Physica A 177 (1991) 373-380
North-Holland
Models for intermediate time dynamics with
two-time boundary conditions
L.S. Schulman
Physics Department, Clarkson University, Potsdam, NY 13699-5820, USA
The consequences of conditioning a stochastic process at two times are examined, in particular
with regard to the behavior at intermediate times. These results clarify discussions on the relation
between the thermodynamic and cosmologicalarrows of time.
I. On the virtues of modeling
This is an article about the arrow of time. It is a field where the word "obviously"
comes in place of proofs and where people of good will - and intelligence - unresolvably disagree. In this arena there is a place for very simple models. By mapping profound, esoteric, cosmological assertions onto the abstract models whose analysis has
become high art in the hands o f Onsager, Fisher ~, and others, one gains two advantages. First, the obvious does become obvious. Discourse is transferred from words to
equations. Secondly, it encourages sharp analysis of the underlying assumptions that
gave rise to the irreconcilably "obvious" assertions.
The model that I will use here has its origins in dynamical implementations [ 1 ] of
the Fisher droplet model [2 ]. For present purposes that implementation has been so
abstracted that one is left with not much more than a general Markov process. Nevertheless, insights from the use o f those implementations for the study of metastability
will smooth our analysis.
2. Background on arrows of time and some "obvious" assertions
The manifest time asymmetry expressed by the second law of thermodynamics is
known as the thermodynamic arrow (TA) of time. The historically perplexing aspect
of TA was that it flew in the face of dynamical laws that were fully time symmetric.
Even CP violation has not led to any mechanism by which this tiny effect could induce the vast thermodynamic asymmetry (although the possibility remains open).
~1 In whose honor this article is written.
0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
374
L.S. Schuhnan /Dynamics with two-time boundary conditions
A second arrow, fully as universally pervasive, but more subtle in its direct consequences, was noticed only in the present century. This is the expansion of the universe, and it is designated the cosmological arrow (CA). CA carries less baggage than
TA, since even at the time of its discovery it fitted into the framework of general
relativity. It is thus tempting to suggest that TA is a consequence of CA and an early
proponent of this idea was Gold [3]. As discussed in ref.[4], I do not buy Gold's
specific arguments but l do accept the thesis itself. However - and here is where obvious is pitted against obvious - others find the idea without merit. In the next few
paragraphs l will give a hand-waving and not entirely consistent argument j o r the
thesis. Then I will quote some of the world's leading thinkers on why the thesis is
(obviously) impossible, Finally I will get to the point of this article and develop a
model in which the detailed workings of the "impossible" are made manifest.
3. C A ~ T A
What you first need is to put the world in a state of low entropy at some time in the
past. Rapid expansion can do this even if the universe did not begin that way. Matter
does not remain in equilibrium in the face of the expansion, and is caught in metastable states, like uranium, diamonds, stars and galaxies. The departure from these
metastable states we see as our pervasive entropy increase. But this is only the beginning of the story, as one would like to understand how the global entropy increase
becomes compulsory in every nook and cranny of the universe (with special consideration given to open systems). The local imposition of the global tendency was the
question addressed by Gold [3], who used an intermediate arrow, the radiation arrow ( R A ) , and developed the chain C A ~ R A ~ T A .
Although the foregoing argument has weaknesses (even some criticized in ref. [4 ] )
1 nevertheless believe it to have a core of truth.
4. The infamous "switch-over"
If C A ~ T A , then if the universe enters a contracting phase the thermodynamic arrow should change direction. To some this gives visions of clocks suddenly running
backwards and this has been considered a powerful argument against C A ~ T A . For
example Davies [ 5 ] says
Further difficulties with Gold's model emerge from more detailed considerations
of the "switch-over" that occurs at the point o f m a x i m u m expansion .... A complete
time symmetry about the point of m a x i m u m expansion would require an instantaneous changeover from diverging to converging waves (or having a mixture of both ).
L.S. Schulman / Dynamics with two-time boundary conditions
375
Weinberg [ 6 ] too is unhappy with this idea:
In particular, it is hard to see how time's arrow could be reversed just at the moment
when R (t) (the radius of the universe) reaches its maximum value...
Likewise, Penrose [ 7 ] finds the idea of the switch-over distasteful. He does note that
for processes that have reached equilibrium this is no problem, but when it comes to
systems out of equilibrium his picture of the switch-over era is as follows:
Otherwise one would have to envisage, it seems to me, a middle state in which
phenomena of the normal sort (e.g. retarded radiation and shattering watches)
would co-exist with phenomena of the "time-reversed" sort (e.g. advanced radiation and self-assembling watches).
I have aimed my arrows at these three individuals both because they are among the
respected savants on these matters and in the hope that their very distinction will
reduce their personal sensitivity. In section 5 I will examine in detail the consequences of two-time boundary conditions on intermediate time behavior. It will not
happen that you can barbecue in the garden while watching the rain fall up. As for
conventional notions, such an event remains unlikely.
Finally, I would like to comment on why it is so easy to find the switch-over business unattractive, and more generally to reject proposals in which entropy decreases.
I believe the source of the confusion to be the Dogma of Initial Conditions. This is
the humanly natural tendency to think in terms of initial conditions. Even Gold fell
prey to this error [4]. With two-time boundary conditions the mystery disappears.
Entropy decreases, but even a casual study of the dynamical framework shows this to
be a trivial consequence of the formulation.
5. Analytic calculation of the effects of two-time conditioning
Although I will use probabilistic language, the arguments also apply to classical mechanics with the sample space measure replaced by a measure on phase space. In this
presentation I will mostly treat time and space state-variables as discrete.
Let X, be the random position (or state) of the system at time t. Its distribution
function or propagator is the conditional probability
G ( x , t; y ) = Pr(X, =xlXo = y ) •
( 1)
We want the system to be in state c~ at t = 0 and state//at t = T. Call this event 8. Two
questions arise: What is the probability distribution of X, and what are the effective
transition probabilities? The basic transition probability is
W ( x , - y ) = Pr(Xt+ ~= x l Xt = y )
(2)
but with the remote conditioning, a different, effective object will appear. First con-
L.S. Schulman /Dynamics with two-time boundary conditions
376
sider the distribution function, which we call F(x, t; 8):
F(x, t: .~)-=Pr(Xt = x [ 8) ---' P r ( X , = x a n d YT=fllXo =ce)
Pr( X r = fl[ Xo =ce)
b Pr(X~ = f l l A ; = x ) P r ( X , = x l X o = o ~ )
Pr( X~ = fll Xo=o~ )
c G(fl, T - t ; x) G(x, t; a )
G(/~, T: a)
=
(3)
Equality a is the conditional probability identity. Equality b uses the Markov property. (This step could break down for classical mechanics if there is coarse graining at
intermediate times.) Step c uses time-translation invariance and invokes the definition of G.
To see the consequences of eq. (3), we will invoke the properties of G. In matrix
notation, ( W),,.= W ( x , - y ) , G is given by
G(x, t; y) = ( W ' ) ......
(4)
For the systems we consider there is a spectral expansion W = Z~.>~o2~,P:., where ,t~, are
the eigenvalues of 144 20>2t>~ .... and P), are the associated projections. Since W is
stochastic, ( 1..... 1 ) is a left eigenvector with eigenvalue 2o = 1. The corresponding
right eigenvector, p, is the equilibrium state of the system. We write 2 ~= e - J/~', so that
r~ is the longest relaxation time. It follows that
W ' = P o + e -'/'' (P, + ~ ) ,
(5)
with the remainder :# involving (22/2 ~)' and other projection operators. For eq. (3)
we require ( W ' ) ...... ( WT-')/~.,. and ( wT)Ij,. Note that (Po),,,= [p( 1..... 1 ) ]~,,=p~,.
This yields our first important - albeit "obvious" - result: Let T and T - t be large
enough to allow neglect of all but the leading term (Po) in (5). Then
F(x,t;?)=
(wT-')a*(W')"~
( WT)/,~
pp(W'),,~ - G ( x , t ; c ¢ ) .
(6)
P/J
F(x, t; ~) has lost all dependence on ,8. This is the "forgetting" of the future. For t
near T, define s=- T - t so that instead of (6) we have
F(x. T-s: 8) = ( W").~,~(
W~-'),,~
(I4,T),,
'
~G(fl, s;x) p~p/J= G ( x , s ; f l ) ,
(7)
where the last equality in ( 7 ) uses detailed balance. The distribution function o f x in
the reverse time variable s therefore looks exactly like the usual propagator, conditioned on the "initial" state//. Finally, for t ~ T/2, e -t/'' can be neglected throughout,
and we have F(x, t; 8) ~p,.. This says that the system is in equilibrium.
Things are more interesting when T/r~ is not enormous. Factoring out the constant
L.S. Schulman / Dynamics with two-time boundary conditions
./(-pp/G(~,
377
T; ~), we have
e(T_t)/z I
F(x,t;g)=Y
l+--P~ax+~
Pp
)
G(x,t;a),
(8)
where ~ now refers to terms decaying more rapidly than r~. To analyze (8), I will use
the experience gained from the studies of metastability in the droplet or cluster models
mentioned above [ 1 ]. What we need is the structure OfPl. For statistical mechanical
systems with long lived metastable states, p has extremely small values on configurations associated with the metastable state. The first "excited" state, the eigenvector
used in P~, has substantial weight on those configurations, although it also has opposite sign values on the "stable" configurations.
The upshot is that Qpx = (P~px/pp) has the following structure: Iffl is taken from
among the metastable configurations, Qax can be large. It will be especially large when
x is itself in the same metastable state. This implies that by requiring the final state of
the system to be a configuration associated with a long lived metastable state, that
system's distribution is distorted and tends to look like the configuration it is destined
to reach. If, in addition, ot (the initial state) is a similar metastable state, then the
system will hardly decay. As usual, the qualitative part of these conclusions could be
drawn easily. Also note that t = T / 2 has no special feature whatever.
The story for transition rates is similar and arises from a slightly more complicated
identity than eq. (3). Without going into details we have
l ~ v = Pr (X,+ l = x l X, = y and Xo = ot and X r = fl)
= W~,.
G(fl, T - ( t + 1 ); x)
G(fl, T - t ; y )
(9)
For variety, I will show what form (9) takes for Brownian motion. This is a continuum application, but that presents no problem. With diffusion coefficient D, the
propagator is G ( x, t; y ) = ( 4nDt ) -1/2 e x p [ - ( x - y ) Z/ 4Dt ] . From (9),
ff.xv= Wxy exp ( ( x - Y ) [ ½ ( x + y ) - f l ] )
2D(T-t)
,
(10)
where we assume the unit time interval is short compared to t and T. Inspection of
( 1O) shows that transition probabilities are biased toward getting the final (t = T)
answer ,6.
Finally, we deal with the t = T / 2 switch. Well, it just is not there. As pointed out
along the way, if the time scale of a process is short compared to T, by T / 2 it is in
equilibrium. Clock hands do not suddenly switch direction, because there is no clock.
On the other hand, for relaxation times comparable to T the system is further from
equilibrium than evolution under G ( x , t; c~) alone would have predicted. So your
clock never does behave normally. Its tendency to equilibrate is slower than an initial
L.S, Schulman / Dynamics with two-time boundary conditions
378
Three state evolution with intial conditions
1
0.9
0.8
0.7
/
/
0.6
0
0.5
0.4
0.3
•
0.2
_--
0.1
0
""
20
4()
60
80
100
time -->
Fig. 1. H i s t o r y o f a t h r e e - s t a t e s y s t e m with t r a n s i t i o n r a t e s ,v +.v: 0.2, +t,+ v 0.02, t ' + : : 0.05, z ,y: 0.005.
Initially Pr(.g~ = x ) = 1. Solid, d a s h e d a n d d o t - d a s h e d lines are respectively the o c c u p a t i o n p r o b a b i l i t i e s
o f x, v a n d z.
Three state evolution with two-time boundary conditions
1
0.9
0.8 i
0.7
+
(1.6
0.5
0.4
8
g
0.3
0.2
0.1
0
20
40
60
80
time -->
Fig. 2. As in fig. 1. except t h a t we also r e q u i r e P r ( X m o = X ) -- 1.
100
L.S. Schulman /Dynamics with two-time boundary conditions
379
condition based calculation would predict. Moreover, the disturbing notion of having
opposite thermodynamic arrows operating side-by-side never materializes. This is because any process slow enough not to be in equilibrium at T/2 will (as remarked) not
be behaving "normally" and one simply never gets the perplexing scenarios imagined
by Davies and Penrose.
A graphical exhibit may be helpful (see figs. 1 and 2). We considered a three-state
({x, y, z} ) stochastic process in which the transition x ~ y is relatively rapid, while
y ~ z is slow. A bias in these rates makes x and y metastable. This is a yet further
simplification of the models considered in ref. [ 1 ]. Boundary conditions in which the
system was required to be in x at t = 0 and t = Twere considered and we took Tlong
enough for one of the processes to equilibrate but not the other. The occupation probabilities were calculated from eq. (3), using eq. (4), and are shown in the figures,
both for the no-final condition case and for the above boundary condition. As can be
seen, entry to the state z is severely suppressed.
The physics in this discussion has been understood for a long time. For a variety of
reasons [4,8 ], the need for two-time boundary conditions was felt and the potential
for experimental implications of future boundary conditions was also studied [ 9,10 ].
The new features of the present article is the analytic formulation of and control over
this problem.
Everything done so far related to classical systems. As noted in ref. [ 11 ], quantum
mechanics is fundamentally different. If the initial and final states involve localization of wave packets, then there is an anti-relaxation effect. The greater T, the worse
things get. I believe that for most purposes of statistical mechanics this difference
would not be noticed. The main consequences are with respect to the quantum measurement problem. However, this is a topic for another day, and it is my hope that the
statistical arguments presented here, valid irrespective of the nature of one's dynamics, will eliminate puzzlement over the "'switch-over" as an objection to theories relating thermodynamic and cosmological arrows of time.
References
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J. Star. Phys. 34 (1984) 35; B. Gaveau and L.S. Schulman, J. Phys. A 20 (1987) 2865; Lett. Math.
Phys. 18 (1989) 201; J. Math. Phys. 31 (1990) 3030.
[2] M.E. Fisher, Physics 3 (1967) 255; M.E. Fisher and B.U. Felderhof, Ann. Phys. (NY) 58 (1970)
176; 58 (1970) 217; B.U. Felderhof and M.E. Fisher, Ann. Phys. (NY) 58 (1970) 268; B.U.
Felderhof, Ann. Phys. (NY) 58 (1970)281.
[3] T. Gold, Am. J. Phys. 30 (1962) 403.
[4] L.S. Schulman, Phys. Rev. D 7 (1973) 2868.
[ 5 ] P.C.W. Davies, The Physics of Time Asymmetry (Univ. of California Press, Berkeley, 1977) p.
194.
[6] S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972) p. 597.
380
L.S. Schuhnan / Dynamics with two-time boundary conditions
[7] R. Penrose, in: General Relativity, S. Hawking and W. Israel, eds. (Cambridge Univ. Press,
Cambridge, 1979) p. 597.
[ 8 ] W.J. Cocke, Phys. Rev. 160 (1967) 1165.
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New York/Israel Univ. Press, Jerusalem, 1975 ).
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