on the relation between the psychological and thermodynamic arrows of time todd a. brun and leonard...
TRANSCRIPT
On the relation between the psychological and thermodynamic arrows of time
Todd A. Brun and
Leonard Mlodinow
Time Flies Like An Arrow...
(...fruit flies, by contrast, like a banana...)
One of the most obvious observations about the world is
that there is something called Time, and that this
something is constantly moving; or perhaps,
everything in the world is constantly moving through
time. Time is separated into two very different
regions: the Past, and the Future, divided by an
infinitesimal moment of Now. Moreover, this motion
through Time only proceeds in one direction—from the
Past to the Future—and never the other way around.
Indeed, this observation is so obvious that for almost all
of history it was never even questioned.
Newtonian Time
This began to change when “motion through time”—that
is, dynamics—began to be formulated mathematically.
Isaac Newton described the behavior of particles by
laws of motion (what we would now call differential
equations). These motions were determined by the
state of the particles (their positions and velocities)
and the forces between them (the laws of motion).
Time is a global parameter t that constantly advances,
at a uniform rate, throughout the universe.
Evolution from Newton’s laws is deterministic, and
complete knowledge of particle positions and velocities
at any time determines their motion at all times.
“We may regard the present state of the
universe as the effect of its past and the
cause of its future. An intellect which at a
certain moment 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 analysis, it would embrace in a
single formula the movements of the
greatest bodies of the universe 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.”--Pierre-Simon Laplace, A Philosophical Essay on Probabilities
Laplace
This picture seems to put time on a more rigorous footing.
But it opens two very puzzling questions.
1. The mathematical description of time evolution doesn’t
single out a moment of ‘Now’: all times are on an equal
footing.
2. Newton’s laws are reversible: if we make a movie of
two particles interacting by Newtonian forces, and run
the movie backwards, both evolutions obey the laws of
motion. Nothing in the laws of motion distinguishes
past from future.
But in our experience this is not the case. Eggs break
when we drop them, but they rarely fly together again.
Moreover, we have a strong perception that the past is
fixed but the future is not. How can we reconcile this
with reversible laws?
Thermodynamics and the arrow of time
During the 19th century the
answer to some of these
questions was discovered. In
studying the properties of
heat, scientists were forced to
postulate a mysterious
quantity called entropy, which
is constantly increasing.
Indeed, this constant increase
is the second law of
thermodynamics. The
symmetry between future and
past is broken.
The direction of increasing entropy is the thermodynamic arrow of time.
Rudolf Clausius
Boltzmann realized that
entropy was, in a sense, a
measure of the disorder of a
system. As time evolves,
systems tend to evolve from
more orderly arrangements
to less orderly
arrangements. In a state of
maximum entropy, a system
is as disorderly as possible.
(Like my office.)Ludwig Boltzmann
Probabilities
Boltzmann realized that the second law of
thermodynamics was essentially just a consequence of
probability theory. There are many more disorderly
arrangements than orderly ones. Therefore, almost all
orderly arrangements will tend to evolve into less
orderly ones. There are a few arrangements that will
become more orderly, but they are much less likely.
The power of this insight is that it does not require any
ability to track the motions of the untold numbers of
particles making up macroscopic systems. Moreover, it
reconciles the irreversible evolution of thermodynamics
with the reversible evolution of Newton’s laws of
motion.
There is a hitch in this argument. Suppose we are now in
a highly orderly state. Then we expect to evolve to a
more disordered state as time goes forward. But if we
run Newton’s laws backwards, we will almost certainly
also evolve to more disordered state!
Sean Carroll, “From Eternity to Here”
In other words—reasoning purely by probability—an
orderly state is much more likely to have evolved as a
spontaneous fluctuation from a less-orderly state than
to have evolved from a more orderly state. This, also,
does not agree with our experience, where disorder
constantly increases. To avoid this paradox, we must
assume that our universe started in a very orderly
state, and this leads to the universal arrow of time.
The initial state of the universe is one of the biggest
problems in the field of cosmology.
Coarse-graining
This puzzle is closely related to the idea of coarse-
graining. In describing a complex, many-body system,
like a volume of gas, we do not try to describe the
positions and velocities of every molecule. Rather, we
use collective quantities: density, pressure,
temperature, momentum density. This description
throws away most of the information about the gas,
but the coarse-grained description is self-contained.
This fails if we run the film backwards. Almost all
microscopic states corresponding to a given coarse-
grained state will increase in entropy in either direction
of time. So clearly the microstates of our world are not
generic.
The independence of the coarse-grained description
corresponds to invariance under small perturbations.
We’ll use this later.
The psychological arrow of time
What about the idea of “Now?” Why do we feel as though
we are moving through time from the past to the
future? Why are our perceptions of the past—fixed and
immutable—so different from our perceptions of the
future—unknown and in constant flux? This perception
is the psychological arrow of time.
The answer to this question seems to boil down to a
simple asymmetry: we remember the past, but we
don’t remember the future. At each time t, we have
memories of events prior to t, but no memories of
future events.
This leads to a new puzzle. Why does the psychological
arrow of time line up with the thermodynamic arrow?
Thanks for the memories
Our intuition may suggest that this question is nonsense.
Of course we remember the past and not the future.
That’s what “remember” means. But the remarks of
Laplace show that this is not quite so straightforward.
Given enough knowledge of the world at the present
time, we can know its state at any time. So why
shouldn’t we be able to remember the future?
The answer seems to be that “enough knowledge” is far
more knowledge than any person could conceivably
have; and even given the knowledge, the ability to
solve the equations of motion is far beyond any
conceivable reason. Even in a deterministic universe
(which ours is not, in reality), chaos will quickly render
any prediction inaccurate.
In fact, the entire discipline of thermodynamics reflects
this inability. Rather than trying to track the myriad
degrees of freedom of the world, we construct an
effective, highly coarse-grained description, and treat
the unknown microscopic state of the system
probabilistically.
Given these difficulties, it is kind of remarkable that we
can even remember the past, let alone the future.
So, what does it mean for a physical system to act as a
memory? And why does the psychological arrow
always line up with the thermodynamic arrow?
Irreversible memories
A natural conjecture is that the asymmetry of memories
follows from the thermodynamic arrow of time itself.
This is exactly what we expect if memories are
irreversible systems.
What does a memory do? Here is a description (Wolpert
1992):
1. The memory begins in an initial “blank” state.
2. The memory physically interacts (directly or indirectly)
with the system to be recorded. The state of the
memory is changed by this interaction, and is
correlated with the system.
3. At a later time, the record can be retrieved by
observing the memory and extracting from its state
information about the system at the earlier time.
Landauer’s principle
So, what in this description requires a memory to
irreversible? The answer is step 1: initializing the
memory must be an irreversible process. This is due to
a physical effect called Landauer’s principle.
Landauer’s principle states that processing information
can, in principle, be done reversibly, so long as no
information is lost; but erasure of information must
inevitably produce an increase in entropy. In other
words, logical irreversibility implies physical
irreversibility.
Erasing one bit of information consumes energy E = kBT
ln(2).
Initializing a memory—preparing the “blank” state—is
clearly an irreversible process, because whatever state
the memory had beforehand is erased. (The
interactions with the system could also be dissipative,
but they need not be.)
Most familiar systems that can serve as memories
certainly do operate irreversibly, so this assumption
seems natural enough. But is it really necessary?
A reversible paradigm
Consider the system below. Particles can pass through
the gap between the two chambers. If one chamber
starts with significantly more particles than the other,
this system will exhibit a thermodynamic arrow of time.
The rotor revolves one slot each time a particle passes through. This rotation can be made to act reversibly.
This rotor can serve at time tf as a memory of the (net)
number of particles to pass from left to right since time
t1:
where r(t) is the position of the rotor, M is the number
of slots, and rref = r(t1). So the rotor operates
reversibly, and it does not require an irreversible
preparation—only knowledge of its state at the initial
time.
But here is a more remarkable possibility: can the rotor
serve as a memory at time t1 of the net number of
particles that will cross by time tf? Can it be a memory
of the future?
What is a memory?
In practice, we cannot use the rotor as a memory of the
future in this way, because we do not know the value of
rref’ = r(tf) ahead of time. But as Laplace pointed out, in
principle that value is determined ahead of time. Why
can we not think of the rotor as recording, not the
number of crosses that have happened, but the number
of crosses that will happen?
This interpretation seems to violate our notion of what a
memory is. We would be treating the rotor as encoding
information about the particles before it ever
interacted with them. Are there reasonable
requirements on the definition of a memory that would
rule out such an interpretation?
Requirements of a memory
The state of the memory and of the system (i.e., the rest
of the world) are determined by their states (s0,r0) at
some reference time T. (But T need not be an initial
time.)
We proposed the following properties for a physical
system to be considered a memory (or record) of
another system:
1. We can define two functions fR(r(t)) (the read-out
function) on the memory system and fS({s(t)},t1≤t≤t2)
(the coarse-graining function) on the system to be
recorded.
2. At some time tread we have fR(r(tread))=fS({s(t)},t1≤t≤t2) .
3. Consider solving the equations of motion for the
system and memory starting from a different reference
state s0. Then there is some nontrivial set of such
states for which condition 2 is still satisfied. The
functions fR(r(t)) and fS({s(t)},t1≤t≤t2) are also not
constant over this set. We call this condition
Generality.
4. Both the read-out and coarse-graining functions are
robust against small changes to (s0,r0) at the reference
time T. This condition is Thermodynamic Robustness.We argue that these requirements rule out the possibility of a memory of the future for systems with a well-defined thermodynamic arrow of time.
Back to the future?
Consider our paradigmatic system again. There are two
ways that it could be interpreted as a memory of the
future. First, we could have T be an initial condition,
and have our read-out function depend on the position
of the rotor at the final time tf.
This type of interpretation violates generality. The read-
out function has to be “fine tuned” to the particular
state of the system being recorded. If the system were
not in the right state, the memory would be wrong.
(This is like a stopped clock being right twice per day—if
you look at it at the right time, it can seem surprisingly
accurate.)
The other possibility is that the reference time T could be
a future condition, rather than an initial condition.
(This would be a strange-looking description, but it’s
logically possible.) In this case, the state of a memory
at a given time could easily be correlated with the
state of a system at a later time.
However! Remember that almost all low-entropy states
will increase in disorder in both time directions. The
state of the memory would have to be very carefully
fine-tuned to avoid disrupting the thermodynamic
arrow of time. So in this case, the memory lacks
thermodynamic robustness.
The curious thing is that this conclusion does not require
the memory itself to be irreversible. But to embed
such a memory into a universe with a thermodynamic
arrow of time, without either disrupting this arrow or
violating our sense of what it means to be a memory,
the memory system can only record the past, not the
future.
What are memories?
This definition of a memory are quite broad, and include
many systems that (in principle) record information,
although (in practice) we may be unable to retrieve it.
This would include, e.g., waves emitted by falling
stones and tracks left by decay products in mica, as
well as human brains, computer memories, written
notes, and other such familiar systems.
We call these in-principle-retrievable memories
generalized records. In fact, most of the events on
earth have left no lasting record here. Charles Bennett
of IBM estimates that the overwhelming majority of
data about past events on earth have escaped with the
soft infrared emissions from the planet. Only rare
events have left a more enduring record.
Memory vs. anticipation
It is still possible for one physical system to be correlated
with the future state of another. We can think of these
as predictions or anticipations of future events.
However, such correlations behave quite differently
from memories. For one thing, such correlations tend
to fall off extremely rapidly. Our ability to accurately
predict is extraordinarily limited.
Records of the past, however—while in general very
incomplete—can endure for very long times with little
degradation. We have detailed knowledge of certain
human events from thousands of years ago, and fossils
from billions of years ago.
Now...and now...and now...
So, in a sense, the feeling of moving through time is an
illusion. At any given time, we remember the events
that have gone before, and can only guess at the
events that are to come. And what is more, we
remember our past selves, and their ignorance of the
events that have since come to pass.
But at any time, we will always have the sense of Now: a
unique moment, on the cusp of the past, and poised on
the brink of tomorrow. We will never visit this moment
again. So I hope it was a good one.
THANK YOU FOR YOUR ATTENTION!