m (7) we see t h a t for d # A '

G(LIA, )

sin (t - - to)(E~-- E~,) 2zlvoo, t' Eo--E ,

2A lVm ,l = ( t - to),

, Em # Era, ,

E,,, = E,~, ,

1 6 0 H. KI~IPS

to order A, where A<G(~)Giz,)22 for any A # A ' , and hence A - + 0 as 22-~0.

Therefore G(Az,~= Gi~,z) and G(zz,>>0 for A # A ' and t>to, to order A. G(zz) is given by ~G(z~,~= 0, which only however holds to order <NA, where N

is the range of A. Hence the degree of approximat ion in evaluat ing G(zz) m ay be as much as NA. The pair of equations which are isomorphic with (1), bu t

for which G(z~,~ are given by (7), and for which t > to and t < to respectively, will be called the (( Pauli equat ion )) and (~ anti-Pauli equat ion ~) respectively. Obviously for t>to and t < to respectively, (1) reduces to the Pauli and anti- Pauh equations respectively, to order A. (Note tha t G<zA> does not appear in (1), which is why the poor accuracy with which G<Az~ is evaluated above, does not affect the accuracy with which (1) reduces to the Pauli or ant i -Paul i equations.)

We will now show tha t the Pauli equation, under certain cont inui ty assumptions, and for (t--to) large, reduces to a Markoff equation, to order A.

Le t the subspace A' correspond to a range A ' E of the eigenvalues of Ho. l

Let @(~) be the number of ]m'} eA' for which H0[m'} ----Elm'}; and let V ~ be the root-mean-square value of the set of V ~ , for which Ira'} eA' and Holm'} ~- Elm'}. Then we have for the G(~,> in the Pauli equation, for A r A',

E 2 s i n ( t - - t o ) ( E ~ - - E ' ) ~EA B'eA'~ E m - - E ~

!

lgow we assume tha t @(~,~ and V~B, ~re continuous functions of E ' , and tha t l

A ' E is small enough so tha t bo th @(E,) =- @(~,) and V ~ , = V ~ , for all E ' e A' E, to within a good approximat ion. In tha t case we have

sin (t - - to)(Em-- E ' ) = E , . - E ' '

to order A. Let us also assume tha t the separations between the eigenvalues of H0 in A'E, are small enough so that , to order A,

= d P , ' sin ( t - to)( m- E')

E m - - E '

If we let d be a constant characterist ic of the separat ion between the eigen-

values of Ho in A ' E then this last condition amounts to (t--to)d<<~, because it is just this which guarantees tha t tha t there are sufficiently m a n y •' e A 'E, contained between successive nodes of [sin(t--to)(Em--E')]/(Em--E'), to just i fy replacement of the summat ion over E ' by an integral. This places an upper limit on the range of t > to.

Lastly, let us assume tha t if the width co~, of the in terval common to AE and A ' E is nonzero, then (t--to) is large enough to ensure a sufficiently

THE MASTER :EQUATION AND THE ARI:t0W OF TIME 1 6 1

quick cut-off of the above integrand so that, to a good approximation and for most E~,

co

2~ sin (t-- to)(F+~-- E') E, , - - E' ' E,~ ~ A' E , J

f dE' sin ( t - - to)(E,~-- E') _ o E ~ - - E'

B'EZJ'E 0, E,~r A'E .

In that case we get, to order A,

m~/I

If on the other hand eo~s,---- 0, then (~<sa,~ -~ 0 to order A is to be the requirement on (t--to). Therefore, in general, we get that, to order A, for {Ar

= Z �9

,m~A ~meA'B

This, however, is just the expression for the transition probability per unit time from A' to 4, to order A (8). Therefore, under the above assumptions, (1) not only reduces to the Pauli equation, but can be further reduced to the Markoff equation, all to order A. (l~ote that under further obvious simplifica- tion we get

in which form the symmetry property G<~z, ~ ----G<z,z> is more obvious.) The above set of assumptions places two contrary set of restrictions on

(t-- to). First we have the condition that (t-- to) << x/d, and then we have the condition that (t--to)>> ~/eoA~,, for any ~0~,r to ensure a quick cut-off of the integrand. The first condition may of course be trivially obeyed by Ho having a continuous spectrum. (In this ease the preceding theory would have to be modified in the usual way for passing from a discrete to a continuous spectrum.) In this ease (t--to) may even become infinite without invalidat- ing the derived form of G<~,~, because ia the limit as (t--to)-+ c~

sin (t-- to)(Em-- E')

(8) L. SCHI~'F: Quantum Mechanics (New York, 1955).

1 6 2 H. KRIPS

In general, however, Ho will not have a continuous spectrum (although it will be seen later tha t H must have at least a part ial ly continuous spectrum) and in this case there is indeed an upper bound imposed on t by ( t -- to) << zt/d. In order tha t the existence of this upper bound does not clash with the lower bound imposed by (t--to)>>z/ozz,, we can see tha t we require e%z,>> d, and hence AE>> d. Also one assumes tha t by the t ime (t ~ to) reaches the upper bound imposed by (t--to)<<n/d, the system will be close enough to equilibrium tha t the distinction between the Markoff equation and Pauli equation is negligible (i.e. of order A; see next Section). Therefore one can assume tha t for (t--to) large enough, the Pauli equation reduces to the Mar- koff equation, to order A.

The Pauli equation (and hence the Markoff equation) is of significance because a system for which the Pauli equation is true exhibits an approach to equilibrium. The s tandard method for showing this is to introduce

E.) = - - ~ p . , j ) In [ p"~) ] ;

and then show tha t J~(t~0 by subst i tut ing for Pt(~, from the Pauli equation and using the symmet ry property G(~ , )= G(~,~. Furthermore, it is shown tha t ~( t )= 0 if and only if~ for all A and A', where G(j~,)~ 0, we h~ve

p,~,) G~,r) '

at which point the system is said to be in equilibrium (9). At equilibrium, E~t~ has its maximum value; and therefore the system can be seen to exhibit a monotonic increase of E,) with t unti l a maximum value is reached at t ime tx, say, after which the system remains in equilibrium (t~ may be infinite).

We now define M to be a Pauli system if it obeys (3) with time-independent H, there exists a to for which ~c~,)= ~P~I m) (m[ (these two conditions

entail tha t M obeys (1), at least for G ~ ) = 1), and (1) is well conditioned for M with A small enough so tha t the p~(~) predicted by the Pauli equation are a good approximation to the actual p~(~) for t > to. (For the t ime being we leave as an open question whether p,~-=p(~) for m e A . ) What is meant here by (( good approximation ~) is, of course, open to question, but we will take it as entailing tha t if (~ is the relative frequency with which the actual Ptc~ deviates from the predicted p~(~ by more than e, then ~ is very small for some very small e, Under this definition, we see tha t if M is a Pauii system

(9) N. vA~r KA~P]~N: :Fundamental Problems in Statistical Mechanics, edited by COH]Z~ (Amsterdam, 1962).

THE M A S T E R E Q U A T I O N AND THE A R R O W OF TIME 163

it will be in quasi-equilibrium for t>tm, where (( M is in quasi-equilibrium af ter t,~ )) means tha t there is an almost uniform probabil i ty distribution among the Ira), for near ly all t>tm. (Mote tha t tm< t~, and hence t~, may be finite even though t~ is infinite.) Similarly, by considering the anti-Pauli equation, to which (1) reduces, to order A, for t < to, we see tha t there exists a t~ < to for which M is in quasi-equilibrium for t < t . ; i.e. for t < to, /~(t) < O.

There is a fur ther point to make here. If H has a purely discrete spec- t rum, then Q(t) is an almost periodic funct ion of t (lo). In part icular , for any

given t, if ~(~o)= ~p(~)E(~), where to<<t, then there exists a t' o for which A

, [I~,t;)--~P~,E,~)II can be made as small as we like, and to>> t. Therefore

A ( )ll in the ease of small ;t~ there exists a t o for which IIG ~ ; ~ - - ~ p ~ E c ~ ~ can zl

be made as small as we nke, because IIGII < c~ (s), and it can therefore be seen tha t we can reduce (1) to the ant i -Paul i equat ion to within the same order A as we reduce (1) to the Pauli equation. This is only possible, however, if bo th the Paul i and anti-Pauli equat ion reduce to the t r ivial /~(~)= 0, to order A. We therefore conclude tha t H must have an at least part ia l ly con- t inuous spectrum for H, if (1) is to reduce to the Yauli equat ion in a non- t r ivial way.

In part icular , we see tha t it is the open-ended nature of systems which al-

lows them to approximate the behaviour characterist ic of a solution of the Paul i equation, over some finite t ime interval ; because for closed systems (i.e. systems enclosed by infinite potent ia l barriers at finite distances apart , and therefore with discrete energy spectra)~ (1) cannot reduce to a Pauli equa- t ion to order A in any bu t the tr ivial case of p~(~ = 0, to order A. This conclusion is seen later to entail tha t a sat isfactory resolution of the classical re- currence paradox (~) must be made within the f ramework of quan tum mechanics, because, classically speaking, all boxes enclosing systems are impene- trable, and therefore effectively const i tute infinite potent ia l barriers.

Final ly it should be noted tha t the E(t) defined above is not analogous to !

the classical en t ropy E ( ~ ) = - f](t)[ln/(~)]dw, at t ime t, where ](t) is the distri- but ion funct ion in phase-space, and dw is an infinitesmal volume element in phase space. In par t icular the analogy does not hold because, whereas EI~ ) is invar iant under t ime-reversal of the state at t, E(~) is not, except as a special

case which can be considered excluded by the choice of boundary condition made to reduce (1) to the Pauli equation. This lack of analogy has the agree- able consequence tha t the classical reversibil i ty paradox (~) does not go over into quan tum mechanics, which is just as well, because the reduct ion of (1) to the Paull equat ion does not involve anyth ing as suspicious as the classical

(lo) I. P~RCIVAL: Journ. MaSh. Phys., 2, 235 (1961). (11) D. TER HAAR: BeY. Mod. Phys., 27, 289 (1955).

164 H. zmPs

s tosszahlansatz on which any paradoxes can be blamed. (The quan t i ty analogous to E(t) is in fact Tr~(t)lnQ,), which is, however, of no use in set t ing up

paradoxes , because of its t ime independence.)

4 . - I n t e r p r e t a t i o n .

So far we have pu t no in te rp re ta t ion a t all on the Ho and I V ; and there-

fore the above resul ts would app ly to a n y sys tem with a Hami l ton ian H. Now,

however, let us ident i fy Ho of the above fo rmal i sm with the Hami l ton ian of a

sys tem M consist ing of N nonin terac t ing molecules in a box whose sides are not infinite po ten t ia l barr iers . We then suppose t ha t there is a weak interac-

t ion (per turbat ion) between the molecules, which we represent b y the 2V of the above formal ism. Assuming M is a Paul i sys tem, and t h a t we observe it a t some t ime a f te r to, we see tha t , a f te r a per iod of t ime, M will r ema in in

quasi-equil ibrium. Hence the theory of the previous Sections serves to vali-

da te the so-called pos tu la te of equal a priori probabil i t ies for a Paul i sys tem

for nea r ly all t which exceed to b y more t h a n some character is t ic re laxa t ion

t ime ~ ---- tin - - to. F r o m this pos tu la te one can then derive, v ia the Darwin-

Fowler me thod (1~), the i m p o r t a n t resul t t ha t M spends near ly all its t ime in

a Maxwel l -Bol tzmann dis t r ibut ion (after t ime t,,) ; and hence we can define the pressure, t empera tu re , etc. for M in the usual way. I t is because of this t h a t the Paul i equat ion can be considered as governing the evolut ion to the rmo- dynamic equi l ibr ium (qua Maxwel l -Bol tzmann distr ibution).

Unfor tunate ly , the above pa ragraph raises more questions t han it answers. I n par t icular , how can the results (viz. the pos tu la te of equal a priori probabilities, for near ly all t > tin) be appl ied to sys tems in pract ice which are not

isolated over periods long enough to include a to on which we can impose the bounda ry condit ion ~cto~ = ~P~I m} (m]. This object ion can be got around, how-

m

ever, b y point ing out t h a t even though M is only isolated over a short period,

there is sense in ta lk ing abou t the s ta te which M would have had a t t ime to

had it been isolated for all t. Admi t t ed ly there seems something a trifle absurd

abou t imposing bounda ry conditions on M at a t ime when M did not exist as a sys t em in its own right, bu t this can be got a round artificially b y propa-

gat ing the bounda ry condition forward in t ime f rom to to a period when M is ac tua l ly isolated.

The second question raised b y the above resul t is why, for the sys tems we observe, is i t never the case t ha t J~t < 0; i.e. why are the to for the var ious

sys tems which we observe correlated in such a way tha t , for a n y t when we observe the systems, t > each of the to. Are we to accept this as a b ru te

(12) E. SC~I~SDINO~R: Statistical Thermodynamics (~ew York, 1960).

THE MASTER EQUATION AND THE ARROW OF TIME 165

fact about nature, as G~u~nhv~ suggests (~), or is there some more fundamental reason behind the correlation? ]~0LTZMA~ in his original discussion of this problem in 1885 (1~), took the latter line, claiming that the correlation was imposed by the laws governing the temporal evolution of the universe as a whole. I will t ry to vindicate Boltzmunn's view, making use of the Rei- chenbach concept of a branch system (~). My answer will also avoid the arti- ficiality of imposing boundary conditions on systems at times when they did not exist as systems in their own right.

Consider the case where M is a Pauli system for which p~ = p(~ for all m e A, where G(~) is large enough and the appropriate continuity conditions hold which guarantee that (1) reduces to a Markoff equation, to order A, for large (t--to). Such an M will be called a (( Markoff system >). ~ow by multi- plying both sides of the Markoff equation by d t > 0, we get

A'

(8b) T(z~. ) = G~A~.~ dt/G(A'),

A

A t A ' ,

where T(4~, ) is the probability that M is found in A at t + dt, given that it is in A' at t. According to (2), (3) (and assuming H time independent), we get

(9) T(~,) ~ [Tr E(d)(exp [--~Hdt]) E(~,)(exp [i l l dt])]

l~ow suppose that during some interval [tl, t2], t 2 > t l and (tl--to) large enough, a subsystem M1 of M is quasi-isolated from M (i.e. M1 is a (~ branch system >~ of M). To see what this means, let j (z , ~ 2 and ~ be the tIilbert spaces associated with -2~1,-71//2 and M, respectively, where M~+ M~----M, so that J r = ~f~XXZ~. Let Try, Tr~ and Tr be the operations of taking the trace in ~ , ~ and ~ , respectively. If ~(t), Q~t) and Q(t) are the density operators at time t, for M1, Ms and M, respectively, then, by yon Neumann's rule, ~l(t)----Tr~@(t), and ~(t)--~ Tree(t). Let I~, I~ and I be the identity operators in 5~f~, ~ and ~f, respectively. If M~ is isolated from M (as opposed to being (~ quasi-isolated )), which we will define shortly) then, by definition,

(lOa)

(13) A. GRUNBAV~: Philosophical Problems o] Space and Time (New York, 1963). (14) L. BOLTZMANN: Nature, 51, 413 (1885). (15) H. REIeHWNBACI~: The Direction o/ Time (Berkeley, 1956).

166 m xmPs

where H~,), H2(t~ are operators in ~ 1 , ~Yf2 respectively. From this one can deduce tha t (cf. (3))

~i(~, = - - i [H~(~), ~i.,] and ~ t , = - - i[H2(t), ~(t,];

and hence that , if {EI(~ } is a complete orthogonal set of projections on ~fl corresponding to the set of subspaces {6} of ~ , then T~(~,~, the transit ion probabil i ty from (~ at t to (~' at t § dt, is given by

(10b) . -EI(~,) ]

T~(~,~ = Trl EI(~ exp [-- ~H~(~ dt] ~ exp [iHl(t) dt]

where G~(t~ is the dimension of (~. 5Tow usually H cannot be split up as in (10a), and therefore (10b) will not

hold, but one can quite generally write

H = (H1(t~ X 12) -Y (I, X H2(t~) -Y V(t>,

where V,) is an operator in ~ . We consequently define M1 as being quasi- isolated from M during [tl, t,], if, for any t and t + dte[tl, t2], we have

( l la) Tr [ E(z) exp [ - iH dtj E(z'~G(z,---~ exp [iH dt]] ----

= Tr {E(z, [exp [-- iHl,t, dr] X I2] [I1 X exp [-- iH2(t, dt]].

�9 E,z,)G(z,, [exp [iHl,t)dt] X 12] [11 X exp [iH2(t)tit]I}

to order #; and we have

. E(~,) [ittl(t) dt]] TI(~,) ----- Trl E(~ exp [-- ~Hl(t) dt] G(~,---~ exp

to order tt', where # and re' are small positive numbers. We now assume tha t each subspace A can be split into the direct product

of two subspaces (~ and 7 in W1 and W2 respectively, so tha t

(12) E(~ --- EI(~ X E2cr) ,

for some (~ and 7, where the sets {(~} and {9'} are complete orthogonal sets of subspaces of ~ 1 and ~ s respectively, with corresponding sets of projection operators {EI(~, } and {]~(~,} respectively.

THE MASTER EQUATION AND THE ARROW OF TIME 167

In par t icular we have

(13) ~ E~(0)= 11 and ~ E,(v, = I , .

We let (Tx(t) and G,(r) be the dimensions of (~ and ? respectively, so tha t

G(A ) = GI(~)~2(V). Subst i tu t ing (2), (9), (12) and (11) into (8a) gives

Tr [(E~,6, • E~(~, ~(,+a,)] ----- ~ {Tr [(El(x, X E,(v,,) ~(,)]} �9 6' W'

�9 Tr {[E~(~, • E~(v,] [exp [-- iHl(,, at] • I , ] [I1 • exp [-- iH~(~)dr]]-

• E~(~,) [exp [ill1,,)dt] • 12] [11 • exp [iH~,,, dt] ] [ .

]

Summing both sides over 7, using (13) and yon ~ e u m a n n ' s rule, we obtain

Tr~ [.El,6)~l(t "~- dt)] --~ ~ Tr~ EI(~) exp [--iHx(t)dt]~(~,~ ex~, [iH~(t)dt] �9 6',y'

�9 {Tr2 [exp [-- iH~(t) dr] ]~(v'------2)G2(7,) exp [ill2(,) dt]] } �9

�9 {Tr [(E~(~,, • E~(v,,)0(o1} = 2 T~<6~,){Tr~ [El(y)0~(o]} , 6'

where, by (11b), Tl(6t. ) is the t ransi t ion probabi l i ty f rom 8' at t to ~ at t + dt, to order #'. Fur the rmore , by (2), we see tha t Tr~[El(6~m+d,)]=p~.,+d,(t) , where PLt+dt(6, is the probabi l i ty of finding M1 in (~ at t Jr dt. Similurly, Trl [EI(~,~I(t,] =pl.t(x~. Therefore, to order A,

(14) Pl.~+d~(~) : ~ TI(~'~Pl.~(e', ,

provided # and # ' are small enough only to affect te rms in (8a) to within order A. Since the TI(tt,~, ~ va 8', are obviously propor t ional to dt, to order A, (14) is equivalent to a Markoff equat ion to order A in the Tl(ot, ~.

We fur ther assume

! HI(, )=H~o+ r~c,), <m(.IVI~,)Im(1)> = O, <m, . IHlo lm( .> = E ~ . ( 1 ) . ~ 1 )

(the {[m,l)}} being a c.o.n, set in ~1, subsets of which span each of the El,t,), and I[ Vl(t)l] sufficiently small for t e [tl, t2], and al,~) large enough, and the up-

168 m ~RIPS

propr ia te cont inui ty conditions, such tha t

to order A, as in Sect. 3. In par t icular l'a(~t,)= TI(~,~ , to A. F r o m this last resul t we can then show tha t there is an approach to quasi-equilibrium as t increases in M1, as well as in M as a whole. This is done by defining

EI(~) = - - ~pl.~(~) In [ pl't(~) ] L GI(~) j

and showing tha t El(t+d~)--El(t)~O with the help of (14) (cf. Sect. 3). The conclusion is therefore tha t under certain plausible restrictions on M and /~1

(where M1 is a subsystem of the isolated system M), if M1 is a branch system of M during any t ime interval [tl, t2], and if (tl - - to) is large enough to ensure tha t M is governed by the Markoff equation, t hen M1 is also governed by the Markoff equation, and both M and M1 approach quasi-equilibrium (( in the same direction ~> (i.e. as t increases beyond to) (16).

We now make the obvious step of ident ifying M of the above formalism with a sufficiently large par t of our surroundings to ensure isolation; and iden- t i fy M1 during [tl, t2] as any branch system of M. The above conclusion then explains why we never observe J~t < 0 in the branch systems contained in our surroundings.

I will now summarize the essential points of the above argument , and comment on how they relate to Grunbaum's position (13). W h a t I have done is to take a sys tem M with Hi lber t space 5/z, and let M1 be any subsystem of M with t t i lber t space 5~1 so tha t ~%f= ~ 1 • I then showed tha t under cer- ta in physically not implausible cont inui ty assumptions, and by imposing a certain boundary condition (5) on the whole of M (including M1) at some t ime to in the dis tant past, the Markoff equations for M and Ms (~ run parallel in t ime ~) during the t ime tha t M1 is quasi-isolated f rom the rest of M (i.e. both M and M1 approach quasi-equilibrium as t increases). Final ly I suggested identi- fying M as a large bu t finite par t of the Universe, and M1 as any subsystem of it.

One possible criticism which may be advanced against the above argument is tha t the final step of ident ifying M and M1 is tagged onto the body of physical theory with the sole purpose and result of get t ing out the Reichenbach

postulate tha t the arrows of t ime in the various branch systems run parallel. As such, m y suggestions amount to nothing more than a fancy res ta tement

(1~) The Darwin-Fowler method can also be used to show that the condition of quasi- equilibrium mentioned here (although coarser than that used in (12)), still entails a Maxwell-Boltzmann distribution (provided G(A)< c~).

T H E I~AST]~R ] ~ Q U A T I O N A N D T H E A R R O W OF T I M E 169

of Grunbaum's proposi t ion tha t , as a ma t t e r of bru te fact, the Reichenbach postulate is t rue .

I would disagree with this criticism for several reasons. First , my suggestions make predictions other t han the Reiehenbach postulate. In part icular they predict t ha t M, if it remains isolated, will suffer Eddington~s thermodynamic dea th for (t--to) large; and tha t , for t ~ to, M exhibits the t ime-reverse of

the process which occurs for t ~ to. Second they suggest the cosmologically interest ing possibili ty tha t there exist other systems M', isolated from M, which have opposite t ime arrows to M. (The cosmological implications will not be dwelt upon here, if only because the theory employed above is non- relativistic, and hence out of place in cosmology wi thout patching up.) Third, my suggestions only involve a boundary condition at one t ime to on a finite sys tem M, whereas Grunbaum's suggestions involve conditions on the Uni- verse as a whole, over all times. Thus my suggestions have advantages of simplicity, ferti l i ty, and added fMsifiability, over Grunbaum~s suggestion; and would therefore qualify well ahead of Grunbaum~s suggestion, as a viable pa r t of theory of macroscopic phenomena. (Grunbaum's suggestion is simply an induct ive generalization, with no theoret ical ly significant ramifications, and as such it is doubtful whether it would ever qualify to join the ranks of a scientific t h e o r y - - j u s t as in the case of the inductive generalization (( all ravens are black ~).)

Even though my suggestions do have the above advantages over Grun- baum's , it m a y still be argued tha t I have made Reichenbach~s postulate dependent on a boundary condition and, as such, a ma t t e r of brute fact. In reply to this I can only question a usage of the t e rm (( brute ~) according to which a fact is not bru te if and only if it is deducible f rom the laws of physics wi thout any boundary conditions. My objections to this usage of (~ brute )) is tha t it ends up making brutes of near ly all the facts deducible f rom the laws of physics. This is because near ly any law of physics involves certain character- istic constants (h in quan tum theory, G in Newton 's gravi ta t ion theory, etc.) whose magnitudes are only determinable experimental ly, and hence become boundary conditions on the laws. Surely it is ra ther the case tha t a fact is not brute , if it is a tes t of a viable pa r t of some scientific theory ; and not just a tes t of some induct ive generalizat ion which has no par t in a scientific theory.

Under this more realistic definition of (( brute ~) it is apparen t tha t al though Grunbaum's suggestion makes Reichenbach's postulate a brute fact~ the sug-

gestions I have presented above do not. Therefore, in the sense tha t I have deduced Reichenbach's postulate f rom something more than just a brute

fact, I consider Bol tzmann 's position vindicated.

The author would like to acknowledge m an y useful discussions with Prof. C. A. HUgST; and to the referee of an earlier paper.

12 - I I Nuovo Gimento B.

170 H. KRIPS

@ R I A S S U N T O (*)

Si t e n t a di r i o t t ene re , p o n e n d o u n a eondiz ione al eon to rno , la de r ivaz ione del la m a s t e r e q u a t i o n svo l t a da Fu l in sk i e K r a m a r c z i k . Ci5 f ae i l i t a l ' i n t e r p r e t a z i o n e fisica. Si f a n n o dei c o m m e n t i a n e h e sul la d i rez ione del t empo .

(*) Traduzione a eura della t~edazione.

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(*) IIepeaec)eno pec)amtuefi.

the master equation and the arrow of time

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