More on arrows of time
Next time: (and beyond)
QM + SR -> Field theory: are there "building blocks"?
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A new conception of "empty“ space
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Cosmology – the beginning and end of time
Announcements:
Any topics that anyone would like to discuss in the remaining
classes?
e.g. postmodernism and science
unification in physics etc.
Black hole firewalls
Entropy and QM
• QM gets rid of some ambiguity about the meaning of entropy. Classically, entropy
had something to do with the number of micro-arrangements consistent with the
macroscopic appearance. This is ambiguous in several respects- one being that
there is no obvious classical way of counting arrangements, which have a
continuum infinity of possibilities. QM avoids this, because we can write a discrete
list of quantum states and their probabilities, given some macroscopic knowledge
of the system. Crudely,
entropy is the log of the number of possible quantum states.
– Technical details: some quantum states will in general be more likely than
others, given known facts about the system. There's still a precise formula for
the entropy in terms of the states’ probabilities.
– Sometimes what's known about the system says not just how likely it is to be in
the various states in your list, but also what particular superpositions of those
states are most likely. (e.g. two-well system, in one well) There's a precise
generalization of the entropy formula which deals with those cases too:
-Tr(rln(r)).
– The entropy comes out independent of what list of quantum states you chose.
• e.g. eigenstates of energy, or eigenstates of some other variable
Maxwell's Demon Revisited
It seemed that one ought to be able to cheat the second law.
•
Consider “Maxwell’s demon,” a hypothetical entity who performs impossible feats. For example, he stands at the door
between two rooms and only lets molecules through one way. This process would reduce entropy, since there's more
ways to place the molecules if they can go on either side than if they're confined to one side.
•
Then you get high pressure on one side, low pressure on the other. You could then use that pressure difference to drive
a piston. Is this possible?
• Before:
• After:
How can the demon see when to open and
close the gate?
Can’t use equilibrium light: no info.
Needs a non-equilibrium flashlight.
E.g. photons small enough to see where
particles are.
Will create more entropy than the demon can
remove.
Entropy and Subjectivity
So QM has also allowed us to clean up the definition of entropy
but the definition of entropy still has a somewhat subjective ring
– If you specify the quantum state, the entropy is zero, because log[1]=0
– Therefore the entropy is still a function of the set of variables specified (energy,
etc.) and of the accuracy of their specification, not just of the state of the
system.
restate the 2cd law
Left to itself, any nearly closed system is
equally likely to end up in any of the allowed quantum states
– consistent with the known energy, number of particles, etc.
• Therefore you will find the system with values of various measurable quantities
which give maximum entropy,
– i.e. are consistent with the maximum possible number of states.
• This remains highly directional in time.
It says what will be, but not what used to be.
Order, disorder, etc.?
Do NOT be misled by various statements about "order".
• The net entropy of water and its surroundings increase
– when an ice-cube melts in warm weather,
– and when an ice-cube freezes in cold weather.
– Energy is released by the freezing, and that energy goes off to make entropy in
the surroundings.
• The temperature is a measure of how much energy is needed to make some
entropy.
– If T is high, entropy is maximized by having the water in the apparently
disordered state.
– But if T is low, the released energy can make lots of entropy, so total entropy is
maximized by the ice forming.
• The ancient tradition (e.g. Aristotle) was full of explanations about why things
tended toward some final state, such as heavy stuff down, absence of motion of
things on the Earth, etc. (teleology) The various teleological principles are now
replaced with this blank "equal probabilities of states", coupled with statistical
mechanics, the techniques for counting states.
Boltzmann's statistical argument for the 2nd law
1. You don't know the actual microstate.
2. So your best guess about the future will be one consistent
with as many microstates as possible, so long as they are
consistent with the current observation.
3. Which can be shown to lead to constant or increasing
entropy.
• What's wrong with that reasoning as an explanation for the
time asymmetry?
– Let’s grant that Boltzmann's mathematical argument for point
(3) is ok.
Boltzmann in reverse
Pointed out by Boltzmann's friend Loschmidt
1.
2.
You don't know the actual microstate.
So your best guess about the past will be one consistent with as many
microstates as possible, so long as they are consistent with the current
observation.
3. Which can be shown to lead to constant or decreasing entropy.
By just the same math as used before.
• A false conclusion!
• So Boltzmann must have snuck in an assumption:
– essentially that the past was ordered in a special way which precludes the use
of simple statistical arguments about it.
– But that is precisely the asymmetry which we wanted to explain.
– It's now clarified, but not at all explained.
Why does S increase, not decrease?
• Boltzmann also argues that it is meaningless to argue about why the past is the
low-entropy direction in time, that given an asymmetry we are bound to use
different names for that direction and the other one.
• The question for Boltzmann is why there's an asymmetry,
not why the "past" has a particular property.
• The implication is that if somehow a low-entropy configuration happened by very
rare accident, the times around that, in which entropy is changing monotonically in
time, will be suitable for experiencing one direction as past and the other as future.
• The explanation for why we are observing one of these rare stretches of time
would then be that no observers could evolve in more ordinary, equilibrium times.
– An early anthropic argument!
• This still does nothing to explain the homogeneity of time's entropic arrow: why
would it be changing the same way throughout the known universe? If the original
low-entropy condition is an accident, we need an explanation for why the whole
universe is part of the same accident. That will require another look at cosmology.
– And also is a first look at a still unsolved problem!
Entropy: the Liouville problem
• There's a classical theorem that the volume in classical "phase space" that some
system might be in does not change in time as it follows the classical equations of
motion. (see picture , Sklar Fig. 3.11)
• But the classical entropy is just the log of the accessible volume in phase space.
• So how can it increase in time?
• Although the net volume in phase pace doesn't change, the possible results change
from being a solid trunk of nearby values (all positions and momenta
approximately known) to a set of small fibers (all the positions and momenta might
be very near any of a large number of very different possibilities).
• If your description is forced to be "coarse grained" that means that you lose all
predictability about the results. I.e. you get some useless information about exactly
what the coordinates are if they are approximately known, but you lose the basic
information about what the approximate values will be.
• How can a basic law of nature depend on our "coarse graining" of the description?
quantum coarse-graining
• There's a quantum analog of the coarse-graining problem.
• There is no linear operator (i.e. linear function of y like the operators
representing all the other physical variables) whose expectation value
always increases in time. Entropy is not like the other physical variables.
• Given some initial probabilities to be in a collection of energy eigenstates,
the linear time dependence says that those probabilities never change in a
closed system. (quantum Liouville)
• Hence the entropy (defined in terms of those probabilities) never changes.
• Only if you choose some cruder description than the formal probabilities of
the states can you get an entropy increasing in time in a closed system.
• Again, how can a universal law of nature depend on the crudeness of our
description?
• The 2cd law is not philosophy, but the key guiding principle in chemistry,
thermal physics, engine design, materials science, etc.
• How come they escape the philosophical problems?
Quantum coarse graining
• The 2cd law says that all the allowed quantum states become equally probable in
a closed system. But we ran into problems, because if you're in some state of a
closed system, you stay in that state.
• The density of states of real systems becomes exponentially huge as you consider
larger systems. The states are extremely close to each other in energy.
• Although external perturbations become small and unsystematic, there is no way
to prevent them from causing transitions to nearby states. They serve to
randomly stir the system. Thus for large systems, the claim that the system
actually is in one particular quantum state can have no practical consequences.
• Also, for large systems, the dependence of the entropy on the details of what's
specified (e.g. whether you specify energy or temperature) becomes very small
compared to the total entropy.
• So in practice, for large systems you can just speak of an entropy, without
worrying about the details of the knowledge of the system.
Entropy’s philosophical problem
• The inevitable tiny interactions with the outside world stir any large system among
its many quantum states. So it's fine to use the concept of entropy in practice.
• But what are we to make of the claim "The entropy of the universe always
increases"?
• Is the universe in a quantum state or not?
– If it is, what is the definition of its entropy?
• Remember, log[1]=0, always.
• Is there some physics, currently unknown to us, which sets an intrinsic coarsegraining scale, i.e. some scale on which the wave-function is not an appropriate
description?
• Is that related to the quantum measurement question of whether there is some
natural scale on which the wave-equation no longer gives the complete description
of the time-dependence?
• Is it related to quantum gravity?
Entropy and Quantum Entanglement
• No system (except possibly the whole universe) is completely closed.
• Interactions cause local systems to become entangled with increasingly remote
systems.
• Start with two weakly interacting remote systems, un-entangled.
Any possible state of A can go with any state of B, so:
– Stot=SA+SB
• The Liouville theorem say that Stot doesn’t change.
– Assuming the fundamental form of QM is correct.
• If they each evolved independently the Liouville theorem would say that each of SA
and SB would stay constant.
• The interaction makes A and B entangled. Some states of A coexist only with some
states of B and vice versa.
– e.g the dead cat in A only coexists with the dog in B seeing a dead cat, the live
cat in A only coexists with the dog in B seeing a live cat.
Second Law and Entanglement
• Say there were 1010 initial states possible for each A and B, or 1020 total for the
whole un-entangled combo.
• After a little entanglement you have 1011 possible states for each. But each can only
coexist with 109 states of the other.
• Still 1020 total.
• Since not every state in A pairs up with every state in B,
Stot< SA+SB .
• So we started with Stot= SA+SB .
• And now have Stot< SA+SB .
• But Stot didn’t change (by theorem).
• So SA+SB must have grown!
• Increasing entanglement is the same as increasing entropy of the various local
parts. There’s a negative entropy of remote entanglement.
• Can this growth of local entropies go on forever?
• That depends on cosmology!
• But before we can really discuss cosmology, we need to know a little more about
the ingredients
– How to combine SR+QM?
Main Arrows of Time
We can’t reduce all these arrows to one, but can explore connections among them.
• Psychology:
– we remember the past, but we calculate the future.
• Entropy:
– always increases (2nd Law)
• Quantum measurement:
– the ambiguity in practice always concerns future states, not past ones.
• Cosmology:
– the universe is expanding, maybe forever.
• Black holes:
– the reversed object, a white hole, has never been found.
• Radiation:
– the E-M radiation field is determined by the sources it comes from, not the
places it's going to.
• The weak nuclear force
– is not reversible.
The one known failure of microscopic time reversal
• Quantum field theory says that CPT (Charge-Parity-Time) symmetry is obeyed.
This means if we film an antimatter world in a mirror, then show it backward, it
will obey all the rules for a film of our own world running forward.
• In principle the separate symmetries (C, P, T) don’t have to work.
• If time reversal invariance (T) holds, then processes should run equally well
forward and backward.
• In elementary particle physics, there is a particle called the Ko meson. It has an
antiparticle partner called the anti-Ko. the meson often decays: K pp.
The reverse process does not have the same rate.
– For practical reasons (other arrows of time!), this is not what is measured.
– CPT is used to infer the reversed rate.
– Other processes involving the weak nuclear
force show T-violation directly.
• The one known microscopic asymmetry plays
no role in ordinary events,
but will be crucial to understanding the
matter-antimatter asymmetry of the universe.
Electromagnetism?
• The radiation field in E-M is given as a function of the past positions, velocities,
accelerations of charged particles. There is no reason in the formalism not to
include future sources as well. Why the asymmetry?
• Actually, a given radiation field can be described either way, in terms of where it's
coming from or in terms of where it's going to. It's a convention to always use the
past sources not the future ones. This convention is strongly motivated by the
other time asymmetries, specifically the entropic one.
• In an equilibrium condition, with particles emitting and absorbing radiation
thermally, there would be no reason whatsoever to pick the past over the future as
the "source" of the radiation.
• So the radiation arrow is really another example of the entropic and (quantum
measurement?) arrow.
What about the "collapse of the wave function"?
• Its standard description sounds highly irreversible.
• Let's assume that the collapse actually happens. That means that a system which
was in a single quantum state, which could turn out to have (say) two
macroscopically distinct outcomes ends up in a state corresponding to only one of
those. It's still in a single quantum state,
although not in one that could have been predicted ahead of time.
• In principle, the wave equation allows the following time-reversed situation: a
system is found in a state which could have arisen from either of two
macroscopically distinct predecessors. Then we could argue about whether both of
those predecessors really existed (a reversed MW picture) or whether there had
been a discontinuous change in a state to make it look just like one that could have
arisen from another state (a time-reversed collapse picture).
• No time-reversed-measurement process has ever been observed.
• Thus QM also has an observed "arrow of time”.
• How does it connect with the entropic and psychological arrows?
Entropy and Quantum Measurement
• Actual QM measurements often allow one
macroscopic world to evolve into any of
Live cat
several possible distinct outcomes.
• The reverse process is not observed.
• Statistical mechanics describes events in Ice statue of cat
which a large number of distinct
possibilities evolve into results which are
Ice statue of dog
so macroscopically similar as to be
practically indistinguishable.
Live cat
Dead cat
puddle
The result is that our world is consistent with only one past but many futures in precise
detail (QM irreversibility) but macroscopic events are often consistent with a huge
variety of macroscopically distinct predecessors, while often giving macroscopically
unique consequences. This combined situation provides a possible arena for memory,
i.e. detailed physical states which give information about the past which is not implicit
in miscellaneous macroscopic events.
Psychology and entropy
(see Sklar, Hawking)
• Despite Sklar's doubts, there seems to be a close connection between our mental
asymmetry and the entropic asymmetry. For starters, in an equilibrated world
(entropy already maximized, no entropic time asymmetry) there is no information
(that's why Maxwell's demon can't function) so there could be no minds whatever.
• The low-entropy property of the past is needed to allow some information about it
to be conveyed with a relatively small number of bits. I.e. a photograph of some
place tells you a lot about it, but a photograph of some equilibrated mush conveys
essentially no information. When we find some highly ordered artifact, (say a fossil)
we use it primarily to find out about the past, not the future. The ability of the past
to leave informative records reflects its order.
• Memories, like pictures, etc. are also an informative record of the past, stored in a
small fraction of the number of particles represented. It is natural to informally
associate the direction in time for which such mental records exist with the
direction in time for which ordinary physical records exist. There is no formal proof
that that direction is the same as the lower-entropy direction, since the near future
is also very far from equilibrium and therefore in principle could also leave traces.
Still, if you accept that the asymmetry between evidence of past and future states
flows from their entropic asymmetry, there is no reason to exclude mental
evidence. Probably the psychological arrow flows from the entropic arrow.
Psychology and quantum measurement.
• Let's imagine a world in which reversed QM measurements occurred. The current
physical state of every object, including our brains, would be a potential outcome
of the evolution of any of a number of macroscopically distinct prior states, just as
in our world it could evolve into any one of a number of macroscopically distinct
subsequent states.
• It would no more be possible to simply remember a unique past than it is now
possible to accurately foresee a unique future in QM experiments (or anything
else). In other words it would be impossible in principle to tell if a time-reversed cat
had been alive or dead previously! Both memories would be equally valid.
• Philosophers would debate whether there really had been multiple distinct pasts,
or if there were anti-collapses, in which the wave function acquired properties
which looked just like those that would have been inherited from non-existent
pasts.
• So the psychological distinction between past and future is closely connected with
the asymmetry of the QM measurement process.
• It is unclear if, without some coherent memory, any mind could develop to the
point of wondering about symmetries. I.e. the QM measurement asymmetry may
be a prerequisite for consciousness.
Cosmology and Black-Holes
• Various types of black holes (e.g. collapsed stars) follow naturally from the
standard cosmological picture, together with the law of increasing entropy. There
is no reason to expect there to be any white-holes (time-reversed black holes,
with ordinary matter pouring out). However, they are also solutions of the G-R
(reversible) equations. So there is presumably some connection between their
absence and the other irreversibilities, but it is not yet elucidated.
Cosmology and entropy
•
•
•
If the increasing-entropy arrow of time is to be consistent, there must be no backwardin time loops in space-time geometry. Any such loops would prevent there from being
ANY consistent global time-ordering for the whole space-time, as we noted when
worrying about causality.
There have been many speculations that somehow the expansion of the universe and
the increasing entropy are connected. Both at least affect all of the universe we know.
However, there is no established reason why a slowly contracting universe would obey
different local laws than a slowly expanding one. If the universe were to someday start
to contract, it seems unlikely that there will be any short-term change in the behavior of
steam-engines, etc. as a result. But we haven’t done the experiment.
Is irreversibility compatible with cyclic cosmologies?
– In certain cases, it seems possible.
• Entropy associated with a given collection of matter goes up as it gets more
entangled with other things.
• Some re-start phenomena (collisions of infinite branes, birth of baby universes
by quantum fluctuations) restarts some subset with low internal
entropy/particle by increasing number of particles, not by entropy loss overall.
Links between between measurement and entropy arrows?
• Start with collection of little distinct non-interacting beads in an isolated container.
Place them all in a small region. Each one's y spreads out. Decoherence (via
“measurement” entanglement with other things) would then split these spreadout y's into different "worlds". After many repetitions of the process, the density
of worlds (or potential worlds, if collapse happens) with any particle near any
position would become uniform. If you haven't opened the box (i.e. let it interact
enough with your mind to know the contents) the best guess you can make about
its contents is based on that uniform probability density- the same one that the
entropy maximization rule says to use.
• So the arrow that says the universe starts with low entanglement between remote
objects is the same as the arrow that ordinary local entropy terms start small.
• If somehow the resolution of the measurement problem were to involve
intrinsically non-linear irreversible violations of Schroedinger dynamics, that same
fundamental irreversibility would also give the Second Law. That would get around
the Loschmidt issue, of how to avoid the time-reversed version of Boltzmann’s
argument. The postulated hypothetical physical process would allow only
probabilistic predictions, but would allow deterministic retrodictions. This
conjecture was left over from trying to deal with QM, and not specially contrived
to account for stat mech.
more speculative connections
The point of these speculations is not to provide reliable information, but rather to
suggest how different aspects of these problems might tie together to form
somewhat testable hypotheses in the future.
• If some many-world picture holds, configuration space must expand if it is not to
acquire a significant density of "worlds"- i.e. distinct blobs of wave-function. If it
does acquire an increasing density of worlds, re-coherence processes start to
become common. Then memory becomes impossible.
• Why would configuration space expand? The Carroll-Chen proposal? So if manyworlds holds, the cosmological expansion may have to connect, at least in the
long run, to the arrow of quantum measurement, which in turn connects to the
psychological arrow.