initial entropy
Dashen
I am reading in the introduction of hep-th/0208013: "the low entropy
starting point is the ultimate reason that the universe has an arrow
of time, without which the second law would not make sense." This was
apparently "emphasized by Penrose many years ago".
Could somebody please explain this statement, or perhaps point me to a
useful place to look?
Presumably by low entropy they simply mean that the initial conditions
look finely tuned (very special). Of course the meaning of "special"
or "finely tuned" is rather meaningless without a measure (the density
of states if you will). So I guess they use the effective field theory
intuition to describe what is tuned and what is not. Is this all?
Now what does this have to do with an arrow of time and the second
law?
Let us assume the universe is describable by a conventional
Hamiltonian and Hilbert space. Let us assume the universe started off
in a pure state. I guess the meaning of an "arrow of time existing"
and "the second law being true" can be loosely transcribed as "at late
enough times the pure state looks classical" - that for late enough
times the dynamics is well approximated by classical mechanics on
large enough length scales. You could have imagined a state which
never looks classical at any time, for example the, ground state of a
double well potential. Or you could have imagined a state in which
really funny stuff happens, like every time you flip a coin it comes
up tails.
Are they claiming that these kinds of states are far more numerous and
far more likely? That the initial states which lead to late time
behaviour which looks classical and in which statistical laws are
obeyed (like a sequence 10000 coin flips will more or less give an
equal number of head and an equal number of tails) are vastly
outnumbered by the ones that don't?
thanks,
Dusan
Phillip Helbig---remove CLOTHES to reply
<038d687e-fa68-4cef...@i76g2000hsf.googlegroups.com>,
Dashen <sim...@gmail.com> writes:
> I am reading in the introduction of hep-th/0208013: "the low entropy
> starting point is the ultimate reason that the universe has an arrow
> of time, without which the second law would not make sense." This was
> apparently "emphasized by Penrose many years ago".
>
> Could somebody please explain this statement, or perhaps point me to a
> useful place to look?
@INPROCEEDINGS{RPenrose89a,
AUTHOR = "Roger Penrose",
TITLE = "Difficulties with Inflationary Cosmology",
BOOKTITLE = "Fourteenth Texas Symposium on Relativistic
Astrophysics",
YEAR = "1989",
EDITOR = "Ervin J. Fenyves",
ORGANIZATION = "",
SERIES = "",
ADDRESS = "New York",
PUBLISHER = "New York Academy of Sciences",
VOLUME = "571",
Also, Penrose discusses this in his books The Emperor's New Mind,
Shadows of the Mind, The Large, the Small and the Human Mind, The Road
to Reality.
> Presumably by low entropy they simply mean that the initial conditions
> look finely tuned (very special). Of course the meaning of "special"
> or "finely tuned" is rather meaningless without a measure (the density
> of states if you will). So I guess they use the effective field theory
> intuition to describe what is tuned and what is not. Is this all?
>
> Now what does this have to do with an arrow of time and the second
> law?
>
> Let us assume the universe is describable by a conventional
> Hamiltonian and Hilbert space. Let us assume the universe started off
> in a pure state. I guess the meaning of an "arrow of time existing"
> and "the second law being true" can be loosely transcribed as "at late
> enough times the pure state looks classical" - that for late enough
> times the dynamics is well approximated by classical mechanics on
> large enough length scales. You could have imagined a state which
> never looks classical at any time, for example the, ground state of a
> double well potential. Or you could have imagined a state in which
> really funny stuff happens, like every time you flip a coin it comes
> up tails.
If you know the wave function of the universe, why aren't you rich?
---Murray Gell-Mann [to James Hartle]
Sorry, couldn't resist. :-|
The problem is a classical problem. Perhaps there is some
quantum-mechanical solution, but framing the problem itself quantum
mechanically might not be the best way to address it.
Arnold Neumaier
Our descriptions of the universe are always coarse-grained, and hence
must be modelled by density matrices which develop in an irreversible
fashion in any time-dependent but bounded region of the universe.
This creates the arrow of time, and according to nonequilibrium
thermodynamics, the entropy increases, hence must have been much lower
at the beginning than it is now.
Arnold Neumaier
Gerald Knizia
> Our descriptions of the universe are always coarse-grained, and hence
> must be modelled by density matrices which develop in an irreversible
> fashion in any time-dependent but bounded region of the universe.
What you point out is correct, but I'm not sure that Penrose's argument is
compatible with that. Entropy is NOT a property of a microscopic physical
system. It is a property of /the coarse grained description/ of that system.
There can be multiple descriptions, for exactly the same system, with
different entropies.
So the state of the universe as a whole doesn't have a entropy per se.
I'm always very sceptical if people try to conclude information concrete
microstates (one could assume the universe is in one) from thermodynamic
arguments.
> This creates the arrow of time, and according to nonequilibrium
> thermodynamics, the entropy increases, hence must have been much lower
> at the beginning than it is now.
But the entropy increases in both directions! If you would flip the time,
and evolve backwards, the entropy of your density matrix description would
also increase. Although the pure state which is finally reached is the
initial state...
--
- C. Gerald Knizia/cgk | #28673212 | this mail was made with intention.
Arnold Neumaier
> Arnold Neumaier wrote:
>> Our descriptions of the universe are always coarse-grained, and hence
>> must be modelled by density matrices which develop in an irreversible
>> fashion in any time-dependent but bounded region of the universe.
>
> What you point out is correct, but I'm not sure that Penrose's argument is
> compatible with that. Entropy is NOT a property of a microscopic physical
> system. It is a property of /the coarse grained description/ of that system.
> There can be multiple descriptions, for exactly the same system, with
> different entropies.
> So the state of the universe as a whole doesn't have a entropy per se.
Yes, but it has one in every specific coarse-grained description.
> I'm always very sceptical if people try to conclude information concrete
> microstates (one could assume the universe is in one) from thermodynamic
> arguments.
A coarse-grained description may still be microscopic.
_Every_ description we have is a coarse-grained description since we
ignore gravitation. Thus all our theories would have to be treated,
strictly speaking, with statistical mechanics...
>> This creates the arrow of time, and according to nonequilibrium
>> thermodynamics, the entropy increases, hence must have been much lower
>> at the beginning than it is now.
>
> But the entropy increases in both directions!
It would increase in both directions if both directions existed.
But the big bang is supposed to be the _beginning_, hence one can go
from there only in one direction.
(Except if one assumes a mirror universe having existed before
the big bang, in which time passes backwards.)
Arnold Neumaier
Chris H. Fleming
The entropy of a pure state is zero, so how is it increasing?
Arnold Neumaier
No subsystem of a bigger system is in a pure state, except if
the dynamics is extremely trivial, or the number of degrees of
freedon is so small that the system can be prepared in such a
state by special action.
As a result, every nontrivially large system is in a mixed state.
(Statistical mechanics relies on this.)
Now all the systems that we have ever observed are subsystems of bigger
systems, hence they are described by mixed states.
By extrapolation, it is likely that the universe as a whole is also
in a mixed state, although it _could_ possibly be in a pure state.
But there is no intrinsic reason why it should be in one, and
whether it is or not is not empirically testable.
A moreserious problem is that the universe is usually supposed to be
a closed system. In that case it should be governed by a Hamiltonian
dynamics, which would imply that the entropy remains constant,
even for a mixed state.
However, in general relativity, there is no well-defined total energy,
so there is probably also no well-defined total entropy.
I guess that the entropy of the universe is infinite.
But there is always local production of entropy, so in some sense,
the entropy increases although always being infinite.
Arnold Neumaier
Chris H. Fleming
wrote:
He specifically said pure state. So again, how can anything evolving
backwards in time into a pure state possibly be increasing in entropy?
Not that you would agree with that.
> As a result, every nontrivially large system is in a mixed state.
> (Statistical mechanics relies on this.)
The mixed state formalism is merely more mathematically convenient.
One can go beyond a mixed state formalism and obtain the stochastic
fluctuations of the pure states.
> Now all the systems that we have ever observed are subsystems of bigger
> systems, hence they are described by mixed states.
>
> By extrapolation, it is likely that the universe as a whole is also
> in a mixed state, although it _could_ possibly be in a pure state.
> But there is no intrinsic reason why it should be in one, and
> whether it is or not is not empirically testable.
Extrapolating there would be a fallacy of composition.
> A moreserious problem is that the universe is usually supposed to be
> a closed system. In that case it should be governed by a Hamiltonian
> dynamics, which would imply that the entropy remains constant,
> even for a mixed state.
>
> However, in general relativity, there is no well-defined total energy,
> so there is probably also no well-defined total entropy.
>
> I guess that the entropy of the universe is infinite.
> But there is always local production of entropy, so in some sense,
> the entropy increases although always being infinite.
There is a Hamiltonian constraint, which is sufficient for first
quantization, a.k.a. the Wheeler-deWitt equation.
Although the operator ordering ambiguity leaves it's solutions
ambiguous in the higher orders of a semiclassical expansion.
But the (theoretical) fact that black holes and cosmologies can have a
well defined entropy, and the fact the one can even derive Einstein's
field equations from an a priori assumption of the laws of
thermodynamics and the Bekenstein entropy formula, would lead one to
at least consider that perhaps a closed system/Hamiltonain/Lagrangian
quantization formulation is not the proper route to take... at least
with the metric and connection variables.
Arnold Neumaier
I was challenging with my remark precisely the fact that he was
talking of a pure system. Assuming a pure state at any time is
inconsistent with nonequilibrium thermodynamics.
This has nothing to do with the direction of time.
>> As a result, every nontrivially large system is in a mixed state.
>> (Statistical mechanics relies on this.)
>
> The mixed state formalism is merely more mathematically convenient.
> One can go beyond a mixed state formalism and obtain the stochastic
> fluctuations of the pure states.
Bot the observable consequences are identical.
The mixed state is the most complete description of a system.
>> Now all the systems that we have ever observed are subsystems of bigger
>> systems, hence they are described by mixed states.
>>
>> By extrapolation, it is likely that the universe as a whole is also
>> in a mixed state, although it _could_ possibly be in a pure state.
>> But there is no intrinsic reason why it should be in one, and
>> whether it is or not is not empirically testable.
>
> Extrapolating there would be a fallacy of composition.
If all states we observe are mixed, wh"y _should_ we assume that
the state of the universe is pure. It doesn't help a thing, and
treating all states in the same way has the advantage of simplicity.
>> A more serious problem is that the universe is usually supposed to be
>> a closed system. In that case it should be governed by a Hamiltonian
>> dynamics, which would imply that the entropy remains constant,
>> even for a mixed state.
>>
>> However, in general relativity, there is no well-defined total energy,
>> so there is probably also no well-defined total entropy.
>>
>> I guess that the entropy of the universe is infinite.
>> But there is always local production of entropy, so in some sense,
>> the entropy increases although always being infinite.
>
> There is a Hamiltonian constraint, which is sufficient for first
> quantization, a.k.a. the Wheeler-deWitt equation.
> Although the operator ordering ambiguity leaves it's solutions
> ambiguous in the higher orders of a semiclassical expansion.
>
> But the (theoretical) fact that black holes and cosmologies can have a
> well defined entropy, and the fact the one can even derive Einstein's
> field equations from an a priori assumption of the laws of
> thermodynamics and the Bekenstein entropy formula, would lead one to
> at least consider that perhaps a closed system/Hamiltonain/Lagrangian
> quantization formulation is not the proper route to take... at least
> with the metric and connection variables.
The relation between black holes and entropy strongly suggests at
least to me that the correct theory of the universe as a whole should
be thermodynamic in nature.
Arnold Neumaier
Chris H. Fleming
wrote:
> Chris H. Fleming schrieb:
>
>
>
An individual classical (or even quantum) trajectory of a Brownian
particle is one realization of the noise among many. This is the kind
of information that the mixed state cannot give you. It is a
mathematical impossibility as unravelings of the mixed states are not
mathematically unique.
The noise average of fluctuations is diffusion, but to ignore the
fluctuations as a physical phenomena and say that the diffusion is a
complete description? A coarse-graining is obviously not the most
complete description of a system.
> >> Now all the systems that we have ever observed are subsystems of bigge
Darwin123
> Hi,
>
> I am reading in the introduction of hep-th/0208013: "the low entropy
> starting point is the ultimate reason that the universe has an arrow
> of time, without which the second law would not make sense." This was
> apparently "emphasized by Penrose many years ago".
The entropy is sometimes defined as Boltzmann's constant times the
natural logarithm of the number of degrees of freedom.
Given this definition, it is obvious. For wavelike excitations,
the number of degrees of freedom is proportional to the volume the
waves are confined in. Consider standing waves in a given volume. In
any case, all energy and mass can be describes as waves. Thus, as the
universe expanded, the number of degrees of freedom increased as the
volume increased.
In physics, freedom is associated with disorder. Instead of
saying, "The disorder increases with time" we could say "the freedom
increases with time." So being an optimist, I would associate entropy
with freedom instead of disorder |:-)
Arnold Neumaier
The classical mixed state of a Brownian particle is a probability
distribution over all possible individual trajectories.
A classical pure state is a particular trajectory. There is neither
noise nor are there fluctuations; the trajectory is completely
deterministic. One therefore cannot even speak of a Brownian
particle anymore, since this concept is _defined_ probabilistically.
> This is the kind
> of information that the mixed state cannot give you. It is a
> mathematical impossibility as unravelings of the mixed states are not
> mathematically unique.
That's precisely why a pure state description is _impossible_ for
a system correctly described by a mixed state.
> The noise average of fluctuations is diffusion, but to ignore the
> fluctuations as a physical phenomena and say that the diffusion is a
> complete description? A coarse-graining is obviously not the most
> complete description of a system.
Bu a probability density is the most complete description of a
stochastic process. It contains _everything_ you can say about
the process.
If you can say more about a system then it is not fully described
by the stochastic process under discussion. In this case, the latter
is only a convenient approximate summary of the former.
What holds for classical processes holds with appropriate adaptations
also for quantum stochastic processes.
Arnold Neumaier
Chris H. Fleming
[Moderator's note: Huge amount of multiply-quoted text snipped. -P.H.]
> >> The mixed state is the most complete description of a system.
>
> > An individual classical (or even quantum) trajectory of a Brownian
> > particle is one realization of the noise among many.
>
> The classical mixed state of a Brownian particle is a probability
> distribution over all possible individual trajectories.
>
> A classical pure state is a particular trajectory. There is neither
> noise nor are there fluctuations; the trajectory is completely
> deterministic. One therefore cannot even speak of a Brownian
> particle anymore, since this concept is _defined_ probabilistically.
>
> > This is the kind
> > of information that the mixed state cannot give you. It is a
> > mathematical impossibility as unravelings of the mixed states are not
> > mathematically unique.
Everything you are saying is wrong.
One can derive the Fokker-Plank equations from a microscopic model.
Corresponding to this microscopic model are Langevin equations for the
pure states.
There most certainly is noise/fluctuation... that is what gives you
diffusion!
In general, there are many unravelings of mixed state, but this is a
mathematical statement and has nothing to do with the physics here. One
can construct unravelings which cannot possibly have any physical
interpretation at all. E.g. probabilities only being preserved on
average.
If you have a microscopic model, then the unravelings are uniquely
determined by the corresponding Hamiltonian dynamics.
> That's precisely why a pure state description is _impossible_ for
> a system correctly described by a mixed state.
>
> > The noise average of fluctuations is diffusion, but to ignore the
> > fluctuations as a physical phenomena and say that the diffusion is a
> > complete description? A coarse-graining is obviously not the most
> > complete description of a system.
>
> Bu a probability density is the most complete description of a
> stochastic process. It contains _everything_ you can say about
> the process.
You know that the unravelings aren't unique. You stated this above. The
Fokker-Plank equation doesn't give you the statistics of the stochastic
process at all.
One can rig all kinds of processes to fit the mixed state dynamics.