Text
arXiv:1310.1095v1 [cond-mat.stat-mech] 18 Sep 2013
On the Relation between the Psychological and Thermodynamic Arrows of Time
Leonard Mlodinow∗
California Institute of Technology,
Pasadena, California, 91125
Todd A. Brun†
Center for Quantum Information Science and Technology,
University of Southern California, Los Angeles, California
(Dated: October 7, 2013)
In this paper we lay out an argument that generically the psychological arrow of time should align
with the thermodynamic arrow of time where that arrow is well-defined. This argument applies to
any physical system that can act as a memory, in the sense of preserving a record of the state of
some other system. This result follows from two principles: the robustness of the thermodynamic
arrow of time to small perturbations in the state, and the principle that a memory should not have
to be fine-tuned to match the state of the system being recorded. This argument applies even if
the memory system itself is completely reversible and non-dissipative. We make the argument with
a paradigmatic system, then formulate it more broadly for any system that can be considered a
memory. We illustrate these principles for a few other example systems, and compare our criteria
to earlier treatments of this problem.
PACS numbers: 05.70.Ln 65.40.gd 05.90.+m
I. INTRODUCTION
The question of why the future is distinguished from
the past has been of interest ever since the nineteenth
century, when physicists began to explore the fact that
the equations governing dynamics are invariant under
time reversal, while our environment obviously is not.
In modern physics, we say that any local, Lorentz invari-
ant quantum field theory with positive energy is invari-
ant under time reversal, if you also reflect in space, and
interchange particles and antiparticles; together those
operations generate the so-called CPT transformation.
Though we now know that C and P can sometimes be
violated, in most practical applications, those violations
play no role, and so the dynamics will be invariant under
T.
Despite the interchangeability of past and future with
respect to the laws of microscopic physics, we humans
have no trouble distinguishing whether time is moving
forward or backward. Smoke rises and disperses from
chimneys, but never gathers and returns; dropped eggs
splatter on the floor, but once dropped they never jump
back into their shell. This time asymmetry can be under-
stood as a result of the second law of thermodynamics:
it follows from the equations of physics governing time
evolution that if a system is confined to a small region
of phase space at some time, then with virtual (but not
total) certainty it will occupy a much larger region of
phase space at other times. The observed asymmetry
of past and future can therefore arise due to a bound-
∗ Current Address: Center for Quantum Information Science and
Technology, USC, Los Angeles, California.
† Email: tbrun@usc.edu
ary condition that, in one direction of time, near the big
bang, the universe was in a state of low entropy. Given
that situation, we can define the “past” of any given mo-
ment as those times that are closer to the big bang, when
the entropy of the universe was lower, and the future as
those times that are further from the big bang, and in
which the entropy of the universe is higher. This is the
“thermodynamic arrow of time.”
Beside watching whether eggs are splattering into
messes or cleaning themselves up, there is another way
we can define a direction in time: we remember the past,
but not the future. From the point of view of human psy-
chology, this is perhaps the key to our feeling for past and
future. After all, though it would cause raised eyebrows
if we observed smoke retreating into chimneys, most of us
would be far more surprised if a memory of that chimney
turned out to reflect the chimney’s future state rather
than its past. Because of this, physicists sometimes refer
to the direction of our memory as the “psychological ar-
row of time” (see, for instance, Hawking, 1985 and 1994)
[1, 2].
The thermodynamic and psychological arrows of time
obviously agree, but why should they? If the equations
that govern a system are agnostic with regard to which
way is the future, then why do systems (memories) arise
in nature that reflect the state of other systems (the sys-
tems being remembered) in one direction of time, but not
the other? Why do streaks in mica correlate with cos-
mic rays that traversed them in the past (as defined by
the thermodynamic arrow) but not the future? Why do
we see contrails from planes past, but not planes yet to
come? Why do we remember the thermodynamic past,
but not the thermodynamic future?
Imagine that our universe were in a state such that, at
some time T in the future (i.e., further in time from the
2
big bang than we are today) the entropy were to start
to decrease. It is extremely unlikely that we would be
in such a state, but it does not violate the fundamental
laws of physics. Then for times greater than T , entropy
would decrease with time rather than increasing. For
t > T would people the world over marvel at chimneys
sucking back plumes of smoke, and broken eggs jumping
into their shells, or would the psychological arrow of time
also reverse itself, with the result that we perceived time
to run in the reverse direction, and the second law still
held true? Our answer is that latter. In what follows, we
will show that the psychological and thermodynamic ar-
rows must point in the same direction: the psychological
arrow follows from the thermodynamic one.
That the psychological and thermodynamic arrows
must align has in the past been argued on the basis of
Landauer’s principle [3], which states that the total en-
tropy of a closed system must increase upon the erasure
or reinitialization of a memory record. According to that
argument [1, 2, 4–6] a realistic memory must allow for
the erasure of records. Erasure, Landauer showed, is in
essence like the smashing an egg—it is a process whose
time reversal violates the second law. A memory, in that
view, must therefore be a dissipative, hence irreversible,
system, and so to remember the future would be a feat
akin to orchestrating the remnants of a smashed egg to
jump back and reassemble inside its shell. We will argue,
on the other hand, that the principle that the psycho-
logical and thermodynamic arrows of time must align
actually arises from a combination of the fundamental
microscopic laws of physics and a reasonable requirement
of what it means for a system to function as a memory.
Therefore, the two arrows must align even for systems
that are time reversible.
In Section II we present a simple paradigm for a mem-
ory embedded in a system with a well-defined thermo-
dynamic arrow of time. The dynamics of this system—
including the memory—are reversible, but we argue that
the direction of memory must still match the thermody-
namic arrow of time. In Section III we give a more precise
definition of what it means for a system to act as a mem-
ory, and give a broader argument that the direction of
memory must align with the arrow of time. In Section
IV we examine a few other examples of systems that can
function as memories, and argue for generic differences
between memory (i.e., correlations with the past) and
anticipation or projection (i.e., correlations with the fu-
ture). In Section V we compare our argument to other
takes on this problem, and discuss their points of com-
monality, and in Section VI we conclude.
II. A SIMPLE PARADIGM
Consider the system depicted in figure 1. A sealed ves-
sel is divided into two chambers, separated by a barrier
with a single narrow gap. Within the vessel are N elastic
particles that can collide with each other or the walls,
FIG. 1. A reversible system with a well-defined arrow of time
and a memory.
but are otherwise free. In the gap between chambers is
a rotor—a reversible counter, or turnstile—that rotates
one position clockwise or counterclockwise as a particle
moves left or right, respectively. This system should be
considered closed—it is not in contact with a thermal
bath—and all the dynamics are reversible. Nevertheless,
generically this system can exhibit a thermodynamic ar-
row of time if at some early time, ti, there is an imbalance
of particles between the two chambers, as is depicted in
the figure. From almost all states with such an imbal-
ance, the system will tend to evolve toward a state in
which the number of particles in the two chambers are
roughly equal. A thermodynamic arrow of time is defined
for systems in which coarse-grained variables—such as,
in this case, the total number of particles on each side—
exhibit a preferential direction of evolution. For times
greater than ti, therefore, the thermodynamic arrow of
time of this system will point toward increasing times,
which we will call the future. This is like the present sit-
uation of our universe: we are in a nonequilibrium state
with a boundary condition that at some time the uni-
verse was in a state of much lower entropy; we call that
direction of time the past.
The rotor in this system can function as a memory that
records the net number of particles that crossed from the
left chamber to the right. Suppose that the rotor has
M positions labelled 0, . . . , M − 1, and that (for simplic-
ity) the number of distinct positions M is much greater
than the average number of particles that will cross be-
tween t = ti and a later time t = tf . We let r(t) be a
coarse-grained variable that takes the values 0, . . . , M −1
corresponding to the rotor position at time t. We also
assume that the number of particles is sufficiently small
(compared to the size of the vessel) that the probabil-
ity of more than one particle crossing at the same time is
negligibly small, and that the average energy of the parti-
cles is sufficiently low that the possibility of fast particles
that might transfer enough momentum to make the rotor
spin (rather than just shifting one position) can also be
neglected.
We ‘read out’ the record of the net passing of particles
between time ti and a time t by using a simple function
fread(r(t)) = r(t) − rref mod M, (1)
3
where rref is the setting of the rotor at the reference time,
rref = r(ti). This read-out function provides a memory
of the past.
Because this whole system is deterministic and re-
versible, it seems as if we could equally well choose a
reference time tf > t and interpret the rotor as a memory
of the future by defining a different ‘read-out’ function:
f ′
read(r(t)) = r′
ref − r(t) mod M, (2)
where r′
ref = r(tf ). At times t < tf , we can now interpret
the rotor as a memory of the net number of particles
that will cross from left to right between now and the
later time tf . Our simple memory seems to have become
a memory of the future.
One might object to describing the rotor as a mem-
ory of the future on the grounds that since tf > t we
would not generally know, at time t, the value of r′
ref ,
and hence could not use the rotor as described in (2).
That argument depends on the implicit assumption that
“time flows,” which immediately implies that we can-
not remember the future because “it hasn’t happened
yet.” That naive view is unsatisfactory, because the mi-
croscopic dynamics are reversible and give no privileged
role to either direction in time. There is actually no a pri-
ori reason that a rotor coupled as we’ve described should
not reflect the state of the particles at future times, just
as it does for the past. In fact, any system with reversible
dynamics encodes both past and future within in it, as
Laplace [7] pointed out over two centuries ago:
We may regard the present state of the uni-
verse as the effect of its past and the cause of
its future. An intellect which at a certain mo-
ment would know all forces that set nature in
motion, and all positions of all items of which
nature is composed, if this intellect were also
vast enough to submit these data to analy-
sis, it would embrace in a single formula the
movements of the greatest bodies of the uni-
verse and those of the tiniest atom; for such
an intellect nothing would be uncertain and
the future just like the past would be present
before its eyes.
Our key observation is that, despite the fact that the
state of a classical system encodes both its past and fu-
ture, when the system being remembered has a well-
defined thermodynamic arrow of time, we can rule out
the possibility of a memory recording its future, because
any such memory could remember only one possible con-
figuration of that system. If the system changed its state
even minutely, the memory would no longer create an
accurate record. We call the requirement that a memory
be capable of remembering more than one fixed state of
a system generality.
Imagine you have a digital camera that contains a chip
capable of recording what the camera’s sensors pick up,
but only for one particular scene. Suppose, further, that
every time you change any aspect of the scene, you have
to insert a new chip designed specifically to record that
precise scene. Heuristically, one would think that a chip
of that sort could hardly be called a memory, for a mem-
ory that must be fine-tuned based on the state of the sys-
tem being remembered is hardly of any use. But there
is another, more fundamental reason to require general-
ity: it allows us to rule out, as memories, systems that
are correlated with other systems but are not causally
related.
Consider the system we described above. We can, in
theory, calculate the state of the particles at all times,
and hence, in place of the rotor, we could have set up
an independent meter that is not coupled to the parti-
cles at all, but which we have designed (based on our
calculation) to always exhibit the correct number of net
particles crossing between the chambers as determined by
our calculation. A meter such as that, which we set up to
reflect the particle crossings, but which does not interact
with the particles in any way, should not be considered a
memory, yet would have the same correlations with the
system of particles as the rotor. The key difference be-
tween the two is that a small alteration in the state of
the particle system would not destroy their correlation
with the rotor, whereas it would destroy the correlation
with the independent meter we set up. Unless we re-
did our calculations and made a compensating alteration
in the meter, it would no longer accurately reflect the
passage of the particles. Correlated systems such as the
meter are excluded from the definition of memories by
the principle of generality: a memory should be capable
of remembering more than one thing.
If we apply this idea to our rotor, we immediately see
that there is a sharp distinction between the two func-
tions fread(r(t)) and f ′
read(r(t)), in (1) and (2), that de-
scribe the rotor’s read-out.
To see this, let us begin by supposing, in the case of
fread(r(t)), that we slightly alter the state of the particles
(but not the rotor) at ti. We alter the coordinates and
momenta of the particles by a small amount, but such
that no particle changes position from one side of the
barrier to the other, or, more precisely, so that the coarse-
grained variable r(ti) remains unchanged. If we then let
the system propagate from ti until t under the action of
the same Hamiltonian as in the unperturbed case, the
system will in general exhibit the same arrow of time,
and though its time evolution will be altered, the read-
out function in (1) will remain an accurate reflection of
the past. In this case, therefore, the rotor functions as a
memory of the past for a large family of states that are
in some sense “near” the unperturbed state we discussed
above.
Now let’s consider the other case. In this case, when
we look for a family of “nearby” states for which the rotor
continues to function as a memory of the future, we find
that, generically, we must reject them all because they do
not preserve the arrow of time—that is, even the most
minute generic changes in the coordinates of the system
4
at tf , when propagated back to earlier times will result
in a system with a more equal distribution of particles
than at tf , rather than a more lopsided distribution.
The unperturbed system, due to our assumption of the
existence of the thermodynamic arrow of time, evolves
forward in time from a more lopsided distribution of par-
ticles at ti to a more evenly distributed system at tf ; the
time reverse of that evolution will therefore propagate
the system from a more even distribution to a less even
one. But that state of the unperturbed system at tf was,
due to the way it arose, a very special state, and one
that is unstable with respect to perturbation. A system
of particles in a generic state at a time t will lead to a
state with a more even distribution of particles at both
later and earlier times. Hence, when it comes to remem-
bering the future, there is no family of “nearby” states
that is both consistent with the existence of an arrow of
time, and for which the rotor continues to function as a
memory of the future. As a memory of the future, the
rotor does not satisfy generality.
These principles can be readily generalized beyond this
simple system, and we do so in the following section. The
conclusion is clear: generically, we expect the ‘psycholog-
ical arrow of time’ (broadly defined for any system that
can act as a ‘memory’) to align with the thermodynamic
arrow of time in any system where that thermodynamic
arrow is well-defined.
III. WHAT IS A MEMORY?
It is difficult to come up with a simple criterion that
is general enough to capture every example of a system
that can function as a memory. However, we will propose
a definition that seems to apply to many examples, and
that satisfies the criterion of “generality” discussed in the
previous section.
Suppose we can divide the world into two subsystems:
the record (or memory) R, whose state is given by a vector
r in some phase space R, and the system S (effectively,
whatever is being recorded plus everything else) whose
state is given by a vector s in some phase space S. We
will assume for the moment that the system and record
are classical and completely reversible. The system and
record evolve together. We can determine (in principle)
the values of s(t) and r(t) for all times t by specifying
the values (s0, r0) at any one time T . At the moment,
we make no assumptions about whether T is in the past
of future of any of the other relevant times (e.g., tread,
t1, and t2 in the conditions below).
For the subsystem R to serve as a memory, it must
satisfy the following requirements:
1. We can define two vectorial functions fR(r(t)) and
fS ({s(t)}, t ∈ I), where fR is a function of r at some
time, and fS is a function of s over some interval or
range of times I.
2. There is some read-out time tread and interval I =
[t1, t2] such that fR(r(tread)) ≈ fS ({s(t)}, t ∈ I).
That is, at the time tread the state of the record
reflects the state of the system at some time, or
range of times.
3. Generality. Suppose that we change the state of
the system s0 at T while keeping the state of the
record subsystem r0 fixed. We require that there
be some nontrivial set of possible s0s such that con-
dition 2 above remains true, without changing the
definitions of fR or fS , and without changing the
time tread or the interval I. We also require that
the functions fR(s(tr )) and fS ({s(t)}, t ∈ I) are not
constant over this set of possible s0s.
4. Thermodynamic robustness. The read out and
coarse graining functions fR(r(t)) and fS ({s(t)}, t ∈
I) must be robust under small perturbations of the
microscopic degrees of freedom (of both the system
and the memory) at T .
Let us examine these conditions one at a time. In
condition 1, the function fR is a read-out function. It
specifies how the record is encoded in the microscopic
degrees of freedom of the subsystem R. Similarly, fS ex-
tracts whatever property of the system S is being stored.
In general, these functions will involve a great deal of
coarse-graining over the fine details of their respective
subsystems.
Condition 2 gives the obvious requirement that the
read-out of the record should indeed correspond to what-
ever quantity is supposed to be recorded. In the condi-
tion here, we have required only that this correspondence
be approximate. Another reasonable way of defining this
would be to require that the two functions correspond for
many states s0, but not necessarily all. Note also that we
have treated a memory that is read at a single moment of
time. It is straightforward, however, to extend this def-
inition to memories that operate over a range of times,
and even to memories whose records evolve as they inter-
act with the system. Here we consider only the simplest
case.
We can see how the rotor system in the previous sec-
tion satisfies conditions 1 and 2. The function fR(r) is
the read-out function fread(r). We can define the corre-
sponding variable fS straightforwardly. Let N (s) be the
number of particles to the right of the partition when
the underlying variables are s. (This is a simple exam-
ple of a coarse-graining.) Then fS ({s(t)}, I) is simply
N (s(t)) − N (s(ti)) in the case of remembering the past,
and N (s(tf )) − N (s(t)) in the case of remembering the
future.
Condition 3 embodies the assumption of generality: In
order to consider R a memory, it must be capable of
recording different values for different states. The re-
quirement that fR and fS are not constant guarantees in
addition that the correspondence between fR and fS is
nontrivial—that is, we exclude the possibility that fR is
5
effectively fixed, and s0 is restricted to a set with some
constant property. Condition 3 thus requires that the
correlation between S and R is not due to fine-tuning.
That is, the system and memory must actually interact,
and the memory must encode some information about
the system, rather than both being correlated with some
common cause.
When we applied condition 3 to the rotor system, we
chose T to be the reference time, ti (in the case of re-
membering the past). That is because if T were chosen
to be any other time, generic perturbations to S at T
would alter the number of particles on either side of the
box at ti, and hence the function fread(r) would have be-
come invalid. As a result, in that example, in order for
the same function fread(r) to work for different s0s, we
chose the reference time T to match the reference time
in Eq. (1).
Condition 4 is a reflection of our assumption that there
exist a thermodynamic arrow of time. The question of
whether a memory can record the future makes no sense
otherwise, since it the the thermodynamic arrow that
we use to define past and future. Condition 4 guaran-
tees this. Consider the first case of the rotor system and
memory, for which Condition 3 required that t = T = ti.
Suppose that, at that time, the system is in a configura-
tion like that pictured, with far more particles on the left
side of the partition. From almost all initial conditions
with such an imbalance, the system will tend to evolve
toward a state in which the number of particles in the
two chambers are roughly equal. For times greater than
ti the thermodynamic arrow of time of this system there-
fore points toward increasing times, which we have called
the future. This is like the present situation of our uni-
verse: we are in a nonequilibrium state, with a boundary
condition that at some time the universe was in a state
of much lower entropy; we call that direction of time the
past.
Though perturbing the system we just described will
generally not destroy the thermodynamic arrow of time,
there are very special conditions for which such a per-
turbation would have that effect. For example, consider
a system which, as time increases from ti, evolves into a
state where the numbers of particles are more unequal.
This is not a highly intuitive choice, but in fact we can
find such conditions (in principle) by following the usual
intuitive evolution, and then “running the film back-
wards.” Because the dynamics are reversible, this is a
legal (albeit unlikely) evolution. We denote this time-
reversed initial condition ̄sT , where we use the bar to
denote the time-reversed version of the phase-space vec-
tor s(T ).
Initial conditions ̄sT are analogous to states in which
shattered tea cups fly back together. Starting from ̄sT ,
the distribution of particles between the two chambers
becomes increasingly more unequal, rather than less.
This system exhibits an arrow of time, but the arrow
for such a system is not robust: almost all states except
for a tiny neighborhood of ̄sT will not exhibit this behav-
ior; and the longer we want this unintuitive behavior to
persist, the smaller this neighborhood must be. Even an
initial condition ̄s′
T that is very close to ̄sT will tend to
exhibit the generic behavior, in which the distribution of
particles becomes more rather than less equal.
For the rotor system, condition 3 thus requires that the
reference time T be chosen to coincide with ti or tf , while
condition 4 eliminates the latter. T must be an “initial
condition” in the sense of the thermodynamic arrow of
time. So if ti is a time exhibiting a strong deviation from
equilibrium (leading to a well-defined thermodynamic ar-
row of time in the direction of increasing t), then T , the
interval I = [t1, t2] over which the system is recorded,
and the read-out time tread, must satisfy
ti ≤ T < t1 < t2 < tread.
In other words, even if the memory subsystem is itself a
completely reversible system, its “psychological” arrow of
time must align with the thermodynamic arrow of time.
If it does not, it will not satisfy the generality or robust-
ness requirements that we have argued must be part of
the definition of a memory.
IV. EXAMPLES OF SYSTEMS THAT
FUNCTION AS MEMORIES
Having introduced the concepts underlying our defini-
tions with the model system in Sec. II, we can now try to
apply these ideas to other examples of systems that func-
tion as memories. While in most cases we cannot exactly
specify the readout and coarse-graining functions fR(r(t))
and fS ({s(t)}, t ∈ I), it is not difficult to argue for the ex-
istence of these functions, nor is it difficult to argue that
they satisfy the requirements specified in section III.
Most obvious examples of memories are synthetic—
that is, they are systems deliberately engineered by hu-
mans to store information of a particular type. We will
briefly consider some canonical examples of this below.
But certain naturally-occurring systems also satisfy the
definition of a memory given above. Most of these ex-
amples depend on irreversible processes, so it is not sur-
prising that they align with the usual thermodynamic
arrow of time [1, 2, 4, 5]. But in principle this is not
necessary—reversible systems can also function as mem-
ories, and when they do so, their arrows of time will also
align with the thermodynamic arrow, by the argument
in section III.
a. Synthetic memories: computer storage. Given
the ubiquity of modern electronic technology, various
kinds of computer memories are all around us. These
include computer RAM; magnetic storage on hard drives
(and, still in a few applications, magnetic tape); flash
memory; and charge-coupled devices (CCDs) and active-
pixel sensors (APSs) in digital cameras.
If we take as an example a single computer bit, the
readout function fR(r(t)) corresponds to a range of av-
erage voltages across the bit. A “high” range might be
6
on the order of 5 V, the “low” range close to 0 V. The
bit stores an input voltage that was applied at an earlier
time, so fS ({s(t)}, t ∈ I) in this case is also an averaged
voltage range. It is clear that these functions satisfy the
requirements in section III: the computer bit can store
either 0 or 1, depending on its input, and no fine tuning
between the input and the bit is necessary. Moreover,
the value of the stored bit is correlated with the input at
an earlier time, not with later inputs, as common sense
would suggest.
b. Reversible and quantum computers. Existing
computer technology is emphatically not reversible: any-
one who has held a laptop computer actually on his or
her lap is quickly aware of how much heat such a com-
puter produces. But it was argued by Landauer [3] and
proven by Bennett [8] that dissipation is not necessary,
in principle, for computation.
Bennett’s proof involved first defining a reversible
model of a Turing machine. A Turing machine is an
abstract model of a computer, and has two subsystems
that could be considered memories by the above defini-
tion. First, the computer has a tape (or more generally
n tapes), divided into unit cells, each cell containing one
symbol from a finite alphabet. The “read/write head” of
the Turing machine is located at a particular cell at any
given time. Second, the computer has an internal state,
which is one of a finite set of possible settings. The Turing
machine is designed with a transition rule, or program:
given the value at the current tape cell and the current
internal state, the machine replaces the symbol with a
new symbol, moves the head at most one position to the
left or right, and transitions to a new state.
In Bennett’s reversible model, the transition rule is
invertible: given the current tape symbol, location and
internal state it is possible to invert the transition rule
and return to the state of the Turing machine at the
previous state. He proved that such a reversible machine
could emulate a standard Turing machine, at the cost of
a relatively small overhead in the amount of tape used.
From the work of Landauer, such a logically reversible
machine can also be made to be physically reversible.
Moreover, in principal such a machine need not dissipate
any power in order to function.
Of course, in practice even such a reversible memory
generally would be used by first setting it in a standard
starting state. For instance, a string of n reversible bits
would be prepared in the 00 · · · 0 state. They could then
reversibly store n input bits x1x2 · · · xn by being XORed
with those bits; this procedure would simply copy the
values x1x2 · · · xn into the memory. This initialization
step would be an irreversible process.
But in principle this initialization step to 00 · · · 0 is
not necessary. Suppose that the reversible bits started
instead in a general state y1y2 · · · yn. Then, after be-
ing XORed with the input bits x1x2 · · · xn, the memory
would be left in the state (x1 ⊕ y1)(x2 ⊕ y2) · · · (xn ⊕ yn).
However, it is still quite possible to recover the stored
bit values simply by redefining the readout function
to take the initial state of the reversible bits into ac-
count. The readout function would be fR(z1z2 · · · zn) =
(z1 ⊕ y1)(z2 ⊕ y2) · · · (zn ⊕ yn), where z1z2 · · · zn is the
state of the reversible bits at the time of readout. We
can see that this readout function satisfies the require-
ments of section III: changing the system being recorded
(in this case, the input bits x1x2 · · · xn) while keeping the
initial state of the memory fixed does not require us to
change the readout function fR.
Reversible computation may be of practical interest for
classical computers to reduce the problem of power dissi-
pation; but it is absolutely vital for quantum computers,
whose unitary operation must be intrinsically reversible.
While our definition in section III probably cannot be
applied to quantum systems without alteration (more on
this below), we believe the general principles requiring
that memories align with the thermodynamic arrow of
time will apply to quantum memories as well.
c. Photographic film. Photographic film is a canon-
ical example of a synthetic, irreversible system designed
(unlike computer memories) to produce a permanent
record. To produce a photograph requires that the film
be prepared in a very special initial state. This prepara-
tion step will clearly be irreversible. The record is pro-
duced by exposing the film to light, thus driving irre-
versible chemical reactions. Since both the preparation
step and the recording step involve irreversible processes,
the “arrow of time” associated with this record must
clearly line up with the thermodynamic arrow of time.
This type of irreversible record is used as a paradigm in
the work of Wolpert [4].
d. Tracks in mica and contrails in the stratosphere.
The examples considered so far are all synthetic systems,
designed to record and retrieve information. Of course,
naturally occurring systems can also function as memo-
ries. Indeed, any two systems that interact have to po-
tential to exchange information about their respective
states.
In most cases, however, this information is spread out
in the form of complicated correlations between many de-
grees of freedom in a way that makes it impractical to re-
trieve. Gell-Mann and Hartle refer to such correlations—
perhaps retrievable in principle, but certainly not in
practice—as generalized records [9], to contrast them with
the type of synthetic records described above.
There are at least a few cases, however, where naturally
occurring systems can store information for some length
of time in a form that is robust and not particularly dif-
ficult to retrieve. A classic example is the existence of
cosmic ray and fission product tracks in naturally occur-
ring crystals of mica [10].
The passage of high-energy particles through regular
crystals of mica disrupts the crystalline structure along
the trajectory of the particle. These tracks can be made
visible by etching the crystal in acid. (In fact, syn-
thetic crystals of mica have been used as simple parti-
cle detectors.) In this case, both the “preparation of the
memory”—that is, the formation of the crystal—and the
7
recording process are irreversible. In this case, however,
the preparation step occurs naturally (attesting to the
nonequilibrium state of the earth’s crust). The read-out
function—the tracks of disruption—record the number of
high-energy particles and their directions, and perhaps
some information about their energy, but not in general
when they occurred.
e. Ripples in water and other outgoing waves. For
completeness, it would be good to consider a naturally
occurring “memory” system that was also reversible.
Most such records are unfortunately transient in their
existence. However, a familiar phenomenon does illus-
trate this type of system—namely, the emission of waves
from moving (or reflective) objects.
The canonical example of this is the spreading of rip-
ples from a stone tossed into a pond. These ripples are
not perfectly reversible, of course, but dissipation is not
essential to their function. Nor, in principle, does the
pond have to be prepared in an exact state—waves are
generically linear, and hence could be separated from a
background.
Our own senses make use of sound and light waves
to acquire information about objects outside ourselves.
Given their finite speed of propagation, any wave we in-
tercept is in reality a record of an earlier event. While
in everyday circumstances, these records are not of long
standing, we are quite capable of seeing light emitted
from stars thousands of years ago—or from other galax-
ies millions or (with the help of telescopes) even billions
of years ago.
The propagation of waves is generally reversible to a
good or even excellent approximation. In principle, the
emission process is also reversible. Nevertheless, wave
emission almost always exhibits a strong arrow of time.
This is because the universe is far from equilibrium,
which in turn is due to the overall thermodynamic arrow
of time arising from the fact the the universe apparently
began in a very low entropy state.
V. DISCUSSION
A. Memory vs. Anticipation
As we have pointed out above, it is quite possible for
a system at a given time to be correlated with the state
of another system in their thermodynamic future rather
than their past. We argue that such correlations are in-
consistent with the properties we expect of a “memory,”
as defined in Section III above. However, even in a sys-
tem with a well-defined thermodynamic arrow of time,
such correlations can and do arise; we would term such a
correlation with the future state of another system “an-
ticipation” or “prediction.”
Of course, for systems with very regular dynamics
the future is highly predictable—these systems exemplify
Laplace’s idea that the present state reveals the future
as well as the past. But even in less trivial systems, it
is possible to anticipate future behavior up to a point.
We expected such correlations to differ from memories,
however, in a number of respects.
Consider our paradigmatic example from Section II.
By counting the net number of particle transitions from
the left half to the right half of the vessel, one can esti-
mate a rough transition rate between the sides. This, in
turn, would allow one to extrapolate the number of par-
ticles on each side in the future. If the numbers of parti-
cles on each side at the reference time T are NL(T ) and
NR(T ), respectively, then we would project the numbers
at some future time t > T to roughly follow an exponen-
tial law:
NL(t) = 1
2
[NL(T ) (1 + e−2γt) + NR(T ) (1 − e−2γt)] ,
NR(t) = 1
2
[NL(T ) (1 − e−2γt) + NR(T ) (1 + e−2γt)] .
The transition rate γ can be estimated from observation.
If we let the system evolve for some length of time ∆t
and observe n net particle transitions, we would estimate
γ ≈ n/∆t(NL(T ) − NR(T )). We expect, however, that
this estimated rate will not be precisely accurate, and
of course even if it were, the actual behavior will devi-
ate from the expectation. Thus, we expect the accuracy
of this future correlation to fall off exponentially, while
the accuracy of the past correlation remains precise. (Of
course, at very long times the system will approach equi-
librium, with a roughly equal distribution of particles
between the two chambers.)
We can compare this difference to our experience of the
real world as well. Information that is stored in a robust
record can be retrieved very well, even after a long period
of time; and how well it can be retrieved is not dependent
on the regularity of the system. For example, we known
many details of the last day of Julius Caesar’s life, more
than 2000 years later. But we are unable to predict in
any detail the days of the most famous of our current
citizens even a short time into the future.
B. When is “Now?”
Our psychological perception of time as “flowing” im-
plies that there is a special time—the present, or “now”—
at which events in the future undergo a profound tran-
sition and become events in the past. This feeling is so
fundamental that it is difficult to conceive of any other
way that time could be experienced. And yet, the phys-
ical description of systems evolving in time does not in-
clude this notion of “now.” All times are treated with the
same status; none is singled out; and no physical princi-
ple seems to imply that time must be experienced in this
way.
The modern interpretation of this dividing line be-
tween the past and the future is that it is psychological
in origin. At all moments, we have memories of the past,
but not of the future, which makes a profound difference
8
to how we regard past and future. As argued in the pre-
vious section, records of the past can remain robust and
reliable for long periods with little loss, while our ability
to anticipate the future falls of extremely rapidly. Psy-
chologically we make a very sharp distinction between
these two processes, which is no doubt an adaptive trait.
However, the notion of “now” as an idealized point be-
tween the past and the future does not really hold up
to scrutiny [5]. The process of registering and recording
a memory will in general take some characteristic time,
so that our conscious impression of when something hap-
pens will in general lag the actual event by some amount.
Our brain does its best to stitch all its sense impressions
together into a seamless whole, so we are in general not
aware of any lag.
Moreover, we have a very good ability to anticipate
highly regular events in the immediate future. Anyone
who has ever caught a ball or stepped onto a moving
escalator has experienced this: to catch, we reach not for
where the ball is, but where it will be. We can therefore
think of the moment “now” as actually being somewhat
spread out in time, with some lag into the (idealized)
past and possibly a small extension into the (idealized)
future.
C. Conclusions
One of our deepest observations about nature is that
we remember the past, but not the future: that is, that
the psychological arrow of time aligns with the thermo-
dynamic arrow of time. In fact, this phenomenon is so
deeply embedded in our experience that it took almost
all of history even to recognize that there was a question
to be answered.
Our modern understanding of the arrow of time recog-
nizes its origins in an unusual, low entropy state of the
universe in the far past. This leads to the thermodynamic
arrow of time, exhibited by irreversible systems, which in-
cludes almost all systems of sufficient size and complexity.
Since almost all physical systems that can function as
memories or records are themselves irreversible—either
in their dynamics, or in the requirement of an irreversible
preparation step, or both—it is perhaps not surprising
that they exhibit an arrow of time that aligns with the
overall thermodynamic arrow.
In this paper, however, we have argued that even com-
pletely reversible systems that can function as memories
must exhibit an arrow of time that aligns with the usual
thermodynamic one. This arises purely from a reasonable
notion of what it means for a system to be a memory:
that it must exhibit correlations with another system;
that these correlations must arise not from careful fine-
tuning of the system and memory together, but from
interactions between them; and that these correlations
must be robust to small perturbations in the system and
memory states. A memory is only a memory if it has the
potential to remember more than one thing. We illus-
trated this argument with a simple paradigmatic system,
and then pointed out how it works in a variety of exam-
ples.
There is a very important open question left in this
work. The models we have chiefly focused on, and our
definition of a memory, take an implicitly classical view
of the world: we assume that the state of the universe
can be factorized, so that the system and memory can
each be treated as having a definite well-defined state at
any given time. But of course, the universe is actually
quantum mechanical. This definition cannot be applied
directly to quantum systems, because of the phenomenon
of entanglement, which implies that the system and mem-
ory states cannot necessarily be factored at all times. We
believe that the essence of this argument will still hold
for quantum systems as well as classical ones; but the
technical details of how to make this argument precise
are work for the future.
In the meanwhile, we believe that our current work
validates our intuition about the arrow of time, and gives
insight into what it means to remember the past, and to
dream about the future.
ACKNOWLEDGMENTS
LM and TAB would like to acknowledge fascinating
and useful conversations over the years with Sean Car-
roll, Murray Gell-Mann, Jonathan Halliwell, Jim Hartle,
Stephen Hawking, and John Preskill. This work was sup-
ported by the hospitality of the Institute for Quantum
Information at Caltech.
[1] S.W. Hawking, “Arrow of time in cosmology,” Physical
Review D, vol. 32, no. 10 pp. 2489–2495 (1985).
[2] S.W. Hawking, “The no boundary condition and the ar-
row of time,” in The Physical Origins of Time Asym-
metry,edited by J.J. Halliwell et al, pgs. 346–357 (Cam-
bridge University Press, Cambridge, 1994).
[3] R. Landauer, “Irreversibility and heat generation in the
computing process,” IBM Journal of Research and De-
velopment, vol. 5, pp.183–191 (1961).
[4] D. Wolpert, “Memory systems, computation, and the
second law of thermodynamics,” International Journal of
Theoretical Physics, vol. 31, No. 4, pp.743–785 (1992).
[5] J.B. Hartle, “The physics of ’now’,” American Journal of
Physics, vol. 73, Issue 2, p.101–109 (2005).
[6] J.B. Hartle, “The quantum mechanical arrows of time,”
arXiv:1301.2844 (2013).
[7] P.S. Laplace, A Philosophical Essay on Probabilities,
translated by F.W. Truscott and F.L. Emory, 6th Edi-
tion, p.4 (Dover, New York, 1951). A similar though
9
somewhat less eloquent statement was actually made 50
years earlier by the philosopher and Catholic priest Roger
Boscovich—see Stephen G. Brush, The Kind of Motion
We Call Heat, vol. 2, p.595 (North Holland, Amsterdam,
1986).
[8] C.H Bennett, “Notes on the history of reversible compu-
tation,” IBM Journal of Research and Development, vol.
32, no.1, pp.16–23 (1988).
[9] M. Gell-Mann and J.B. Hartle, “Strong decoherence,”
in Proceedings of the 4th Drexel Conference on Quan-
tum Non-Integrability: The Quantum-Classical Corre-
spondence, ed. by D.-H. Feng and B.-L. Hu (International
Press of Boston, Hong Kong, 1998).
[10] P.B. Price and R.M. Walker, “Electron microscope obser-
vation of etched tracks from spallation recoils in mica,”
Physical Review Letters, vol. 8, pp.217–219 (1962).
