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!