Boltzmann's H Theorem
Kieran Mullen
boning up on kinetic theory, which is not my field of expertise. I'm
using Kerson Huang's textbook on Thermodynamics and Statistical
Mechanics.
In it, (as in many other books) he develops Boltzmann's H-Theorem. But
then he goes on to claim that there's no big mystery about its lack of
reversibility and the reversible laws of mechanics. However, Garrod
in his book seems to imply that there are still a lot of subtleties and
some things that are not understood. What is the consensus on this?
Is there consensus?
Huang's explanation is that the monotonic decrease in H rests on the
"molecular chaos" assumption. The system only intermittently achieves
molecular chaos, and so about that point H has a peak (forward in time
due to the H-Theorem, and back about that point due to reversibility).
However, in doing so it seems he's thrown out the value of the H-Theorem
since he cannot claim it applies to the long term behavior and the
approach to equilibrium.
It seems to me that to the extent one explains away the apparent
irreversibility in Boltzmann's kinetic theory, you also vitiate its
ability to explain the approach to equilibrium.
Older books (Wannier's, for example) imply that the issue is all
quite murky. Huang seems to say it's all understood.
I'm not asking for the general theory of Time, the Universe and
Everything here. I'm just asking any experts on modern kinetic theory
for the consensus opinion of the value of Boltzmann's H-theorem and
what it proves (and what it doesn't).
Oh - a second quick question. Does anyone know what has Boltzmann's
inspiration to define H as he did? Is it an obvious thing to do? In
retrospect it's clear you need to take a logarithm since we want entropy
to be additive and yet probabilities combine by multiplication. Was
there some standard trick that yielded H=f*log(f), or was it an inspired
guess?
---------------------------------------------------------------------------
Kieran Mullen email: kie...@ou.edu
Dept. of Physics and Astronomy phone: (405) 325-3961
The University of Oklahoma FAX: (405) 325-7557
Norman, OK 73019, USA http://www.nhn.ou.edu/~kieran/
john baez
> I'm teaching a grad course on Statistical Mechanics and I have been
>boning up on kinetic theory, which is not my field of expertise. I'm
>using Kerson Huang's textbook on Thermodynamics and Statistical
>Mechanics.
I haven't read that book. I love his book "Quarks, Leptons and Gauge
Fields" - a nice no-nonsense introduction to the Standard Model. For
statistical physics my favorite has always been Reif's "Statistical
and Thermal Physics". But for foundational issues, like Boltzmann's
H-theorem, I really urge you to look at Zeh's "The Physical Basis of the
Direction of Time" and Huw Price's "Time's Arrow & Archimedes' Point:
New Directions for the Physics of Time". Many physics textbooks
downplay the real mystery of the second law of thermodynamics. These
books delve into it!
>In it, (as in many other books) he develops Boltzmann's H-Theorem. But
>then he goes on to claim that there's no big mystery about its lack of
>reversibility and the reversible laws of mechanics. However, Garrod
>in his book seems to imply that there are still a lot of subtleties and
>some things that are not understood. What is the consensus on this?
>Is there consensus?
There's a lot of controversy, but the people who take the trouble to think
about these things carefully usually avoid the silliest mistakes, so there's
at least some consensus among experts. For example, one consensus among
experts is that you cannot derive a time-asymmetric law (e.g. that entropy
increases with time) from time-symmetric equations of motion without some
time-asymmetric assumption.
In the case of the H-theorem, this time-asymmetric assumption is the
what Boltzmann called the Stosszahlansatz (collision number assumption).
So the question is: why, and under what conditions, is this assumption
correct - or at least approximately so?
That is where things get interesting.
> Huang's explanation is that the monotonic decrease in H rests on the
>"molecular chaos" assumption. The system only intermittently achieves
>molecular chaos, and so about that point H has a peak (forward in time
>due to the H-Theorem, and back about that point due to reversibility).
>However, in doing so it seems he's thrown out the value of the H-Theorem
>since he cannot claim it applies to the long term behavior and the
>approach to equilibrium.
Huang's explanation sounds rather odd. The H-theorem shows that under
certain assumptions the time derivative of entropy is greater than or
equal to zero. If these assumptions only held at moments when the
entropy is at a minimum, the H-theorem wouldn't be all that useful.
You certainly don't need the H-theorem to see that the entropy of a
finite-sized box of atoms will occaisionally have minima and that it
will descrease before these minima and increase afterwards!
(For anyone wondering, entropy is minus Boltzmann's constant times what
Boltzmann called "H", so decreasing H is the same as increasing entropy.)
> Oh - a second quick question. Does anyone know what has Boltzmann's
>inspiration to define H as he did? Is it an obvious thing to do? In
>retrospect it's clear you need to take a logarithm since we want entropy
>to be additive and yet probabilities combine by multiplication. Was
>there some standard trick that yielded H=f*log(f), or was it an inspired
>guess?
I don't know what inspired Boltzmann to define H the way he did, but
the work of Gibbs makes it perfectly clear that entropy should be
defined the way it is. Do you want me to explain Gibbs' reasoning or
are you particularly interested in what was going through Boltzmann's
mind when he defined H?
BillyFish
<< Older books (Wannier's, for example) imply that the issue is all
quite murky. Huang seems to say it's all understood.
>>
Go older. Look at Toman's book. First class!
Aaron Bergman
>In article <36A5A27B...@ou.edu>, Kieran Mullen <kie...@ou.edu> wrote:
>> I'm teaching a grad course on Statistical Mechanics and I have been
>>boning up on kinetic theory, which is not my field of expertise. I'm
>>using Kerson Huang's textbook on Thermodynamics and Statistical
>>Mechanics.
>
>I haven't read that book. I love his book "Quarks, Leptons and Gauge
>Fields" - a nice no-nonsense introduction to the Standard Model. For
>statistical physics my favorite has always been Reif's "Statistical
>and Thermal Physics". But for foundational issues, like Boltzmann's
>H-theorem, I really urge you to look at Zeh's "The Physical Basis of the
>Direction of Time" and Huw Price's "Time's Arrow & Archimedes' Point:
>New Directions for the Physics of Time". Many physics textbooks
>downplay the real mystery of the second law of thermodynamics. These
>books delve into it!
I'll look into those. I was looking through my little brother's
Serway and it contained statements about the heat death of the
universe that seem completely unjustified as far as I can tell (I
posted about this many moons ago.)
>
[...snip...]
>
>In the case of the H-theorem, this time-asymmetric assumption is the
>what Boltzmann called the Stosszahlansatz (collision number assumption).
I think this is generally termed the assumption of molecular
chaos in the text books I've seen.
>So the question is: why, and under what conditions, is this assumption
>correct - or at least approximately so?
IIRC, (Huang is unfortunately 3K miles away right now), this is
the heart of Huang's exposition.
>
>That is where things get interesting.
>
>> Huang's explanation is that the monotonic decrease in H rests on the
>>"molecular chaos" assumption. The system only intermittently achieves
>>molecular chaos, and so about that point H has a peak (forward in time
>>due to the H-Theorem, and back about that point due to reversibility).
>>However, in doing so it seems he's thrown out the value of the H-Theorem
>>since he cannot claim it applies to the long term behavior and the
>>approach to equilibrium.
>
>Huang's explanation sounds rather odd. The H-theorem shows that under
>certain assumptions the time derivative of entropy is greater than or
>equal to zero. If these assumptions only held at moments when the
>entropy is at a minimum, the H-theorem wouldn't be all that useful.
But, it necessarily has to hold. As everything in sight is time
reversible, if, at a point where molecular chaos holds, H must be
at a peak as the slope must be negative on either side.
Of the few expositions on entropy I've read, Huang's has made the
most sense, especially regarding the popular statements of the
second law.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
BillyFish
>>
I should have spelled it correctly. Tolman.
By the way, it is an excellent quantum mechanics text as well.
William Buchman
john baez
Aaron Bergman <aber...@princeton.edu> wrote:
>As everything in sight is time
>reversible, if, at a point where molecular chaos holds, H must be
>at a peak as the slope must be negative on either side.
This doesn't make sense to me, for a number of reasons.
0. It's not grammatical - I don't see what the "if" is doing there.
1. The distinguishing feature of a peak is that the slope is
zero at the peak, negative on the right side, and positive on
the left side. So I don't understand the part about "at a peak...
the slope must be negative on either side".
2. The molecular dynamics is reversible but the Stosszahlansatz
is not reversible; it is time-asymmetric. When it is true, the
slope of the H-function is either zero or negative. So I don't
understand the part about "everything in sight is time reversible".
Gary A Pajer
stopped thinking about it cuz it hurt, and because I'm an
experimentalist. :) I *do* remember that Wannier, in his book, gives
a description of a small finite analogue, the "Kac circle". I think it
allows for "molecular chaos", and it leads to time-asymetric behavior.
I told myself that if I ever wanted to understand H, I would start with
the Kac circle. Maybe you (we) should take a look at the Kac circle.
Gary
Andrzej Pindor
>
>................. But for foundational issues, like Boltzmann's
> H-theorem, I really urge you to look at Zeh's "The Physical Basis of the
> Direction of Time" and Huw Price's "Time's Arrow & Archimedes' Point:
> New Directions for the Physics of Time". Many physics textbooks
> downplay the real mystery of the second law of thermodynamics. These
> books delve into it!
Unfortunately, Price's book is full of inconsistencies and confused
logic. Three weeks ago I wrote about it in sci.physics, giving a few
examples of my objections to Price's reasoning (I can provide many more
in case anyone is interested). Here is what I wrote:
....
1. On page 26 the author observes that to prove H-theorem (i.e. to show
that entropy of a system increases with time) one needs to make an
assumption known as 'stosszahlansatz', which is a form of Principle
of Independence of Incoming Influences (PI^3). He then goes on to
criticize this assumption as unjustified and biased since it is clearly
time-asymmetric. This is fine, except that on page 39 he writes:
"What the discussion in the second half of nineteenth century revealed
is that thermodynamic equilibrium is a natural condition of matter, .."
and later "The statistical considerations suggest that a future in which
entropy reaches its maximum is not in need of explanation;"
Well, for the mentioned discussions and considerations to reach the
stated conclusions 'stosszahlansatz' is required, so either PI^3 is invalid
and he should not be using the stated conclusions as a support for his
arguments which follow, or he accepts the conclusions as valid but then his
critique of PI^3 is inconsistent. He cannot have it both ways.
2. On pages 71 and 72, when discussing the arrow of radiation he writes:
"...large-scale coherent sources of radiation are common..." and "What
remains to be explained is why there are large coherent sources - processes
in which huge numbers of tiny transmitters all act in unison - and why
there are no large coherent absorbers...". He repeats the same statement on
page 148 where he writes: "The universe as we know it contains big coherent
sources of radiation".
Maybe I am totally mistaken, but I know nothing of plentiful
"large-scale coherent sources of radiation" in the universe. In fact,
from the context it seems that according to the author, stars and galaxies
are such sources. Of course we know that they are not and the author
attempts to explain a problem which only exists in his imagination.
3. On page 144 the author attempts to demonstrate dangers of introducing
accidently time direction bias, taking as an example Arntzenius
discussion of transition probablities. He writes "Suppose that we have
100 identical fair coins, each of which is to be tossed once at a
time 't'. Then the probability that an arbitrary coin will be heads
after 't' given that it was, say, heads before 't', is independent of
the initial distribution - that is a number of coins that initially
showed heads. Not so the reverse "transition probability", such as
the probability of heads before 't' given heads after 't': if 99
of the coins initially showed heads than this latter probability is .99
(assuming fair coins);"
Now, how is the probability of finding heads before 't', given that
we know how many coins showed heads initially, a "transition probability"??
What transition? Such probability does not depend on coins being fair,
or whether there is a transition at all! Interestingly, the author puts
the words "transition probablity" in the quotes himself, but this is another
example where he wants to have things both ways - on one hand he pretends
that he is talking about transition probabilities and on the other he
implies that he is aware that the quantity he talks about is not really
a "transition probability".
4. On page 145 he writes: Consider two friends who maintain their
friendship entirely on chance encounters. ... On occasions on which
they do meet, their prior movements are correlated (they both happen to
enter the same cafe at roughly the same time, for example)".
If the meetings above are an evidence that the prior movements of
these two people are correlated, how are these "chance encounters"? The
author seems to have a strange notion of what 'correlated' means or
what 'chance' means. Extending his reasoning we can easily argue that
everything is 'correlated' with everything else.
5. On page 180 the author writes: "The argument (for advanced action -
my comment) will have to rely on nonobservational considerations: facts
such as simplicity, elegance and symmetry, for example. This doesn't mean
that the issue is nonemprical, or metaphysical in the disparaging sense."
I find it amazing that the author, who claims that his reasoning
reveals the aspects of the structure of nature which are independent of
observing agent's bias wants to base his reasoning on such purely
subjective, agent's dependent notions as simplicity or elegance. As for
symmetry, why should the preference for the symmetry over the lack of it
be considered as anything else than a bias resulting from the way we, as
agents, perceive the world?
Andrzej
--
Dr. Andrzej Pindor The foolish reject what they see and
University of Toronto not what they think; the wise reject
Information Commons what they think and not what they see.
andrzej...@utoronto.ca Huang Po
Phone: (416) 978-5045
Fax: (416) 978-7705
Aaron Bergman
>In article <slrn7ageg7....@treex.Stanford.EDU>,
>Aaron Bergman <aber...@princeton.edu> wrote:
>>As everything in sight is time
>>reversible, if, at a point where molecular chaos holds, H must be
>>at a peak as the slope must be negative on either side.
>
>This doesn't make sense to me, for a number of reasons.
>
>0. It's not grammatical - I don't see what the "if" is doing there.
>
>1. The distinguishing feature of a peak is that the slope is
>zero at the peak, negative on the right side, and positive on
>the left side. So I don't understand the part about "at a peak...
>the slope must be negative on either side".
Gak. That came out quite poorly. Let me try again. Assume we have
a point where molecular chaos holds. By the H-theorem, H must
decrease. However, the laws of physics are time-reversal
invariant, H has to be locally at a peak. Does that make any
better sense?
Huang probably explains it much better than I do.
[Moderator's note: Penrose discusses this in his various writings on
quantum physics and consciousness. The basic idea is that one cannot
explain the arrow of time by saying that things move from a more to a
less ordered state, since, by time symmetry, that would imply that
things would move from a more to a less ordered state AS ONE MOVES INTO
THE PAST as well. Thus one has the puzzle of the extremely well-ordered
state of the beginning of the universe. This can explain the arrow of
time but one needs to understand why the universe started in such an
improbable state. I believe this is what the poster is referring to. -P.H.]
Kieran Mullen
> Gak. That came out quite poorly. Let me try again. Assume we have
> a point where molecular chaos holds. By the H-theorem, H must
> decrease. However, the laws of physics are time-reversal
> invariant, H has to be locally at a peak. Does that make any
> better sense?
>
> Huang probably explains it much better than I do.
>
> [Moderator's note: Penrose discusses this in his various writings on
> quantum physics and consciousness. The basic idea is that one cannot
> explain the arrow of time by saying that things move from a more to a
> less ordered state, since, by time symmetry, that would imply that
> things would move from a more to a less ordered state AS ONE MOVES INTO
> THE PAST as well. Thus one has the puzzle of the extremely well-ordered
> state of the beginning of the universe. This can explain the arrow of
> time but one needs to understand why the universe started in such an
> improbable state. I believe this is what the poster is referring to. -P.H.]
>
No, I don't think that's what Aaron meant. He's referring to an
argument in Huang's book, to which I referred when I started this
thread. Huang argues that when molecular chaos (MC) holds, we know
dH/dt is negative definite (the H-theorem). But if at that instant we
reverse all the velocities, we get that H must have just increased to
that point. So at the instant of MC, we get a peak. (This is Huang's
claim.)
John Baez writes:
>2. The molecular dynamics is reversible but the Stosszahlansatz
>is not reversible; it is time-asymmetric. When it is true, the
>slope of the H-function is either zero or negative. So I don't
>understand the part about "everything in sight is time reversible".
It is true that boundary conditions are an important part of
understanding what's going on. And to do justice to this I need to
think a bit more carefully. I just wanted to point out that in Huang's
book, he argues we can apply it *locally*, looking at how H is changing
in the next instant of time, and that if you reverse time, propagating
backwards you get that H must also decrease in time. So he gets a very
jagged curve for H(t). John seems to be saying that Huang is
misapplying the H-theorem, which may well be true. But we don't need to
apply the BC in the infinite past, we can choose the current instant and
then Huang's argument makes sense to me.
Gary Pajer writes:
>It's been a long time since I thought about this. I remember that I
>stopped thinking about it cuz it hurt, and because I'm an
>experimentalist. :) I *do* remember that Wannier, in his book, gives
>a description of a small finite analogue, the "Kac circle". I think it
>allows for "molecular chaos", and it leads to time-asymetric behavior.
>I told myself that if I ever wanted to understand H, I would start with
>the Kac circle. Maybe you (we) should take a look at the Kac circle.
I just been over that, reading Wannier as I wrote my lecture notes.
His discussion of classical kinetic theory is pretty clear. (Huang
starts out with quantum mechanics which seems to muddle the notation
more to me.) Wannier describes the Kac ring, a deterministic system
which from a given, ordered state proceeds to a more mixed one. The
process is reversible, and in addition, the recurrence time is finite.
The model seems a bit of a straw man to me: it has to do with marbles
making a clocklike progression on a ring, and change color when they
encounter a set of "scatterer" points as they march around the ring.
While cute, it's not (to me) quite the same as real kinetic theory.
Most of the texts on kinetic theory seem a bit old. When the mention
the Poincare' recurrence time, they seem to think this means the motion
is periodic. This seems false to me, since even if you come arbitrarily
close to your initial condition, the dynamics are chaotic, so you won't
reproduce your earlier time evolution. Is this accepted in stat mech
circles?
(Yes, I know that chaos is not the same thing as the arrow of
time...)
Kieran Mullen
--
john baez
Aaron Bergman <aber...@princeton.edu> wrote:
>Let me try again. Assume we have
>a point where molecular chaos holds. By the H-theorem, H must
>decrease.
More precisely, the derivative of H must be negative, or possibly
zero, at any time at which the assumption of molecular chaos holds.
>However, the laws of physics are time-reversal invariant, H has
>to be locally at a peak.
Eh? While it's *possible* for H to be at a local maximum at
a point where the assumption of molecular chaos holds, the idea
is that usually H will be decreasing - its derivative will be
negative. After all, Boltzmann's reason for proving the H-theorem
was to explain why entropy is increasing (hence H is decreasing).
If you could somehow show that in fact the assumption of molecular
chaos only held at a maximum of the H function, you would completely
destroy our interest in the H-theorem, since we don't need any fancy
"H-theorem" to prove that a function decreases as we move away from a
local maximum!
Actually, lots of people think the H-theorem is unsatisfactory because
it leads immediately to the question: "why should the assumption of
molecular chaos hold?" Since this is a time-asymmetric assumption, it
simply *assumes* the existence of an arrow of time. And why should this
be true?
So some of these people say: "Forget the H-theorem. All that really
matters is that entropy increases as you move from a local minimum."
Huw Price
>john baez wrote:
>>
>>................. But for foundational issues, like Boltzmann's
>> H-theorem, I really urge you to look at Zeh's "The Physical Basis of the
>> Direction of Time" and Huw Price's "Time's Arrow & Archimedes' Point:
>> New Directions for the Physics of Time".
>Unfortunately, Price's book is full of inconsistencies and confused
>logic. Three weeks ago I wrote about it in sci.physics, giving a few
>examples of my objections to Price's reasoning (I can provide many more
>in case anyone is interested). Here is what I wrote:
>
>1. On page 26 the author observes that to prove H-theorem (i.e. to show
>that entropy of a system increases with time) one needs to make an
>assumption known as 'stosszahlansatz', which is a form of Principle
>of Independence of Incoming Influences (PI^3). He then goes on to
>criticize this assumption as unjustified and biased since it is clearly
>time-asymmetric. This is fine, except that on page 39 he writes:
>"What the discussion in the second half of nineteenth century revealed
>is that thermodynamic equilibrium is a natural condition of matter, .."
>and later "The statistical considerations suggest that a future in which
>entropy reaches its maximum is not in need of explanation;"
> Well, for the mentioned discussions and considerations to reach the
>stated conclusions 'stosszahlansatz' is required, so either PI^3 is invalid
>and he should not be using the stated conclusions as a support for his
>arguments which follow, or he accepts the conclusions as valid but then his
>critique of PI^3 is inconsistent. He cannot have it both ways.
I don't want to have it both ways. My discussion of PI^3 and the thermodynamic
asymmetry goes like this:
1. I point out (as many others have) that the source of the asymmetry in
Boltzmann's H-Theorem is (what I call) PI^3. If PI^3 held in both directions
-- i.e., if motions were independent in the relevant sense both before and
after interactions -- then the H-Theorem would imply that entropy is
non-decreasing in both directions (in other words, that it is constant). So
the H-Theorem simply shifts the puzzle of the T-asymmetry of the second law
from one place to another: the new puzzle is that of why PI^3 holds "before"
but not "after" interactions.
2. I claim that it turns out that the *really* puzzling half of this puzzle is
why PI^3 *doesn't* hold *after* interactions -- i.e. why the world exhibits
the kind of post-interactive correlations associated with a low entropy past
(the kind of correlations which would look "miraculous" to us if they happened
the other way around). A statistically "natural" world would be one in which
PI^3 held in both directions -- i.e. in which entropy was always high. (The
justification for this claim doesn't rest on the H-Theorem -- which would
indeed introduce the kind of circularity Dr Pindor is worried about -- but on
Boltzmann's statistical considerations.)
3. I point out that no one would think that entropy is low in the past
*because* PI^3 fails after interactions. Everyone thinks the explanation goes
the other way: the low entropy past *explains* the correlations after
interactions. This suggests (I claim) that there is a double standard at work
if we take lack of correlations *before* interactions to explain why entropy
doesn't decrease in the future: What we should say is that to the extent that
PI^3 holds before interactions, that is *because* there is no low entropy
future boundary condition. So the basic orientation of the H-Theorem is
misconceived: it gets the explanatory priority the wrong way round. But this
doesn't matter, because the question the H-Theorem purports to answer -- "Why
does entropy increase towards the future?" -- turns out to be the wrong
question to ask, if one's interest is in the T-asymmetry of thermodynamics.
The right question is: "Why does entropy decrease towards the past?"
>
>2. On pages 71 and 72, when discussing the arrow of radiation he writes:
>"...large-scale coherent sources of radiation are common..." and "What
>remains to be explained is why there are large coherent sources - processes
>in which huge numbers of tiny transmitters all act in unison - and why
>there are no large coherent absorbers...". He repeats the same statement on
>page 148 where he writes: "The universe as we know it contains big coherent
>sources of radiation".
> Maybe I am totally mistaken, but I know nothing of plentiful
>"large-scale coherent sources of radiation" in the universe. In fact,
>from the context it seems that according to the author, stars and galaxies
>are such sources. Of course we know that they are not and the author
>attempts to explain a problem which only exists in his imagination.
To get a visible ripple on a still pond, we need to move lots of water
molecules in the same direction at the same time. The oscillation of the
molecules needs to be in phase, or "coherent". That's all I mean by the term.
(It is first introduced on p. 57, I think.) It's the same for all kinds of
radiation: without big coherent sources of this kind, there wouldn't be any
observable asymmetry of radiation. Of course, many such sources are doing this
at lots of frequencies and/or phases at the same time, so are not coherent in
the sense that lasers are.
Dr Pindor's next two objections concern one of the more philosophical parts of
my book. They are also highly specific to particular examples used in the
book, and hence likely to be opaque to anyone who doesn't have a copy at hand.
Unfortunately, the same will be true of my responses. Still, with the
indulgence of the moderator, here goes ...
>
>3. On page 144 the author attempts to demonstrate dangers of introducing
>accidently time direction bias, taking as an example Arntzenius
>discussion of transition probablities. He writes "Suppose that we have
>100 identical fair coins, each of which is to be tossed once at a
>time 't'. Then the probability that an arbitrary coin will be heads
>after 't' given that it was, say, heads before 't', is independent of
>the initial distribution - that is a number of coins that initially
>showed heads. Not so the reverse "transition probability", such as
>the probability of heads before 't' given heads after 't': if 99
>of the coins initially showed heads than this latter probability is .99
>(assuming fair coins);"
> Now, how is the probability of finding heads before 't', given that
>we know how many coins showed heads initially, a "transition probability"??
>What transition? Such probability does not depend on coins being fair,
>or whether there is a transition at all! Interestingly, the author puts
>the words "transition probablity" in the quotes himself, but this is another
>example where he wants to have things both ways - on one hand he pretends
>that he is talking about transition probabilities and on the other he
>implies that he is aware that the quantity he talks about is not really
>a "transition probability".
This objection and the next concern a section in which I am discussing the
problem of causal asymmetry -- i.e., of the difference between cause and
effect, and temporal orientation. My own view is that the difference between
cause and effect is conventional -- a kind of projection of our own temporal
asymmetry. I point out that anyone who wants to defend a more "objective"
account must be careful not to "pass the buck" -- i.e. to think they have
solved the problem, when in fact they've simply analysed causation in terms of
some notion whose own temporal asymmetry is just as problematic. The notion of
probability is particularly tricky: some versions of it (e.g., some accounts
of chance) have time asymmetry built in, and some versions of it (e.g. pure
evidential or frequency accounts) do not.
I use an argument by Frank Arntzenius to illustrate these points. Being a
clever chap (who well deserves his Latin suffix), Arntzenius is well aware of
the danger of putting the asymmetry in by hand, and therefore wants to allow
transition probabilities in both directions. Nevertheless, I argue that the
asymmetry he claims to find rests on the fact that it focuses on *initial*
rather than *final* boundary conditions. I convert things into evidential
probabilities to make this clear. So I don't "pretend to be talking about
transition probabilities". Rather I argue that anyone who wants to claim that
the analogous diagnosis doesn't hold for "genuine" transition probabilities
will end up passing the buck, in one way or another.
>
>4. On page 145 he writes: Consider two friends who maintain their
>friendship entirely on chance encounters. ... On occasions on which
>they do meet, their prior movements are correlated (they both happen to
>enter the same cafe at roughly the same time, for example)".
> If the meetings above are an evidence that the prior movements of
>these two people are correlated, how are these "chance encounters"? The
>author seems to have a strange notion of what 'correlated' means or
>what 'chance' means. Extending his reasoning we can easily argue that
>everything is 'correlated' with everything else.
What reasoning? Here's a way to construct your own version of my example:
Choose your favourite example of a chancey process. Imagine it hooked up to a
random number generator, picking numbers from 0 to 9, say. Run two of these
devices in parallel, so that each picks one number per day. Arrange things so
that on days when both pick 9, something otherwise fairly unusual happens:
your grandmother receives a large bunch of daffodils, perhaps. (If this
happens to your grandmother a lot, substitute gerberas.) Then the following
will turn out true: In years in which your grandmother receives more bunches
of daffodils than usual, there are likely to be more days than usual on which
both chance devices pick a 9.
That's all my example involves. It's trivial in itself, though useful in the
context in which I use it. Where's the "inconsistency", or "confused logic"?
>
>5. On page 180 the author writes: "The argument (for advanced action -
>my comment) will have to rely on nonobservational considerations: facts
>such as simplicity, elegance and symmetry, for example. This doesn't mean
>that the issue is nonemprical, or metaphysical in the disparaging sense."
> I find it amazing that the author, who claims that his reasoning
>reveals the aspects of the structure of nature which are independent of
>observing agent's bias wants to base his reasoning on such purely
>subjective, agent's dependent notions as simplicity or elegance. As for
>symmetry, why should the preference for the symmetry over the lack of it
>be considered as anything else than a bias resulting from the way we, as
>agents, perceive the world?
This objection does touch on an interesting issue. Why do we prefer theories
with "virtues" such as simplicity and symmetry? Why do we subscribe to Occam's
Razor, for example? Without such principles we're in trouble in science, for
there are always plenty of alternative theories compatible with all the
observational data. Without these principles, we simply have no way to choose
between the alternatives. Some people take refuge in instrumentalism at this
point (or the "shut up and calculate" approach, as it is called in QM), but
this has problems of its own -- for example, the problem of specifying a
non-theoretical language in which to describe observations.
Fortunately, I can ignore these hard problems for the purposes of my book. All
I want to claim is that advanced action looks like a good option in QM, by the
standards taken to be appropriate everywhere else in science. (What if those
standards ultimately turned out to be subjective? Wouldn't my position be
inconsistent? No, because there is room for a distinction between good
subjective and bad subjective: someone who thinks that all morality has a
subjective basis doesn't have to allow that anything goes, for example. But
that's a long story, and this isn't the place to tell it.)
Thanks to John Baez for calling my attention to Dr Pindor's comments.
Huw Price.
--
School of Philosophy
Main Quad, A14
University of Sydney
NSW Australia 2000
Tel: +61 2 9351 4057
Fax: +61 2 9351 6660
URL: http://plato.stanford.edu/price
John Collier
[Huge wads of unnecessary quoted text deleted. Please, people, note
that the moderator is unable to post articles containing more quoted
text than new text. - jb]
: 1. On page 26 the author observes that to prove H-theorem (i.e. to show
: that entropy of a system increases with time) one needs to make an
: assumption known as 'stosszahlansatz', which is a form of Principle
: of Independence of Incoming Influences (PI^3). He then goes on to
: criticize this assumption as unjustified and biased since it is clearly
: time-asymmetric. This is fine, except that on page 39 he writes:
: "What the discussion in the second half of nineteenth century revealed
: is that thermodynamic equilibrium is a natural condition of matter, .."
: and later "The statistical considerations suggest that a future in which
: entropy reaches its maximum is not in need of explanation;"
: Well, for the mentioned discussions and considerations to reach the
: stated conclusions 'stosszahlansatz' is required, so either PI^3 is invalid
: and he should not be using the stated conclusions as a support for his
: arguments which follow, or he accepts the conclusions as valid but then [...]
In one case he is talking about the history, and uses the techniques of
an historian, and in the other of a logician. Context matters.
: [...] he writes: "The universe as we know it contains big coherent
: sources of radiation".
: Maybe I am totally mistaken, but I know nothing of plentiful
: "large-scale coherent sources of radiation" in the universe.
There are mega clusters of galaxies. Note the he is _not_ talking
about "coherent radiation", which is something else.
: He writes "Suppose that we have
: 100 identical fair coins, each of which is to be tossed once at a
: time 't'. Then the probability that an arbitrary coin will be heads
: after 't' given that it was, say, heads before 't', is independent of
: the initial distribution - that is a number of coins that initially
: showed heads. Not so the reverse "transition probability", such as
: the probability of heads before 't' given heads after 't': if 99
: of the coins initially showed heads than this latter probability is .99
: (assuming fair coins);"
: Now, how is the probability of finding heads before 't', given that
: we know how many coins showed heads initially, a "transition probability"??
This is standard talk in discussions of Markov processes. Again, context
matters. Huw used the quotes to indicate the problem.
: 4. On page 145 he writes: Consider two friends who maintain their
: friendship entirely on chance encounters. ... On occasions on which
: they do meet, their prior movements are correlated (they both happen to
: enter the same cafe at roughly the same time, for example)".
: If the meetings above are an evidence that the prior movements of
: these two people are correlated, how are these "chance encounters"?
Huw was pointing out the problems with the usual use of "chance' and
of "uncorrelated". This is the sort of thing philosophers do. You
should be puzzled. That is his job.
: I find it amazing that the author, who claims that his reasoning
: reveals the aspects of the structure of nature which are independent of
: observing agent's bias wants to base his reasoning on such purely
: subjective, agent's dependent notions as simplicity or elegance.
I don't think Huw regards these as subjective. I don't.
--
John Collier Email: pl...@alinga.newcastle.edu.au
Philosophy -- U. of Newcastle Fax: +61 49 216928
Callaghan, NSW, AUSTRALIA 2038 http://bcollier.newcastle.edu.au
john baez
>I just wanted to point out that in Huang's
>book, he argues we can apply it *locally*, looking at how H is changing
>in the next instant of time, and that if you reverse time, propagating
>backwards you get that H must also decrease in time. So he gets a very
>jagged curve for H(t). John seems to be saying that Huang is
>misapplying the H-theorem, which may well be true.
I haven't read Huang's book, but what you are describing sounds rather odd.
Perhaps different people have different ideas of what the H-theorem is and
what the "assumption of molecular chaos" is. My understanding of the
H-theorem comes mainly from the books by Zeh and Price, both of whom claim
to be adhering pretty closely to Boltzmann's original work. The idea behind
this H-theorem is to find an assumption that *holds and keeps holding*, and
which implies that entropy *increases and keeps increasing*. Boltzmann
wrote down such an assumption and called it the Stosszahlansatz. Whether
this is the same as Huang's "molecular chaos", I don't know - but Huang's
talk of a jagged curve for H sounds very different from what Boltzmann was
trying to explain, namely, a continuing decrease in H - or in other words,
a continuing increase in entropy. Boltzmann's Stosszahlansatz is an
time-asymmetric assumption - if it holds, it will not necessarily hold
when you make the substitution t -> -t - and it's designed to yield time-
asymmetric conclusions, namely the increase of entropy. So Huang's idea
of applying the H-theorem both forwards and backwards in time in a
rapidly alternating manner sounds very different from anything Boltzmann
would have ever dreamt of.
Does Huang really argue that entropy keeps wiggling in a jagged way, so
the 2nd law of thermodynamics is wrong every other microsecond or so?
I've seen people argue that the derivative of entropy should change
sign every Poincare recurrence time or so, but the Poincare recurrence
time is *very long*..
> Most of the texts on kinetic theory seem a bit old. When the mention
>the Poincare' recurrence time, they seem to think this means the motion
>is periodic. This seems false to me, since even if you come arbitrarily
>close to your initial condition, the dynamics are chaotic, so you won't
>reproduce your earlier time evolution. Is this accepted in stat mech
>circles?
The notion of Poincare recurrence time does not only apply to systems
with periodic or quasiperiodic motion; it applies to lots of chaotic
systems too. It says that whenever time evolution is given by a
continuous flow on a compact phase space, if you start the system in
any state and wait long enough it will come back arbitrarily close to
its initial state. It may never come back *exactly*. This stuff is
well-understood by the statistical mechanics intelligentsia.
However, it's worth noting that when you *quantize* a classical system
that has a compact phase space, you almost always get a Hamiltonian with
a discrete spectrum, so the time evolution of quantum states is
quasiperiodic. This leads to the question "where does chaos go when you
quantize?" and thence to the whole fascinating topic of quantum chaos.
john baez
Aaron Bergman <aber...@princeton.edu> wrote:
>But, if you were to reverse the momenta of every particle, molecular chaos
>still would hold, so the conditions of the H-theorem still hold.
No!
As I've said a couple of times, the assumption of molecular chaos
is time-asymmetric. At least this is true for the assumption that
Boltzmann used to prove his H-theorem. Boltzmann called it the
Stosszahlansatz, or "collision number assumption". Perhaps you're
talking about some *other* assumption of molecular chaos?? Someone
on this thread has assured us that the assumption of molecular chaos
and the Stosszahlansatz are one and the same - but I'm getting a bit
worried, since your opinion about it is so different than mine!
So: what's the Stosszahlansatz?
I'll give a summarized answer that leaves out a lot of important
details. If it makes no sense I can give more details.
Suppose we have a homogeneous gas of particles and the density of
them with momentum p is f(p). Consider only 2-particle interactions
and let w(p1, p2; p1', p2') be the transition rate at which pairs of
particles with momenta p1, p2 bounce off each other and become
pairs with momenta p1', p2'. To keep things simple we assume
w(p1, p2; p1', p2') = w(p1', p2'; p1, p2).
The Stosszahlansatz says:
df(p1)/dt =
integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
This is very sensible-looking if you think about it. Using this,
Boltzmann proves the H-theorem. Namely, the derivative of the
following function is less than or equal to zero:
H(t) = integral f(p) ln f(p) dp
The proof is an easy calculation and you can find it on page 38 of
Zeh's "The Physical Basis of the Direction of Time".
Now: since the output of the H-theorem is time-asymmetric, and all
the inputs are time-symmetric except the Stosszahlansatz, we should
immediately suspect that the Stosszahlansatz is time-asymmetric!
And it is!
Let's see what happens when we reverse the direction of time...
The density f(p) is really a function of time, too, so we can write
it as f(p,t) to be precise. Reversing time corresponds to replacing
f(p,t) with f(-p,-t), since we reverse all the momenta when we reverse
the time. Let
g(p,t) = f(-p,-t)
be the density function in this time-reversed world.
Now, you seem to be claiming that if the Stosszahlansatz holds for
f at t = 0, then it also holds for g at t = 0. But if you work it
out, you'll see this is not true. Instead, the Stosszahlansatz for
f at t = 0 implies the following statement for g at t = 0:
dg(p1)/dt =
- integral w(p1, p2; p1', p2') [g(p1')g(p2') - g(p1)g(p2)] dp2 dp1' dp2'
Note the crucial minus sign!
I worked this out just now, using a change of variables for all the
momenta - I hope I've got it right. But regardless of whether I have
the formula exactly right, I *know* the Stosszahlansatz must be time-
asymmetric, because there is no way you can pull a time-asymmetric
rabbit out of a time-symmetric hat! The only way you can show entropy
increases is using a time-asymmetric assumption somewhere. And then
the question becomes: why should *this* assumption be true?
After giving the proof of the H-theorem, Zeh writes:
"This success seems to have been responsible for the myth of the
statistical foundation of the thermodynamical arrow of time. But
how can the asymmetric Stosszahlansatz itself be justified [...] ?"
And that's where things start getting interesting....
Andrzej Pindor
>
> >Dr. Andrzej Pindor wrote:
>
> >john baez wrote:
[Moderator's note: Lots of stuff deleted. Please trim quoted material
when you post. If you don't cut quoted material judiciously, I'll do
it injudiciously, as exhibited here. More quoted stuff deleted
below. -TB]
Irrespective of whether I agree with what you write above or not, you
still have not responded to my objections. Let me repeat again: I object
to you making the statements
....
> >"What the discussion in the second half of nineteenth century revealed
> >is that thermodynamic equilibrium is a natural condition of matter
......
and
......
"The statistical considerations suggest that a future in which
> >entropy reaches its maximum is not in need of explanation;"
.....
after claiming that the PI^3 (or the assumption of molecular chaos, as
it is usually refered to in statistical physics) is unjustified since
to show that equilibrium is a natural condition of matter and that
entropy always incerases you _have_ to make the assumtion of molecular
chaos. At least this is what my knowledge of statistical physics tells
me. I am ready to learn something new so please indicate how the above
statements can be true _without_ the assumption of molecular chaos.
> >
> >2. On pages 71 and 72, when discussing the arrow of radiation he writes:
> >"...large-scale coherent sources of radiation are common..." and "What
> >remains to be explained is why there are large coherent sources - processes
> >in which huge numbers of tiny transmitters all act in unison - and why
> >there are no large coherent absorbers...". He repeats the same statement on
> >page 148 where he writes: "The universe as we know it contains big coherent
> >sources of radiation".
> > Maybe I am totally mistaken, but I know nothing of plentiful
> >"large-scale coherent sources of radiation" in the universe. In fact,
> >from the context it seems that according to the author, stars and galaxies
> >are such sources. Of course we know that they are not and the author
> >attempts to explain a problem which only exists in his imagination.
>
> To get a visible ripple on a still pond, we need to move lots of water
> molecules in the same direction at the same time. The oscillation of the
> molecules needs to be in phase, or "coherent". That's all I mean by the term.
And this is what coherence means, also in the case of lasers. However,
the ripples on the pond are not dependent on a coherent source. They
would arise even if you started with one molecule being driven in some
way up and down (with enough energy put in so that this energy would
transfer to many other molecules to make visible ripples). In the
actual case of, say, a stone falling into water you do have a number of
molecules stimulated in phase, but this is because molecules of the
stone are tightly bound one to another and strike the water in unison.
In the similiar fashion you can create a coherent incoming wave (e.g. by
striking a side of a cup of coffe with a spoon, try it). This in fact
invalidates another of your claims that we never see incoming spherical
waves.
The stone striking the water is considered a point source since the
water molecules are excited locally (even if not exactly in one point)
so this is not thought of as a coherent source. To get a spherical
incoming wave you need to produce an excitiation which is extended in
space (like a rim of the coffe cup) and all these spatially distant
sources (molecules of the rim) have to act in phase. This is why this is
a coherent source. Incoming spherical waves require a coherent source,
outgoing spherical waves do not. Since a point excitation is much more
likely than a coherent exitation in many distant points, we often see
outgoing spherical waves but very rarely incoming spherical waves
(although they happen too, see the coffe cup case).
> (It is first introduced on p. 57, I think.) It's the same for all kinds of
> radiation: without big coherent sources of this kind, there wouldn't be any
> observable asymmetry of radiation. Of course, many such sources are doing this
> at lots of frequencies and/or phases at the same time, so are not coherent in
> the sense that lasers are.
>
Let me repeat it - stars and galaxies are _not_ coherent sources in
the sense which you use above. Their atoms do not radiate in phase.
Please ask any astrophysicist about it. And if you are refering to other
"common large-scale coherent sources", please say what they are. I think
that you are misinterpretting "observable asymetry of radiation"
(which is about lack of incoming spherical waves as opposed to the
abundance of the outgoing spherical waves, and not about the lack of
coherent sinks as opposed to the abundance of coherent sources) to fit
your theories.
> I use an argument by Frank Arntzenius to illustrate these points. Being a
> clever chap (who well deserves his Latin suffix), Arntzenius is well aware of
> the danger of putting the asymmetry in by hand, and therefore wants to allow
> transition probabilities in both directions. Nevertheless, I argue that the
> asymmetry he claims to find rests on the fact that it focuses on *initial*
> rather than *final* boundary conditions. I convert things into evidential
> probabilities to make this clear. So I don't "pretend to be talking about
> transition probabilities". Rather I argue that anyone who wants to claim that
> the analogous diagnosis doesn't hold for "genuine" transition probabilities
> will end up passing the buck, in one way or another.
Again you do not address my objection which is that the question of
the probability of finding a head when we have 99 heads and one tail
has nothing to do with a temporal direction one way or another. What is
the tempral process to which this refers? A transition is a temporal
process since you can talk about situation before and after transition.
What is the relevance of the above probability in the context of talking
about temporal directions and temporal biases? And why do you call this
"a transition probability" (in quotes or not)? The probability you talk
about has no relation to any cause and this is why I claim that you are
being logically incoherent talking about it in the context you are
claiming to discuss i.e. the problem of causal direction.
> >4. On page 145 he writes: Consider two friends who maintain their
> >friendship entirely on chance encounters. ... On occasions on which
> >they do meet, their prior movements are correlated (they both happen to
> >enter the same cafe at roughly the same time, for example)".
> > If the meetings above are an evidence that the prior movements of
> >these two people are correlated, how are these "chance encounters"? The
> >author seems to have a strange notion of what 'correlated' means or
> >what 'chance' means. Extending his reasoning we can easily argue that
> >everything is 'correlated' with everything else.
>
> What reasoning?
The reasoning that if they meet by chance in a cafe then their prior
movements are correlated. This is like saying that if sometimes on those
days that I go for pizza you go to the movies then our movements are
correlated on those days. If this was the case the word "correlated"
would be meaningless since everything would be correlated with
everything else and the word would carry no useful information.
>............Here's a way to construct your own version of my example:
> Choose your favourite example of a chancey process. Imagine it hooked up to a
> random number generator, picking numbers from 0 to 9, say. Run two of these
> devices in parallel, so that each picks one number per day. Arrange things so
> that on days when both pick 9, something otherwise fairly unusual happens:
> your grandmother receives a large bunch of daffodils, perhaps. (If this
> happens to your grandmother a lot, substitute gerberas.) Then the following
> will turn out true: In years in which your grandmother receives more bunches
> of daffodils than usual, there are likely to be more days than usual on which
> both chance devices pick a 9.
If you _arrange_ this so as above you may be in fact introducing a
correlation. However, I do not see any deliberate arrangement in the
chance meeting of two people in your example. If no deliberate
arrangement is made I do not see any correlation. Please point out one
if you see it.
> That's all my example involves. It's trivial in itself, though useful in the
> context in which I use it. Where's the "inconsistency", or "confused logic"?
I do not have an argument with what you want to say (i.e. the context).
I am just pointing out that your claim of an existence of correlation in
the above case is false and hence the example does not contribute
anything to what you want to say. It is in the same way as the
probability discussed above does not contribute anything to the question
of temporal direction bias since it does not involve two points in time.
I have found many such instances in your book (those we discuss here are
just a small selection) where to argue for a certain claim you point to
things which have no relationship to the claim or make statements which
are in fact false, as with the correlation above.
> Fortunately, I can ignore these hard problems for the purposes of my book. All
> I want to claim is that advanced action looks like a good option in QM, by the
> standards taken to be appropriate everywhere else in science. (What if those
> standards ultimately turned out to be subjective? Wouldn't my position be
> inconsistent? No, because there is room for a distinction between good
> subjective and bad subjective: someone who thinks that all morality has a
> subjective basis doesn't have to allow that anything goes, for example. But
> that's a long story, and this isn't the place to tell it.)
There may be a dstinction between "good subjective" and "bad subjective"
but what is inconsistent is your claim that using subjective,
agent-dependent arguments you can arrive at agent-independent "truth"
about the structure of nature. I have no objections to you indicating
that advanced action may be a different way of interpretting microscopic
phenomena, which avoids paradoxes implied by QM - randomness and
nonlocality. However, lacking observational evidence supporting your
view we are at a level of discussing whether a bottle is half-full or
half-empty. Lack of consistency and often of coherence in your arguments
did nothing to convince me that the advance action is more palatable
than ramdomnsess and nonlocality. In fact I find it much less palatable
since it implies determinism even though you argue that it does not
(again I found your arguments here lacking).
Xavier de HEMPTINNE
>
> Gak. That came out quite poorly. Let me try again. Assume we have
> a point where molecular chaos holds. By the H-theorem, H must
> invariant, H has to be locally at a peak. Does that make any
> better sense?
>
> Huang probably explains it much better than I do.
>
>
May I suggest that the readers of this thread download and read a
short
text on this matter I sent some years ago to the
<http://xxx.lanl.gov> database "chao-dyn" with reference
"chao-dyn/9502004". Reactions would also be most gratefully welcomed.
More details concerning the subject may be read in "Physica D" (vol.
112, (1998) pg.258-274) and of course in:
(http://www.wspc.com.sg/books/physics/1622.html)
Yours,
--
Xavier de Hemptinne
Prof. em. Cath. U. Leuven
(priv:) Duivenstraat 78,
3052 Blanden, Belgium
tel/fax +32 (0)16 40.07.49
mailto:Xavier.de...@chem.kuleuven.ac.be
Huw Price
>.................. Let me repeat again: I object
>to you making the statements
>....
>> >"What the discussion in the second half of nineteenth century revealed
>> >is that thermodynamic equilibrium is a natural condition of matter
>......
>and
>......
>"The statistical considerations suggest that a future in which
>> >entropy reaches its maximum is not in need of explanation;"
>.....
>
>after claiming that the PI^3 (or the assumption of molecular chaos, as
>it is usually refered to in statistical physics) is unjustified since
>to show that equilibrium is a natural condition of matter and that
>entropy always incerases you _have_ to make the assumtion of molecular
>chaos. At least this is what my knowledge of statistical physics tells
>me. I am ready to learn something new so please indicate how the above
>statements can be true _without_ the assumption of molecular chaos.
Two points:
1. You say that I am "claiming that PI^3 ... is unjustified". If you mean that
I am claiming that PI^3 is false, then you've misread me: I think that PI^3 is
(at least approximately) true (at least in our region). (Why the
qualifications? Because, since we don't yet understand why entropy is so low
in the past, we shouldn't rule out the possibility that it might be low for
the same reason in the distant future. If it were, then this might have subtle
observable effects even now. For such phenomena, PI^3 would fail. I certainly
believe that it is "unjustified" to *assume* PI^3 in order to rule out such
phenomena, but this is different from believing that PI^3 is false.)
2. The reason for thinking equilibrium is a "natural" condition of matter
which doesn't appeal to PI^3 is Boltzmann's familiar statistical argument:
under a natural way of counting possibilities, there are many more microstates
compatible with a given macrostate such that entropy increases, than
microstates such that it decreases. (I prefer to put this in terms of
histories, or trajectories: among trajectories compatible with the given
macrostate at the given time, trajectories in which entropy increases are
vastly more probable than those in which it decreases.)
I can see two places at which it might be thought that this familiar argument
slips in PI^3, in a way which would create problems. First, one might think
that in order to use observations to discover the appropriate probability
measure in real systems, we need to assume that PI^3 holds in the cases we
observe -- otherwise, the correlations might be "disguising" the true
probabilities. (Analogy: in deciding by observation what's "normal", we have
to assume that what we observe isn't systematically "abnormal".) Something
like this is true, I think, but not a source of any vicious circularity in my
account. (Remember, I'm not denying that PI^3 is true, in ordinary cases.)
Second, it might be thought that we need what amounts to PI^3, in order to use
the probability measure to make predictions -- i.e. that we need to assume
that the initial microstate is chosen at random (which, it might be claimed,
amounts to assuming PI^3). I think this thought is mistaken. It amounts to
saying that we need to assume that equiprobable microstates are equiprobable.
But that's a tautology, and so we don't need to assume it. (It wouldn't be a
tautology if there were two different kinds of probability involved, but why
think *that*?)
>> To get a visible ripple on a still pond, we need to move lots of water
>> molecules in the same direction at the same time. The oscillation of the
>> molecules needs to be in phase, or "coherent". That's all I mean by the
>> term.
>
>And this is what coherence means, also in the case of lasers. However,
>the ripples on the pond are not dependent on a coherent source. They
>would arise even if you started with one molecule being driven in some
>way up and down (with enough energy put in so that this energy would
>transfer to many other molecules to make visible ripples).
Using just one molecule doesn't make any essential difference. Imagine there
are N molecules around the edge of our circular pond, and that each is to
receive one standard "jiggle" from the outgoing ripple. To produce a ripple
big enough to do this, we need to supply N "jiggles" *coherently* at the
source. In principle, we could do it by giving one jiggle to N molecules, N
jiggles to one molecule, or something in between. No matter: the N jiggles
need to be coherent.
Note that the factor N stems from the nature of the geometry: more sinks than
sources. If we arrange things to avoid this, e.g. by putting the emitter and
absorber at the two foci of an ellipse, so that there's just one of each,
then there is no such imbalance (and no asymmetry of radiation).
>..... To get a spherical
>incoming wave you need to produce an excitiation which is extended in
>space (like a rim of the coffe cup) and all these spatially distant
>sources (molecules of the rim) have to act in phase. This is why this is
>a coherent source. Incoming spherical waves require a coherent source,
>outgoing spherical waves do not. Since a point excitation is much more
>likely than a coherent exitation in many distant points, we often see
>outgoing spherical waves but very rarely incoming spherical waves
>(although they happen too, see the coffe cup case).
This is a nice example of a temporal double standard at work. Time direction
aside, the "coherent exitation in many distant points" is exactly the same in
the outgoing and incoming cases: one is the temporal mirror of the other. So
you can't use a probability argument to exclude the incoming case, unless you
give us a time-asymmetric principle to account for the fact that the same
argument doesn't exclude the outgoing case. (This is essentially my objection
to Popper on pp. 54-7.)
>Let me repeat it - stars and galaxies are _not_ coherent sources in
>the sense which you use above. Their atoms do not radiate in phase.
>Please ask any astrophysicist about it.
The basic point of the water case above applies in this case too. If we want
to say that one sun produces coordinated jiggles in N absorber particles, then
must be (in effect) N coordinated jiggles in the sun to do the trick. Of
course, if we want to say that the coordination in the absorber particles
disappears when we look closely enough, then the need for coordination in the
source disappears too. But now we are no longer talking about outgoing
circular wavefronts.
>>............Here's a way to construct your own version of my example:
>>Choose your favourite example of a chancey process. Imagine it hooked up to
>>a random number generator, picking numbers from 0 to 9, say. Run two of these
>>devices in parallel, so that each picks one number per day. Arrange things so
>>that on days when both pick 9, something otherwise fairly unusual happens:
>>your grandmother receives a large bunch of daffodils, perhaps. (If this
>>happens to your grandmother a lot, substitute gerberas.) Then the following
>>will turn out true: In years in which your grandmother receives more bunches
>>of daffodils than usual, there are likely to be more days than usual on which
>>both chance devices pick a 9.
>If you _arrange_ this so as above you may be in fact introducing a
>correlation. However, I do not see any deliberate arrangement in the
>chance meeting of two people in your example. If no deliberate
>arrangement is made I do not see any correlation. Please point out one
>if you see it.
The fact that it is deliberate is beside the point. I just wanted a graphic
case to illustrate what I meant by "correlation" in this context. In any case,
to deny that there are non-deliberate cases of the same thing, you would have
to deny that there are *any* natural cases in which a conjunction of certain
outcomes of two independent chance processes produces an otherwise unusual
result. In any such case, the relative frequencies have the above structure.
>I do not have an argument with what you want to say (i.e. the context).
>I am just pointing out that your claim of an existence of correlation in
>the above case is false and hence the example does not contribute
>anything to what you want to say.
Given what *I* mean by correlation in this context, the claim is true, as you
just acknowledged wrt the daffodil case.
>I have found many such instances in your book (those we discuss here are
>just a small selection) where to argue for a certain claim you point to
>things which have no relationship to the claim or make statements which
>are in fact false, as with the correlation above.
Well, nobody's perfect. However, there's another possible explanation for the
data (i.e. for your view of the book), viz. that you've misunderstood me. My
fault either way, I suppose, but I think I'd rather be right-but-obscure than
wrong-but-clear!
>There may be a dstinction between "good subjective" and "bad subjective"
>but what is inconsistent is your claim that using subjective,
>agent-dependent arguments you can arrive at agent-independent "truth"
>about the structure of nature.
Where did I claim this?
>I have no objections to you indicating
>that advanced action may be a different way of interpretting microscopic
>phenomena, which avoids paradoxes implied by QM - randomness and
>nonlocality. However, lacking observational evidence supporting your
>view we are at a level of discussing whether a bottle is half-full or
>half-empty.
Do you think there's any point in discussing whether the bottle exists when it
is not actually being observed? If "no", then you are a hard core
instrumentalist, and it doesn't surprise me that you find appeals to symmetry,
etc., unconvincing. (If I were to argue with you, it would be about the
general difficulties facing instrumentalism.) If "yes", then you've already
shown a willingness to go beyond immediate observational evidence, and you
need to explain to me why the arguments I take to favour advanced action in QM
are not simply more of the same.
Harry Johnston
> However, historically, lots of people didn't notice the time-asymmetry
> of Boltzmanns's Stosszahlansatz, so they were first pleased and then
> nervous about getting "something for nothing" - a proof that entropy
> is increasing that doesn't use any time-asymmetric assumptions!
I don't see any fundamental philosophical problem here.
The phase space of the entire Universe [1] is really really big.
We can in principle roughly divide it into various sections according
to the behaviour of entropy.
Section (A) is almost certainly the biggest part; in it, entropy is at
or near a maximum and there isn't any clearly defined arrow of time.
Section (B) is the part in which we find ourselves; the arrow of time
points forwards.
Section (C) is the time-reversal of (B) and the arrow of time points
backwards.
Section (D) is the mixture; in some respects or in some locations the
arrow of time points forwards and in others it points backwards.
In order to wind up with the 2nd law we need to show that, given some
reasonable sounding assumptions:
1) Area (D) is very small compared to (B) and (C); or
1a) A state in area (D) quickly evolves into one in area (A).
2) That most states in area (B) don't evolve into (A), (C), or (D)
except after a really long time.
Neither of these criteria are time-asymmetric, so it should in
principle be possible to demonstrate them without time-asymmetric
assumptions.
Given these assumptions we argue: if the Universe were in (A) there
wouldn't be any lifeforms about to ask questions, so we can invoke the
anthropic principle (?) to dismiss (A) from consideration. This also
allows us to dismiss (D) if we chose to use (1a); if we preferred (1)
we just quote the odds.
That just leaves (B) and (C) both of which follow the 2nd law, from
the point of view of any occupants; QED.
Harry.
[1] I'm considering a classical Universe here because I don't want to
risk confusing myself.
---
Harry Johnston, om...@ihug.co.nz
"He more or less based his whole reputation as a
technologist on the fact that he couldn't get his
PC to work." - Angus McIntyre on Jerry Pournelle
Andrzej Pindor
>So: what's the Stosszahlansatz?
>
>I'll give a summarized answer that leaves out a lot of important
>details. If it makes no sense I can give more details.
>
>Suppose we have a homogeneous gas of particles and the density of
>them with momentum p is f(p). Consider only 2-particle interactions
>and let w(p1, p2; p1', p2') be the transition rate at which pairs of
>particles with momenta p1, p2 bounce off each other and become
>pairs with momenta p1', p2'. To keep things simple we assume
>
>w(p1, p2; p1', p2') = w(p1', p2'; p1, p2).
>
>The Stosszahlansatz says:
>
>df(p1)/dt =
>
>integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
Actually (see Huang, Statistical Mechanics) assumption of "molecular
chaos" (as "Stosszahlansatz" as is refered to in this book) says that a
probability of finding a particle 1 with momentum p1 and particle 2 with
momentum 2 is given by f(p1)f(p2) (this is better reflected in the
German name which means "assumption about the number of collisions"),
i.e that particles are not correlated . The equation above uses of
course this fact. Time assymetry lays not in this assumption but indeed
in the equation above. This is because the number of particles which
enter the collision is a product of particle densities for 1 and 2 (no
correlation), but the number of particles emerging from the collision
is not a product of one-particle desities. Even though in the above
equation it is written as f(p1')f(p2'), because of the conservation of
momentum we have p2'=p1+p2-p1' and f(p2') is not independent of p1'.
It does not seem to me that there is anything controversial here, so I
would say that the time asymetry is an unaviodable consequence of the
assumption of molecular chaos. Huw Price tries to question this
assumption in his book discussed here, but accepts results of
statistical mechanics which can only be derived with this assumption and
which are in agreement with observable physical phenomena. Then the
questionm would rather be "why do we have molecular chaos?" (and
hence time asymetry).
john baez
Harry Johnston <om...@ihug.co.nz> wrote:
>I don't see any fundamental philosophical problem here.
>[...] we can invoke the anthropic principle (?) [...]
You're right, if we invoke the anthropic principle there is not
much problem with explaining the thermodynamic arrow of time: if
entropy wasn't increasing, nothing remotely resembling life as we know
it could exist [1]. This has been known for a long time. The fun
problem is to try to explain the arrow of time *without* invoking the
anthropic principle, since doing *this* might lead to deep new insights
into physics.
If time evolution is given by a continuous flow on a compact phase
space, the Poincare recurrence theorem says that given any initial
conditions, if you wait long enough the system will return arbitrarily
close to those initial conditions. "What goes around, comes around."
This idea was familiar to Nietzsche; he called it the "eternal
recurrence of the same". In a sufficiently complex system of this
sort we can expect very long stretches of boring high-entropy "heat
death" interspersed by little dips in the entropy, and, more rarely,
bigger dips. Life could only happen during sufficiently big dips,
with the arrow of time pointing in the direction of increasing entropy,
whichever way that happens to be at the time.
Quantum mechanics doesn't change this very much. The analogue of
a continuous flow on a compact phase space is time evolution given
by a Hamiltonian with discrete spectrum; here all states evolve in
an "almost periodic" fashion, so an even stronger version of Poincare
recurrence holds.
If this is all there were to it, we would have gotten bored with the
arrow of time puzzle by now. But general relativity drastically changes
the picture! General relativity seems to say that the universe has
*not* been going on forever: instead, it seems to have begun with a
big bang about 10 billion years ago. Entropy seems to have been going
up ever since. Is there a physical reason for this? Is the 2nd law of
thermodynamics perhaps trying to tell us something about this apparent
"beginning of time"? Is there some reason why *gravity* is so deeply
involved in the birth of the universe (the big bang), the formation
of the big free energy sources we see (the stars and galaxies), and the
final death of these stars and galaxies in black holes? Right now we don't
understand how gravity and thermodynamics fit together - could this be
important? These are some questions physicists ask today when they ponder
the arrow of time. We shouldn't let the anthropic principle lull us into
a complacent state of mind where we stop asking these questions.
[1] Actually some people have argued that the anthropic principle only
guarantees the existence of a small patch of spacetime where entropy
is increasing - not the enormous amount of it we see through our telescopes.
Someone who really wants to push the anthropic explanation of the 2nd
law has to deal with this one way or another.
john baez
Andrzej Pindor <pin...@ic-unix.ic.utoronto.ca> wrote:
>John Baez wrote:
>>The Stosszahlansatz says:
>>
>>df(p1)/dt =
>>
>>integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
>Actually (see Huang, Statistical Mechanics) assumption of "molecular
>chaos" (as "Stosszahlansatz" as is refered to in this book) says that a
>probability of finding a particle 1 with momentum p1 and particle 2 with
>momentum 2 is given by f(p1)f(p2) (this is better reflected in the
>German name which means "assumption about the number of collisions"),
>i.e that particles are not correlated. The equation above uses of
>course this fact. Time assymetry lays not in this assumption but indeed
>in the equation above.
The equation I wrote down is time-asymmetric. The equation you describe
is time-symmetric. From certain additional assumptions one can derive
my equation from yours, but as long as these additional assumptions are
time-symmetric, one can also derive the *time-reversed version* of the
equation I wrote down. Using both the equation I wrote down and its
time-reversed version, one can prove the time derivative of entropy is
greater than or equal to zero, and also less than or equal to zero. So
it's zero!
In short, from certain time-symmetric assumptions one can prove that
the derivative of entropy is greater than or equal to zero - but only by
proving it is zero!
This is of no real use in deriving the 2nd law of thermodynamics, because
even though the second law *says* dS/dt >= 0 , the interesting thing about
it is that sometimes dS/dt > 0. A proof that dS/dt = 0 is proving too much:
it must have unphysical assumptions.
The equation I wrote down is enough to prove dS/dt >= 0 but not that
dS/dt = 0. It is explicitly time-asymmetric and is a reasonable candidate
for the kind of assumption we should make to derive the 2nd law.
Calling both the equation I wrote down and the one you mention by the
same name is a sure way to cause confusion.
Andrzej Pindor
> Two points:
>
> 1. You say that I am "claiming that PI^3 ... is unjustified". If you mean that
> I am claiming that PI^3 is false, then you've misread me: I think that PI^3 is
> (at least approximately) true (at least in our region). (Why the
> qualifications? Because, since we don't yet understand why entropy is so low
> in the past, we shouldn't rule out the possibility that it might be low for
> the same reason in the distant future. If it were, then this might have subtle
> observable effects even now. For such phenomena, PI^3 would fail. I certainly
> believe that it is "unjustified" to *assume* PI^3 in order to rule out such
> phenomena, but this is different from believing that PI^3 is false.)
>
It is of course possible that I have misread you on this point. However,
you spend a whole chapter (Chapter 5) arguing that there is no
justification to the principle of mu-Innocence (i.e. that it does not in
fact hold) and this principle, in your own words (end of Chapter 2), is
very close relative to PI^3. So does PI^3 hold or not?
> 2. The reason for thinking equilibrium is a "natural" condition of matter
> which doesn't appeal to PI^3 is Boltzmann's familiar statistical argument:
> under a natural way of counting possibilities, there are many more microstates
> compatible with a given macrostate such that entropy increases, than
> microstates such that it decreases. (I prefer to put this in terms of
> histories, or trajectories: among trajectories compatible with the given
> macrostate at the given time, trajectories in which entropy increases are
> vastly more probable than those in which it decreases.)
On page 30 of your book you point out (rightly) that the purely statistical
arguments have a serious problems. For instance, on their basis one might
also conclude that entropy was high in the past. In other words, they
cannot tell us why entropy always increases and systems keep their
equilibrium states. In some other place in the book (I cannot find the
exact page at the moment) you say for instance that such statistical
considerations cannot explain why gas escapes from a bottle but does not go
in - if we take a state of expanded gas and reverse all velocities, the
resulting state is as probable as the one before the reversal, so why
doesn't the gas go back into the bottle occasionally?
In fact at the bottom of p. 30 you write: "Boltzman ... saw that the
statistical treatement of entropy makes it extremely puzzling why entropy
is not higher in the past". Consequently this treatment does not seem to be
able to explain why entropy _increases_. Incidently here is another
circularity in your reasoning: if the veolcities are reversed, the gas
would go back into the bottle _only_ if the molecular collisions were
deterministic (no quantum randomness). However, the lack of quantum
randomness is what you are trying to argue for. You cannot then assume it
in your considerations.
> Using just one molecule doesn't make any essential difference. Imagine there
> are N molecules around the edge of our circular pond, and that each is to
> receive one standard "jiggle" from the outgoing ripple. To produce a ripple
> big enough to do this, we need to supply N "jiggles" *coherently* at the
> source. In principle, we could do it by giving one jiggle to N molecules, N
> jiggles to one molecule, or something in between. No matter: the N jiggles
> need to be coherent.
The fact that the wave at the rim of the pond is much weaker than closer to
the source is totally irrelevent for coherence. You seem rather to be
saying that the problem is that sources may have high intensity whereas
sinks absorb small amount of radiation. Or that sources tend to be compact
and sinks extended in space. Well, interstellar clouds radiate too and
black holes absorb radiation in large amounts. And galaxes can hardly be
called compact sources, can they? Note also that to have a source of
radiation you have to heat matter and this happens when density of matter
grows due to gravitational forces. Absorbtion can take place at low
teperatures and densities. I should perhaps also mention that Earth absorbs
radiation from the Sun and hence is a more compact absorber than the Sun is
a source.
> Note that the factor N stems from the nature of the geometry: more sinks than
> sources. If we arrange things to avoid this, e.g. by putting the emitter and
> absorber at the two foci of an ellipse, so that there's just one of each,
> then there is no such imbalance (and no asymmetry of radiation).
>
> >..... To get a spherical
> >incoming wave you need to produce an excitiation which is extended in
> >space (like a rim of the coffe cup) and all these spatially distant
> >sources (molecules of the rim) have to act in phase. This is why this is
> >a coherent source. Incoming spherical waves require a coherent source,
> >outgoing spherical waves do not. Since a point excitation is much more
> >likely than a coherent exitation in many distant points, we often see
> >outgoing spherical waves but very rarely incoming spherical waves
> >(although they happen too, see the coffe cup case).
>
> This is a nice example of a temporal double standard at work. Time direction
> aside, the "coherent exitation in many distant points" is exactly the same in
> the outgoing and incoming cases: one is the temporal mirror of the other. So
> you can't use a probability argument to exclude the incoming case, unless you
> give us a time-asymmetric principle to account for the fact that the same
> argument doesn't exclude the outgoing case. (This is essentially my objection
> to Popper on pp. 54-7.)
First of all, this would only be true if the pond was perfectly spherical.
Secondly, if you consider a circle around the source of the wave on water,
molecules at this circle absorb incoming wave _coherently_. Then they
reradiate it in all directions, but due to the interference of these
indivi- dual waves, the resulting wave is a spherical wave moving more
outwardly. Except for black holes anything that absorbs energy, can later
radiate this energy out, so the water molecules reradiating the energy
absorbed from the incoming wave should not be a problem.
Thirdly, I gave you an example of coherent generation of an incoming wave -
coffe cup (or a plate of soup, you can do the experiment on your own). So I
am not _excluding_ the incoming wave. The problem is that to create an
outgoing wave a localized (in space and time) perturbation is enough, but
to create an incoming wave you need a perturbation which is highly
correlated in space and time. Don't you think that the former is much more
likely than the latter on purely statistical grounds? There is no "temporal
double standard here", but this is a nice example of you misinterpreting
physical phenomena
.........
That it is _deliberate_ is indeed beside the point, but the point is that
in your example there indeed is a correlation - daffodils arrive to the
grandmother not purely randomly but on the days when 9 is picked. In fact,
on page 143 you write correctly what correlation of two events means
P(AB)>P(A)*P(B), but then make this false claim that there is correlation
between the movements of these two people before they by chance meet in a
cafe. The only correlation there is the one between being in the cafe and
entering it, but this is true for each of them separately, there is no
correlation _between_ their movements. I have pointed out to you that if
you selectively choose events, you can correlate anything with anything.
May I suggest that you think this over some more? You seem to have this go
past your ears (or eyes :-)). Without such selective choice the above
criterion for the correlation does not apply in the case where you claim
the correlation. If you disagree, please show how the above inequality
applies (on the days when they meet probabilities of them entering cafe,
inidivdualy and for both of them, is 1, and it is not true that 1>1). And
if you want to say that in one context you mean one thing by correlation
and then something else in another context, then this is a very good
example of inconsistency I crticise.
................
> >There may be a dstinction between "good subjective" and "bad subjective"
> >but what is inconsistent is your claim that using subjective,
> >agent-dependent arguments you can arrive at agent-independent "truth"
> >about the structure of nature.
>
> Where did I claim this?
Well, for instance on the page 4 you say that you want to demonstrate a view
from 'nowhen', which is independent of pecularities of our own perspective,
do you not? In another place (again I do not have the page at hand) you claim
that the flow of time is due to us, as agents, perceiving the nature and
not to the structure of nature itself, which you are aiming to reveal (time
symmetry, allowing for advanced action). If this is not the same as the claim
above, what is the difference?
...........
>
> Do you think there's any point in discussing whether the bottle exists when it
> is not actually being observed? If "no", then you are a hard core
> instrumentalist, and it doesn't surprise me that you find appeals to symmetry,
> etc., unconvincing. (If I were to argue with you, it would be about the
> general difficulties facing instrumentalism.) If "yes", then you've already
> shown a willingness to go beyond immediate observational evidence, and you
> need to explain to me why the arguments I take to favour advanced action in QM
> are not simply more of the same.
I do not want to get sidetracked, let's stick to what you say in the book.
However, whatever problems facing instrumentalism, they pale in comparison
with problems which arise when when we divorce from what can be observed. I
definitely think that arguing about 'truth' or 'falsity' of statements
about nature which cannot be decided by observation, is a waste of time as
far as our interaction with nature goes (by definition), even though it may
have some entertainment value,.
--
Dr. Andrzej Pindor The foolish reject what they see and
University of Toronto not what they think; the wise reject
Information Commons what they think and not what they see.
andrzej...@utoronto.ca Huang Po
Phone: (416) 978-5045
Fax: (416) 978-7705
[Moderator's note: Quoted text has been slightly trimmed and lines
reformatted to within 80 columns. Please try to keep lines of text
under 75 columns or so. -MM]
Aaron Bergman
>Using both the equation I wrote down and its
>time-reversed version, one can prove the time derivative of entropy is
>greater than or equal to zero, and also less than or equal to zero. So
>it's zero!
Or at a peak. Huang's point, as I see it, is that H need not have a
continuous first derivative, so you can have cusps.
John Collier
: It does not seem to me that there is anything controversial here, so I
: would say that the time asymetry is an unaviodable consequence of the
: assumption of molecular chaos. Huw Price tries to question this
: assumption in his book discussed here, but accepts results of
: statistical mechanics which can only be derived with this assumption and
: which are in agreement with observable physical phenomena. Then the
: question would rather be "why do we have molecular chaos?" (and
: hence time assymetry).
This, of course, is the real question. I know of little work on the problem
that isn't circular.
John
Chris Hillman
closely as I'd like :-( but think I can answer Kieran Mullen's question:
> Most of the texts on kinetic theory seem a bit old. When the mention
> the Poincare' recurrence time, they seem to think this means the motion
> is periodic. This seems false to me, since even if you come arbitrarily
> close to your initial condition, the dynamics are chaotic, so you won't
> reproduce your earlier time evolution. Is this accepted in stat mech
> circles?
I can't speak for the statistical mechanicians, but I -can- say that it is
definitely the case that the Poincare and Kac recurrence theorems do -not-
imply anything like "periodicity". The theorems say that in a (discrete
time) ergodic dynamical system, -if- you wait long enough (Kac's theorem
tells you how long you can expect to wait), eventually you'll return to
within any small neighborhood of your original position in the phase
space, but of course because of the "exponential divergence of initially
nearby trajectories" in a chaotic system, this doesn't imply anything
like "periodicity" in the sense of repeating history.
See for instance Petersen, Ergodic Theory, Cambridge U Press, 1983, for
the statement and proofs of these theorems --- they are actually pretty
easy to prove, so you don't need to read very much in this book to follow
along, provided you have had some exposure to measure theory.
Someone please correct me if I'm wrong, but I think that in the context of
Hamiltonian dynamics (continuous volume preserving flow on a phase space)
one applies these theorems (about an ergodic measure preserving map on a
probability measure space) by taking some kind of cross section of the
flow and asking how long it will take to return to a same neighborhood of
the point where your trajectory first hit this cross section (under the
assumption that the trajectory is "typical"). E.g., think of the phase
space as a torus and the section as some curve segment tranverse to the
flow. The probability measure would just be the "area" normalized so the
section has measure one. Then Kac's theorem says that if the discrete map
defines an ergodic discrete dynamical system or ergodic cascade, you can
expect to pass through the section within epsilon of the place you first
passed through it after about (length section)/epsilon passages through
the cycle. Note that this number may not have a simple relationship to
physical time!
Chris Hillman
Huw Price
> Huw Price wrote:
>> Two points:
>> 1. You say that I am "claiming that PI^3 ... is unjustified". If you
>> mean that I am claiming that PI^3 is false, then you've misread me:
>> I think that PI^3 is (at least approximately) true (at least in our
>> region)....
> It is of course possible that I have misread you on this point. However,
> you spend a whole chapter (Chapter 5) arguing that there is no
> justification to the principle of mu-Innocence (i.e. that it does not in
> fact hold) and this principle, in your own words (end of Chapter 2), is
> very close relative to PI^3. So does PI^3 hold or not?
A close relative, yes, but not the same thing! In fact, I devote much of
Chapter 5 to arguing that mu-Innocence is *not* the version of PI^3
relevant in thermodynamics (and therefore, in particular, that observational
evidence for the latter is not evidence for mu-Innocence). So the fact that
I think that mu-Innocence may well be false (with interesting implications
in QM) has no bearing at all on the case we were discussing.
Incidentally, there's an improved version of the argument from Chapter 5
in my paper "The Role of History in Microphysics", available at:
http://plato.stanford.edu/price/preprints/history.html
...
>>>..... To get a spherical
>>>incoming wave you need to produce an excitiation which is extended in
>>>space (like a rim of the coffe cup) and all these spatially distant
>>>sources (molecules of the rim) have to act in phase. This is why this is
>>>a coherent source. Incoming spherical waves require a coherent source,
>>>outgoing spherical waves do not. Since a point excitation is much more
>>>likely than a coherent exitation in many distant points, we often see
>>>outgoing spherical waves but very rarely incoming spherical waves
>>>(although they happen too, see the coffe cup case).
>> This is a nice example of a temporal double standard at work. Time
>>direction aside, the "coherent exitation in many distant points"
>>is exactly the same in the outgoing and incoming cases: one is the
>>temporal mirror of the other. So you can't use a probability argument
>>to exclude the incoming case, unless you give us a time-asymmetric
>>principle to account for the fact that the same argument doesn't
>>exclude the outgoing case. (This is essentially my objection to
>>Popper on pp. 54-7.)
> First of all, this would only be true if the pond was perfectly spherical.
This is irrelevant. In both ingoing and outgoing examples, one can adjust for
variable geometry in the edge of the pond (or cup or whatever) by adjusting
timing of the excitations.
> Secondly, if you consider a circle around the source of the wave on water,
> molecules at this circle absorb incoming wave _coherently_. Then they
> reradiate it in all directions, but due to the interference of these
> individual waves, the resulting wave is a spherical wave moving more
> outwardly. Except for black holes anything that absorbs energy, can later
> radiate this energy out, so the water molecules reradiating the energy
> absorbed from the incoming wave should not be a problem.
Fine, but this doesn't give you any justification for distinguishing the
incoming and outgoing cases, in your original argument.
> Thirdly, I gave you an example of coherent generation of an incoming wave -
> coffee cup (or a plate of soup, you can do the experiment on your own). So I
> am not _excluding_ the incoming wave. The problem is that to create an
> outgoing wave a localized (in space and time) perturbation is enough, but
> to create an incoming wave you need a perturbation which is highly
> correlated in space and time. Don't you think that the former is much more
> likely than the latter on purely statistical grounds? There is no "temporal
> double standard here", but this is a nice example of you misinterpreting
> physical phenomena
You're confusing the purely statistical issues (which are symmetric) with
issues about what it is possible for *us* to achieve by manipulation. It is
true that in both cases we have to manipulate the 'earlier' end of the
sequence of events, and that this is easier in the outgoing case (when the
earlier event is localised) than in the incoming case (when it is not). But
why should nature care about what we can manipulate?
...
Let's keep it simple. Assume there are just two places where guys A and B
meet, cafe 1 and cafe 2. Each of them picks one cafe or other at random
for lunch each day at noon, so they meet every second day, on average.
Let P(A1) be the probability that A goes to cafe 1, etc. Normally,
P(A1.B1) = P(A1)P(B1) = 1/4. But if we're told that they meet on a
particular day, then, relative to that evidence, P(A1.B1) = P(A2.B2) = 1/2,
while it remains true that P(A1) = P(B1) = 1/2. So P(A1.B1) > P(A1)P(B1) =
1/4.
Actually, this is irrelevant. It is possible to delete the entire sentence in
which the occurrence of the word "correlated" which bothers you occurs,
without significantly changing the argument of that paragraph. Please feel
free to do so in your copy.
Huw Price.
[PS This correspondence is now closed, at least at my end. HP]
Andrzej Pindor
>
> In article <36B9EA...@ic-unix.ic.utoronto.ca>,
> Andrzej Pindor <pin...@ic-unix.ic.utoronto.ca> wrote:
>
> >John Baez wrote:
> >>The Stosszahlansatz says:
> >>
> >>df(p1)/dt =
> >>
> >>integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
>
> >Actually (see Huang, Statistical Mechanics) assumption of "molecular
> >chaos" (as "Stosszahlansatz" as is refered to in this book) says that a
> >probability of finding a particle 1 with momentum p1 and particle 2 with
> >momentum 2 is given by f(p1)f(p2) (this is better reflected in the
> >German name which means "assumption about the number of collisions"),
> >i.e that particles are not correlated. The equation above uses of
> >course this fact. Time assymetry lays not in this assumption but indeed
> >in the equation above.
>
> The equation I wrote down is time-asymmetric. The equation you describe
> is time-symmetric.
I am not sure what you refer to in the last sentence. I have not
described any equation, I quoted a definition of "molecular chaos" given
by Huang, check equation (3.26) therein. There is only one time in there
so it cannot possibly by either time-symmetric or time-asymmetric.
The equation you quote has a time derivative and indeed is
time-asymmetric.
..........
> Calling both the equation I wrote down and the one you mention by the
> same name is a sure way to cause confusion.
You are perfectly right, this is why I refered to a definition in a well
known book.
My point is that it is not the assumption of molecular chaos which
is time-assymetric, but rather the analysis of the molecular collision
event (which is described by the equation you quote).
Andrzej
john baez
>john baez wrote:
>> Andrzej Pindor <pin...@ic-unix.ic.utoronto.ca> wrote:
>> >Actually (see Huang, Statistical Mechanics) assumption of "molecular
>> >chaos" (as "Stosszahlansatz" as is refered to in this book) says that a
>> >probability of finding a particle 1 with momentum p1 and particle 2 with
>> >momentum 2 is given by f(p1)f(p2) [...]
>> The equation you describe is time-symmetric.
>I am not sure what you refer to in the last sentence. I have not
>described any equation, I quoted a definition of "molecular chaos" given
>by Huang, check equation (3.26) therein.
I meant this equation: "the probability of finding particle 1 with
momentum p1 and particle 2 with momentum p2 is equal to f(p1)f(p2)".
If this equation holds for all p1, p2, it also holds when we reverse
the sign of time, thus multiplying the momenta p1, p2 by -1. That's
what I mean by saying that this equation is time-symmetric.
On other hand, the equation that I cited:
df(p1)/dt =
integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
is not time-symmetric, because when we reverse the sign of time, and
thus multiply all momenta by -1, we get another, inequivalent, equation.
To summarize one final time:
1) Calling both these equations the "assumption of molecular chaos" is
bad because they are different equations, and one is time-symmetric
while the other is not.
2) If the particles in a box of gas satisfy time-symmetric equations
of motion, any proof that dS/dt >= 0 which does not also prove that
dS/dt <= 0 must involve assumptions that are not time-symmetric.
Therefore:
3) Since this assumption is time-symmetric:
"the probability of finding particle 1 with momentum p1 and particle
2 with momentum p2 is equal to f(p1)f(p2)"
we cannot use it alone to show dS/dt >= 0 without also being able to
show dS/dt <= 0. This assumption cannot explain why entropy increases.
However, this assumption can:
df(p1)/dt =
integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
Indeed, this assumption is the basis of Boltzmann's H-theorem.
Andrzej Pindor
[Moderator's note: Quoted text deleted, here and again below. -TB]
> 1) Calling both these equations the "assumption of molecular chaos" is
> bad because they are different equations, and one is time-symmetric
> while the other is not.
I couldn't agree more. This was the point of my posting - calling the
equation you wrote down the "assumption of molecular chaos" might lead
to confusion since the very popular statistical physics handbook (Huang)
uses this term for something else.
> 3) Since this assumption is time-symmetric:
>
> "the probability of finding particle 1 with momentum p1 and particle
> 2 with momentum p2 is equal to f(p1)f(p2)"
>
> we cannot use it alone to show dS/dt >= 0 without also being able to
> show dS/dt <= 0. This assumption cannot explain why entropy increases.
>
> However, this assumption can:
>
> df(p1)/dt =
>
> integral w(p1, p2; p1', p2') [f(p1')f(p2') - f(p1)f(p2)] dp2 dp1' dp2'
Isn't the above what I have said ? Here is the quote from my previous
posting, which somehow droped out from your reply:
...........
My point is that it is not the assumption of molecular chaos which
is time-asymmetric, but rather the analysis of the molecular collision
event (which is described by the equation you quote).
...........
> Indeed, this assumption is the basis of Boltzmann's H-theorem.
Absolutely.