| Topic: |
Science > Physics |
| User: |
"Craig Franck" |
| Date: |
18 Aug 2005 06:43:32 PM |
| Object: |
Entropy question |
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in or if they are behaving in the most likely way.
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "" |
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| Title: Re: Entropy question |
22 Aug 2005 12:54:53 PM |
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Craig Franck <craig.franck@verizon.net> wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this?
Start with a uniform gas in an otherwise empty region. If the gas
is big enough or dense enough -- if its volume is larger than roughly
the cube of the Jeans length -- it will begin to collapse into lumps.
This dynamical instability has been known since Jeans' work in 1902.
These days it can be watched quite carefully in many-body computer
simulations.
What happens to entropy during this process? The modern work on
this apparently began with a 1962 paper by Antonov, but it's in an
obscure Russian journal, and I confess I haven't read it. The usual
reference is a paper by Lynden-Bell and Wood, Mon. Not. R. Astr. Soc.
138 (1968) 495. When you do the calculation, you find that the
entropy of each individual clump decreases, but the entropy of the
system as a whole increases. This is not a contradiction -- it is
possible because during the process of gravitational collapse,
individual particles and small clumps get flung out of the collapsing
regions at very high speeds. The "extra" entropy ends up in a hot,
thin ``interstellar'' gas.
If you think about the energetics, you'll see that this has to happen.
As the gas collapses, it heats up, which stabilizes the collapse. To
collapse further, it has to get rid of energy; something has to carry
energy away from the lumps. One way to describe this is to say that a
gravitationally collapsing object has negative specific heat -- as it
radiates energy, its temperature increases. This leads to unusual
thermodynamic behavior, and in particular your intuition of entropy as a
measure of "disorder" is likely to fail. (A large uniform, homogeneous
distribution of self-gravitating gas actually has a fairly low entropy,
while a black hole has a huge entropy.) The details of the gravitational
interaction are not so important in the analysis; what matters is that
the interaction is long range and purely attractive.
This effect can already be seen in simulations with uniform spheres.
In realistic structure formation, the energy is mainly carried off by
photons, I think, and the entropy mostly goes into a thin photon gas.
One of the standard problems in modeling structure formation is to
understand how, and how fast, collapsing structures can be cooled.
You can, if you like, call this an increase in the entropy of the "rest
of the Universe." But the "rest of the Universe" here need not be
some separate system; it can be a portion of the gas you started out
with. In particular, you don't need to have something else to carry
off the energy, although the clumping will happen faster if you do.
In fact, you can imagine putting the whole system inside a perfectly
reflecting box, so the "interstellar gas" can't escape. If the box is
big enough, you will still get a "gravothermal catastrophe"---the gas
will segregate into a contracting central core and a surrounding halo.
This sort of dynamics is important in understanding the structure of
globular clusters, for instance, and the detailed mechanisms for the
"radiation" of small clumps has been studied numerically.
You might want to look at the beginnning of chapter 5 of Zeh's book
_The Physical Basis of the Direction of Time_ for a short but more
mathematical description.
Steve Carlip
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| User: "Craig Franck" |
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| Title: Re: Entropy question |
22 Aug 2005 06:57:06 PM |
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<carlip-nospam@physics.ucdavis.edu> wrote
When you do the calculation, you find that the
entropy of each individual clump decreases, but the entropy of the
system as a whole increases. This is not a contradiction -- it is
possible because during the process of gravitational collapse,
individual particles and small clumps get flung out of the collapsing
regions at very high speeds. The "extra" entropy ends up in a hot,
thin ``interstellar'' gas.
This actually seems intuitively true because it helps explain how the
large-scale structures of the universe can form.
What's also interesting about gravitational collapse is when stars start
to run out of hydrogen fuel, they get hotter, not cooler.
You might want to look at the beginnning of chapter 5 of Zeh's book
_The Physical Basis of the Direction of Time_ for a short but more
mathematical description.
I'll look for that. I currently have "The Direction of Time" by Reichenbach
and "What is Mathematics?" by Courant et al on order from Amazon.
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "Edward Green" |
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| Title: Re: Entropy question |
23 Aug 2005 06:42:35 PM |
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wrote:
Craig Franck <craig.franck@verizon.net> wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this?
Start with a uniform gas in an otherwise empty region. If the gas
is big enough or dense enough -- if its volume is larger than roughly
the cube of the Jeans length -- it will begin to collapse into lumps.
This dynamical instability has been known since Jeans' work in 1902.
These days it can be watched quite carefully in many-body computer
simulations.
What happens to entropy during this process? The modern work on
this apparently began with a 1962 paper by Antonov, but it's in an
obscure Russian journal, and I confess I haven't read it. The usual
reference is a paper by Lynden-Bell and Wood, Mon. Not. R. Astr. Soc.
138 (1968) 495. When you do the calculation, you find that the
entropy of each individual clump decreases, but the entropy of the
system as a whole increases. This is not a contradiction -- it is
possible because during the process of gravitational collapse,
individual particles and small clumps get flung out of the collapsing
regions at very high speeds. The "extra" entropy ends up in a hot,
thin ``interstellar'' gas.
If you think about the energetics, you'll see that this has to happen.
As the gas collapses, it heats up, which stabilizes the collapse. To
collapse further, it has to get rid of energy; something has to carry
energy away from the lumps. One way to describe this is to say that a
gravitationally collapsing object has negative specific heat -- as it
radiates energy, its temperature increases. This leads to unusual
thermodynamic behavior, and in particular your intuition of entropy as a
measure of "disorder" is likely to fail. (A large uniform, homogeneous
distribution of self-gravitating gas actually has a fairly low entropy,
while a black hole has a huge entropy.) The details of the gravitational
interaction are not so important in the analysis; what matters is that
the interaction is long range and purely attractive.
This effect can already be seen in simulations with uniform spheres.
In realistic structure formation, the energy is mainly carried off by
photons, I think, and the entropy mostly goes into a thin photon gas.
One of the standard problems in modeling structure formation is to
understand how, and how fast, collapsing structures can be cooled.
You can, if you like, call this an increase in the entropy of the "rest
of the Universe." But the "rest of the Universe" here need not be
some separate system; it can be a portion of the gas you started out
with. In particular, you don't need to have something else to carry
off the energy, although the clumping will happen faster if you do.
In fact, you can imagine putting the whole system inside a perfectly
reflecting box, so the "interstellar gas" can't escape. If the box is
big enough, you will still get a "gravothermal catastrophe"---the gas
will segregate into a contracting central core and a surrounding halo.
This sort of dynamics is important in understanding the structure of
globular clusters, for instance, and the detailed mechanisms for the
"radiation" of small clumps has been studied numerically.
You might want to look at the beginnning of chapter 5 of Zeh's book
_The Physical Basis of the Direction of Time_ for a short but more
mathematical description.
Without wishing to trivialize your beautiful exposition, I still think
one could get the idea that this is some subtle and hard to understand
property specific to gravitational collapse: subtle it may or may not
be, but it is not limited to gravitational collapse. The same
trade-off between reduced configurational entropy of a part of the
system for increased overall entropy is seen anytime a system decides
it is in its thermodynamic interest to condense. We might ask
ourselves the same question of condensing steam.
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| User: "Sam Wormley" |
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| Title: Re: Entropy question |
18 Aug 2005 06:55:18 PM |
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Craig Franck wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in or if they are behaving in the most likely way.
Entropy
http://scienceworld.wolfram.com/physics/Entropy.html
Look at the rolls of heat and temperature.
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| User: "Craig Franck" |
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| Title: Re: Entropy question |
18 Aug 2005 07:52:32 PM |
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"Sam Wormley" wrote
Craig Franck wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in or if they are behaving in the most likely way.
Entropy
http://scienceworld.wolfram.com/physics/Entropy.html
Look at the rolls of heat and temperature.
Thanks for the link.
The conceptual problem I'm having is Penrose is saying the relatively
low entropy of the sun (considered as it being so far from thermal
equilibrium) "comes from a huge reservoir of low entropy potentially
available in the uniformity of the gas." But that's only if the gas has
the potential of gravitating together. Otherwise, it's in a state of high
entropy, just like the gas in the container.
So if the gas is in some chaotic state where it may condense or may
distribute so the gravitational attraction is not the dominant factor,
is the amount of entropy bouncing back and forth between very high
as in a gas on earth and very low as potential star-forming material?
It seems the entropy is coming from a potential path that the system
may take in the future.
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
19 Aug 2005 03:38:35 AM |
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Hi, Craig,
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well.
The constant factor is that, as time goes by, a closed system always
evolves to states of higher entropy.
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in
You're quite right, they don't. They know nothing.
There is in fact a difference in scale between your two examples.
Consider an initially uniform sample of gas undergoing a small
perturbation. Because of the invariance in space and time, the
eigenmodes are plane waves, and to understand the response of the
system to the perturbation, you only have to understand the response to
plane-wave perturbations of local density. The density perturbation
gives rise to forces, which can act either to intensify, or to weaken,
the perturbation. There are two components to the force, the one due to
Boyle's law, and the one due to self-gravitation, respectively. The
Boyle's law component always acts to weaken the perturbation, while the
self-gravitation component always acts to intensify it. For small
perturbation wavelengths, Boyle's law always wins, but for sufficiently
large wavelengths, it is the self-gravitation that wins.
or if they are behaving in the most likely way.
You've got it.
The conceptual problem I'm having is Penrose is saying the relatively
low entropy of the sun (considered as it being so far from thermal
equilibrium) "comes from a huge reservoir of low entropy potentially
available in the uniformity of the gas."
The first thing to learn is not to believe everything you read or hear.
Penrose has made lots of money by writing popularizations of science.
Some of the things he writes are perfectly OK, but his logic is often
suspect. Caveat emptor.
But that's only if the gas has
the potential of gravitating together. Otherwise, it's in a state of high
entropy, just like the gas in the container.
So if the gas is in some chaotic state where it may condense or may
distribute so the gravitational attraction is not the dominant factor,
is the amount of entropy bouncing back and forth between very high
as in a gas on earth and very low as potential star-forming material?
It seems the entropy is coming from a potential path that the system
may take in the future.
Yes, well, protons do have the option available to them of fusing
together to form heavy nuclei. That process leads to an increase of
entropy for the system as a whole.
Never forget that the "laws" of physics were invented to explain what
is observed, not the other way around. If the laws don't explain
things, then ultimately they get traded in for a new model.
Cheers,
Zigoteau.
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| User: "" |
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| Title: Re: Entropy question |
18 Aug 2005 07:12:29 PM |
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You can't quantify order
-- Light Falls --
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| User: "NightsoilDalits@RyugyongHotel" |
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| Title: Re: Entropy question |
18 Aug 2005 07:36:59 PM |
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<macromitch@internetCDS.com> wrote in message
news:1124410349.210257.166780@z14g2000cwz.googlegroups.com...
You can't quantify order
-- Light Falls --
You can weigh it. If light falls, it weighs something, and is quantifiable.
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| User: "Gregory L. Hansen" |
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| Title: Re: Entropy question |
19 Aug 2005 10:53:34 AM |
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In article <1124410349.210257.166780@z14g2000cwz.googlegroups.com>,
<macromitch@internetCDS.com> wrote:
You can't quantify order
-- Light Falls --
It depends on how you define "order". Folded laundry versus rumpled
laundry has no thermodynamic meaning. But you can quantify the number of
states a system may assume that gives the same thermodynamic quantities.
--
"Out of the way, you slime, a physicist is coming!"
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| User: "Edward Green" |
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| Title: Re: Entropy question |
19 Aug 2005 11:10:01 AM |
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Gregory L. Hansen wrote:
In article <1124410349.210257.166780@z14g2000cwz.googlegroups.com>,
<macromitch@internetCDS.com> wrote:
You can't quantify order
-- Light Falls --
It depends on how you define "order". Folded laundry versus rumpled
laundry has no thermodynamic meaning. But you can quantify the number of
states a system may assume that gives the same thermodynamic quantities.
Would be totally impossible to quantify the measure of configurational
states consistent with "folded" vs "rumbled"? People who study protein
conformation probably have something to say about this.
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| User: "Edward Green" |
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| Title: Re: Entropy question |
19 Aug 2005 11:05:40 AM |
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Craig Franck wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
I know enough to recognize that yours is a standard question, but not
enough to spit out the standard answer. You are asking how it is in a
gravitating system that sometimes condensed states have higher entropy
than (are more stable than) non-condensed states.
Actually this formulation is not quite right, because what we expect to
have higher entropy overall is the closed system -- as Zigoteau
mentioned. And in this regard, condensation can of course be sometimes
favored in non-gravitating systems as well -- as an isolated volume
containing some water vapor and some liquid. So I'd perhaps first
wonder if there is anything fundamentally different about the treatment
of entropy in gravitationally condensing systems and in, say, van der
Walls condensing systems. Shouldn't we be equally bemused if we have
an insulated vessel containing a condensed phase in equilibrium with
some gas?
The solution must lie in the observation that there is more than matter
in such systems ... there is energy (aside from the rest mass of the
matter). And so the overall entropy is a trade off between the
positional entropy of the mass centers and the entropy associated with
the distribution of the remaining energy; and it must happen sometimes
that the spontaneous state is driven by the invisible second term.
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in or if they are behaving in the most likely way.
Whatever the expression in the N particles, it obviously effectively
leaves out the second factor. Non-interacting ideal gasses do not
condense: gravity is an interaction.
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| User: "Gregory L. Hansen" |
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| Title: Re: Entropy question |
19 Aug 2005 10:51:00 AM |
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In article <E29Ne.3016$zb.2582@trndny04>,
Craig Franck <craig.franck@verizon.net> wrote:
I remember reading an account of the fact that when a gas expands
in a container entropy goes up, but when gas and dust clouds in
space do the opposite and condense, entropy increases as well. (I
think it was in a Roger Penrose book.)
Why exactly is this? From a statistical POV, I don't see how N
number of particles know which kind of system they are participating
in or if they are behaving in the most likely way.
A flip way of defining entropy is that things do what they do, and they
won't undo it. If you drop a rock, it falls to the ground, and won't
levitate back up. If you drop a helium-filled balloon it will rise, and
won't come down until the helium leaks out of it. If you drop a
helium-filled balloon on the Moon it will hit the ground because there's
no atmosphere to bouy it up.
There's enough gas over large regions in interstellar space that it forms
a considerable gravitating mass, and it "falls" together. Unless the
gravity goes away, the gas won't disperse again without some new
influence. The amounts of gas we deal with in the laboratory are much
too small for self-gravitation to have any effect. But we can compress
ten moles of gas to 2000 psi, and gravity can't.
--
"The polhode rolls without slipping on the herpolhode lying in the
invariable plane." -- Goldstein, Classical Mechanics 2nd. ed., p207.
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
20 Aug 2005 06:00:56 AM |
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Hi, Craig,
Having read all the replies, I see what's going on.
No, you're not quite there yet
The gas in space
heats up as entropy increase, while the gas in the laboratory cools
down as entropy increases. And when the gas gets hot enough, it
can access nuclear forces as an energy source.
I did warn you not to believe everything that Penrose writes.
The two most basic theories of modern physics are general relativity
and quantum field theory. The fit between GR and QFT is a bit ragged,
but for most systems there is no conflict. GR and QFT are theories of
mechanics, and the two together allow you in principle to work out how
any given system will behave.
I say 'in principle', because in practice systems with very large
numbers of particles cannot be analyzed in a reasonable length of time.
That's where statistical mechanics comes in, as an approximation
method. It's not the only approximation method in physics. GR is often
approximated by Newtonian gravitation, and QFT is often approximated by
Newtonian mechanics and Maxwellian electrodynamics. Since full-blown GR
and QFT require enormous computing power, they are typically only used
when only they will give the required accuracy. There are ways of
determining when the simpler theories give answers which are good
enough.
Modern statistical mechanics is an approximation to QFT. It subsumes
all of classical thermodynamics, and explains a lot more besides.
Statistical mechanics relies heavily on the idea of thermodynamic
equilibrium. It can come up with answers as long as your system is
closed, and you can break it up into a number of weakly-interacting
subsystems, each of which is near equilibrium. The answers it comes up
with are good for all practical purposes, but you must remember that
the statistical method is not exact. If there is ever a conflict
between mechanics and statistical mechanics, the answer from mechanics
is the right one. To repeat, statistical mechanics is only an
approximate theory. It really only works for systems much smaller than
the solar system.
If a theory treats space and time on a different footing, then you know
it is not relativistic. In special relativity, the momentum p and the
energy together form the energy-momentum 4-vector. The energy is the
time component of the energy-momentum 4-vector, and may be symbolized
p_t. When observers in different inertial frames measure the
energy-momentum 4-vector, they get different answers, say p' and p",
but p' and p" are Lorentz transforms of one another. A central concept
of statistical mechanics is the partition function Z, which is the sum
over all states r of the quantity exp(-E_r/kT). The Helmholtz free
energy F of the system is equal to kTln(Z). All other thermodynamic
parameters of the system can be derived from F: in particular the
entropy S is (dF/dT)|V (partial derivative wrt temperature, keeping the
volume constant). Since statistical mechanics concentrates on p_t and
ignores p_x, p_y and p_z, you know immediately that it is not even
special relativistic.
In a system at thermodynamic equilibrium, there can be no macroscopic
relative motion. In the solar system, relative velocities are of order
1e4 m/s. A near-earth asteroid has a kinetic energy three orders of
magnitude greater than its internal thermal energy. The situation is
extremely far from thermodynamic equilibrium, and relativistic effects
mean that, strictly speaking, you cannot add the entropies of the
different parts to get the overall value. OK, the discrepancy is only
of order 0.01%, but the result is not exact, not fundamental, just a
rough-and-ready way of calculating.
As I tried to explain in my previous post, in a small container full of
gas, the uniform state is stable with respect to small perturbations.
However when the volume of gas is of astronomical size, the uniform
distribution can become unstable. Local concentrations of gas grow
spontaneously, and ultimately form stars. This process is nowhere near
equilibrium. The method of analysis I outlined for you involving
Fourier decomposition of perturbations is essentially the interplay of
Newtonian gravitation and mechanics. Entropy is irrelevant for
determing the largest-scale features of the process. Thermodynamics
might come in at a lower level, for example in determining the
temperature of the local concentration of gas, but there is no way it
can be thought of as driving the process. At most, you can bring the
entropy in as a way of monitoring what is happening, but it is not
guaranteed to be a very good way.
To say the gas in space (1) has lower entropy before it gravitates together
just means (2) it has a great deal of potential energy that could be liberated
under the right conditions.
Statement (2) is possibly meaningful. Statement (1) is just handwaving.
What was conceptually challenging is the fact that, thinking in terms
of entropy doing work, the gas needs to get to a higher entropy state
(form a star) before it can access the all the potential lower-entropy.
Not to say conceptually challenged. I think that Penrose often puts the
cart before the horse. Of course, there's no law against philosophical
musings about vast events in the Universe which dwarf our puny selves,
and it's morally superior to getting paralytic on a Saturday night and
beating up old ladies, but it's more like religion than science.
Cheers,
Zigoteau.
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| User: "Andy Resnick" |
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| Title: Re: Entropy question |
22 Aug 2005 08:12:41 AM |
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Zigoteau wrote:
<snip>
I say 'in principle', because in practice systems with very large
numbers of particles cannot be analyzed in a reasonable length of time.
That's where statistical mechanics comes in, as an approximation
method.
<snip>
Yes, but saying statistical mechanics is an "approximation" to some
deeper truth isn't exactly right right either. That's like believing in
hidden variables in quantum mechanics. Statistical mechanics also makes
a strong statement that microscale physics is not always relevant to the
macroscale. That's where entropy comes in, and irreversibility.
<snip>
As I tried to explain in my previous post, in a small container full of
gas, the uniform state is stable with respect to small perturbations.
However when the volume of gas is of astronomical size, the uniform
distribution can become unstable. Local concentrations of gas grow
spontaneously, and ultimately form stars. This process is nowhere near
equilibrium. The method of analysis I outlined for you involving
Fourier decomposition of perturbations is essentially the interplay of
Newtonian gravitation and mechanics. Entropy is irrelevant for
determing the largest-scale features of the process. Thermodynamics
might come in at a lower level, for example in determining the
temperature of the local concentration of gas, but there is no way it
can be thought of as driving the process. At most, you can bring the
entropy in as a way of monitoring what is happening, but it is not
guaranteed to be a very good way.
What you describe is similar to a phase transition (maybe that was
intentional). Thermodynamics may not be able to provide any details
about phase transitions, but statistical mechanics has a lot to say.
Statistical mechanics can't say what the density (or temperature) of the
gas will be at a specified (x,y,z,t) coordinate, but in terms of the
global dynamics, that may not be relevant anyway. To be sure, it's
possible to create a system with strong dependence on the specific
microstate (turbulence, for example), so that's where the current
research on dynamics is.
<snip>
and it's morally superior to getting paralytic on a Saturday night and
beating up old ladies, but it's more like religion than science.
That gives me an idea... create a religion where it's morally superior
to "getting paralytic on a Saturday night and beating up old ladies"!
:) Think there's a market for that? I need beer money.....
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
22 Aug 2005 02:34:40 PM |
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Hi, Andy,
Yes, but saying statistical mechanics is an "approximation" to some
deeper truth isn't exactly right right either. That's like believing in
hidden variables in quantum mechanics. Statistical mechanics also makes
a strong statement that microscale physics is not always relevant to the
macroscale. That's where entropy comes in, and irreversibility.
I can see where you're coming from, but I'm sticking to my guns.
Statistical mechanics reconciles classical thermodynamics with
mechanics, either Newtonian or quantum. You are right that for a long
time people believed, on the basis of analyses of systems with a small
number of degrees of freedom, that time had no inherent direction in
Newtonian mechanics. In thermodynamics it definitely does have one.
Statistical mechanics explains why such analyses are misguided for
systems where the number of degrees of freedom is very large.
Cheers,
Zigoteau.
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| User: "Andy Resnick" |
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| Title: Re: Entropy question |
22 Aug 2005 03:11:59 PM |
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Zigoteau wrote:
I can see where you're coming from, but I'm sticking to my guns.
Statistical mechanics reconciles classical thermodynamics with
mechanics, either Newtonian or quantum. You are right that for a long
time people believed, on the basis of analyses of systems with a small
number of degrees of freedom, that time had no inherent direction in
Newtonian mechanics. In thermodynamics it definitely does have one.
Statistical mechanics explains why such analyses are misguided for
systems where the number of degrees of freedom is very large.
I don't know if you read these 2 papers by Jaynes mentioned in another,
similar, thread:
Phys. Rev. 106, 620-630 (1957)
Phys. Rev. 108, 171-190 (1957)
But there's an interesting section on the density matrix. Jaynes draws
a distinction between a "complete" and a "sufficient" density matrix.
It plays into how entropy increases regardless of the temporal ordering
of measurement events.
His example is to consider a measurment on a stream of spin 1/2
particles passing through a magnetic field: the object is to measure if
the particles are 'up' or 'down'.
A "sufficient" density matrix is the 2 x 2 matrix specified for a single
particle, while the "complete" matrix for the entire experiment is a 2N
x 2N matrix, where N is the number of particles measured.
In any case, we may consider the sufficient matrix, and interpret the
measurement results in terms of a statistical distribution. This is
what is usually done. Alternatively, we could use the complete matrix,
but then we have only performed a single experiment, and cannot use
probability arguments (in addition to the matrix being rather unwieldy)
to interpret the result.
His point is that when the sufficient matrix is used, we are losing
information: in this case, the possible correlations which occur between
different particles (among other possibilities). And, it is this loss
of information which makes the process irreversible. Now the
interesting thing is that it can be shown that the entropy goes up as
information is lost, regardless of the temporal ordering of events- he
uses multiple experimenters to illustrate this point.
All I'm saying is that there's more to statistical mechanics than just
statistics and mechanics. There seems to be a way to generate
higher-level structure: emergence is the current buzzword, I think.
It's funny how the interesting stuff is learned after formal schooling
is over.....
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
22 Aug 2005 04:06:27 PM |
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Hi, Andy,
I don't know if you read these 2 papers by Jaynes mentioned in another,
similar, thread:
Phys. Rev. 106, 620-630 (1957)
Phys. Rev. 108, 171-190 (1957)
Thanks for the references, which I will read. However, are you saying
that statistical mechanics supersedes normal mechanics? I cannot go
along with that.
All I'm saying is that there's more to statistical mechanics than just
statistics and mechanics. There seems to be a way to generate
higher-level structure: emergence is the current buzzword, I think.
IMHO "emergence" is the second-last refuge of the scoundrel.
Yes, structures emerge, usually when they have been designed in
(including evolution as the blind watchmaker). They can be explained
adequately by statistical mechanics, but the systems actually obey the
standard laws of mechanics, which cannot be discarded if you want to
understand the kinetics of the process.
Cheers,
Zigoteau.
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| User: "" |
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| Title: Re: Entropy question |
22 Aug 2005 04:22:29 PM |
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Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
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| User: "T Wake" |
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| Title: Re: Entropy question |
22 Aug 2005 04:35:38 PM |
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<donstockbauer@hotmail.com> wrote in message
news:1124745749.679918.256000@g43g2000cwa.googlegroups.com...
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
Easy enough really.
Is the probability non-zero? If so, then given a large enough sample
(infinite universe) and long enough it will happen.
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| User: "Sbharris[atsign]ix.netcom.com" |
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| Title: Re: Entropy question |
22 Aug 2005 08:48:02 PM |
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T Wake wrote:
<donstockbauer@hotmail.com> wrote in message
news:1124745749.679918.256000@g43g2000cwa.googlegroups.com...
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
Easy enough really.
Is the probability non-zero? If so, then given a large enough sample
(infinite universe) and long enough it will happen.
COMMENT:
The only problem is there's no reason to think the universe is
infinite. And whether it is or not, there's certainly not enough time
since the big bang to be "long enough" for most interesting things to
happen by chance alone.
It turns out that the universe has to be quite finally tuned to get
even long-lived stars and thus a stable source of low-entropy energy.
And you probably need a stable source of low-energy entropy to get life
(self-reproducing molecular machines) out of ordinary matter. The whole
thing is an amazing series of rim-shots to get stars, then life, then
life with brains complicated enough to make culture, then a secondary
process of evolution of cultural ideas themselves on THAT substrate
(ie, you do it all over again in software instead of hardware, too
boostrap yourself to higher effective intelligence). That is us.
Our star is a third the age of the universe, and brains big enough to
hold spoken culture are maybe 1/100,000th the age of our star. Writen
culture (with attendent increase in pace of cultural evolution) is a
tenth of THAT time. Going from horse and buggy to moon rockets in less
than a century is merely the external mechanical menifestation of
cultural evolution, which proceeds at Moore's law speed once culture is
connected and transmitted at the speed of light.
Computers and digitization of culture is a tenth of written-culture
time. Computer power itself has been doubling every few years since
computers were invented-- a direct consequence of the fact that
computer power proceeds at nearly the pace of cultural evolution (human
software evolution). In a comparable amount of time since invention of
the digital computer, we expect to see digital processing power
increase to the same power as that of our natural brains, making
computers potentially capable of (for the first time) participating
fully in the processing of "human" culture. At some point not too long
after THAT, we reach a cusp at which processing platforms (either
organic or non-organic, or a gemish of both) become completely
designable and constructable by software running on the same
substrates.
You see what happens then-- no more bottleneck. Presently,
computer-hardware evolution has been locked to human cultural
evolution, which in turn has been locked to the maximal pace at which
organic brains can process information, with writen and primitive
digital computational assistance, and speed of light knowledge
transfer. But at the point that computer hardware becomes capable of
FULLY processing cultural software, including the full design of better
computers, both cultural/technical and computer improvement processes
now take off at a rate which is the square of the old one, because now
each process (computer evolution, cultural evolution) is now (for the
first time) capable of fully driving the other.
Presently, computer design improvements procede with a rate-limiting
step connected with thinking on present human brains. When this is
removed, something transformative occurs. We've been headed in this
direction now for billions of years, but each step is so much faster
than the last. Very soon, within 50 or at most 100 years, comes
something like a chain reaction; an explosian.
Stay tuned. Each few years of this will bring a sea change. Can you
feel it? Cell phones, the internet, google, i-PODs. Every few years,
look to some large chunk of human cultural information reduced to
something digital you can hold in your hand and take with you. Until
all of it is, so that all of human knowledge and much of human culture
sits in your "i-POD-like" thingamajig, linked to others, and possibly
increasing directly to your brain. Then look to transformative changes
in access to this, and processing of chunks of this, again every few
years. Until the system doesn't need human brains anymore to continue
to evolve. So you hope that at least some brains are tuned into or
connected to it in some fashion, in order to go along with the ride.
Shortly after that, we get destruction or else something close to
god-hood. Something unimaginable. The old organic-only style brains are
due to go the way of vinyl records, eventually. We'll keep the meat for
awhile-- we always do keep the old stuff around. But the real action
will be increasingly elsewhere.
SBH
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| User: "Craig Franck" |
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| Title: Re: Entropy question |
23 Aug 2005 07:12:10 PM |
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"Sbharris[atsign]ix.netcom.com" wrote
T Wake wrote:
<donstockbauer@hotmail.com> wrote
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
Easy enough really.
Is the probability non-zero? If so, then given a large enough sample
(infinite universe) and long enough it will happen.
COMMENT:
The only problem is there's no reason to think the universe is
infinite. And whether it is or not, there's certainly not enough time
since the big bang to be "long enough" for most interesting things to
happen by chance alone.
Perhaps, but existence itself could be infinite. Imagine an infinite number
of finite universes.
It turns out that the universe has to be quite finally tuned to get
even long-lived stars and thus a stable source of low-entropy energy.
I'd say the universe being infinite is more in accordance with physical
theory than it being finely tuned.
We've been headed in this
direction now for billions of years, but each step is so much faster
than the last. Very soon, within 50 or at most 100 years, comes
something like a chain reaction; an explosian.
The problem with this reasoning is, even if secondary processes like
human culture move toward greater complexity, I can't imagine
evolution itself having a direction toward the devotement of such a
kind of culture.
Shortly after that, we get destruction or else something close to
god-hood. Something unimaginable. The old organic-only style brains are
due to go the way of vinyl records, eventually. We'll keep the meat for
awhile-- we always do keep the old stuff around. But the real action
will be increasingly elsewhere.
You sound like a big fan of Frank Tipler.
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "Edward Green" |
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| Title: Re: Entropy question |
23 Aug 2005 07:12:33 PM |
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Sbharris[atsign]ix.netcom.com wrote:
What a pleasure to hear your erudite e-voice.
{...}
Comments:
Standard comment that unbounded optimism puts most evolution on a pure
exponential, but that in reality quasi-exponentials are merely the rise
of a logistics curve yet to reach inflection.
My favorite dull example: cars vs. computers. Computers still appear
exponential, but cars inflected 50 years ago. That said, we still seem
to be in the ramp-up of artificial information processing, with the
inflection point nowhere in sight.
On transfer of intelligence to different substrate: one will want to
clarify the mapping of consciousness to the physical -- at least, if
one believes there is such a mapping, is not a pure processist, and
wants to go on thinking and not merely be efficiently simulated. I
realize the processists tend to view the expectation of an intimate
mapping of consciousness to the physical brands one a loose-thinking
new-age kook. Funny, I don't _feel_ like a new age kook.
Then, there is sleep. Shakespeare already noticed the connection of
sleep to death -- perhance to dream, and etc. -- and supposing a
simulation of us could be undertaken on a substrate that fufilled
whatever physical/conscious mapping might satify those so-inclined, and
the mucilid us were killed and the simulation woke up, and thought and
felt just like we did, how would that differ from sleeping and waking?
I argue without conviction, and would be a transporter refusenik on
Startrek.
Oh yeah ... I think it was C.S. Lewis ... argued that our _children_
are very like these affective simulations, will think and feel very
much like us, if not identically, and so we should take comfort in this
immortality. Yeah, right. And yet I turn off the switch without
regret each night, and am happy to do so; and who knows that the
conscious ME of that day does not become as extinct as an instance of a
program, to be reloaded the next.
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
24 Aug 2005 04:40:42 AM |
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Hi Edward,
Right on! I can't add anything to that.
Cheers,
Zigoteau.
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| User: "Edward Green" |
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| Title: Re: Entropy question |
24 Aug 2005 04:42:58 PM |
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Zigoteau wrote:
Hi Edward,
Right on! I can't add anything to that.
You just did: a pleasant word. :-)
Thanks for that.
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| User: "Andy Resnick" |
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| Title: Re: Entropy question |
23 Aug 2005 08:00:09 AM |
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wrote:
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
There's a difference between "not being able to understand it now" and
"not ever being able to understand it". I'm not a fan of putting
arbitrary physical phenomena beyond the reach of science.
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
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| User: "Craig Franck" |
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| Title: Re: Entropy question |
23 Aug 2005 06:52:16 PM |
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"Andy Resnick" wrote
donstockbauer@hotmail.com wrote:
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
There's a difference between "not being able to understand it now" and
"not ever being able to understand it". I'm not a fan of putting
arbitrary physical phenomena beyond the reach of science.
I generally agree with this, however, when it comes to minds, I would
not consider this an arbitrary case. I've read somewhat convincing
arguments that mental states cannot in principle be totally understood
in terms of physical processes.
"X believes Y" is problematic in that the phenomenological component
of a belief is not a property you can ascribe to a physical system. As a
bare minimum, you need to be able to translate mental-statements into
brain-statements, which means they are two logically different domains.
(Granted, this may be a linguistic problem: consciousness could be
a "global property" of a brain, but this seems to simply be a purge of
scare words.)
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "Andy Resnick" |
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| Title: Re: Entropy question |
24 Aug 2005 02:03:40 PM |
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Craig Franck wrote:
"Andy Resnick" wrote
donstockbauer@hotmail.com wrote:
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
There's a difference between "not being able to understand it now" and
"not ever being able to understand it". I'm not a fan of putting
arbitrary physical phenomena beyond the reach of science.
I generally agree with this, however, when it comes to minds, I would
not consider this an arbitrary case. I've read somewhat convincing
arguments that mental states cannot in principle be totally understood
in terms of physical processes.
"X believes Y" is problematic in that the phenomenological component
of a belief is not a property you can ascribe to a physical system. As a
bare minimum, you need to be able to translate mental-statements into
brain-statements, which means they are two logically different domains.
(Granted, this may be a linguistic problem: consciousness could be
a "global property" of a brain, but this seems to simply be a purge of
scare words.)
But the problem with your argument is that one must accept the existence
of a non-physical entity: the soul, for lack of a better word. And this
'soul' is by definition not detectable or measurable by physical
methods, and yet it (presumably) interacts with a physical object: the
body.
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
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| User: "Craig Franck" |
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| Title: Re: Entropy question |
24 Aug 2005 06:32:57 PM |
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"Andy Resnick" wrote
Craig Franck wrote:
"Andy Resnick" wrote
donstockbauer@hotmail.com wrote:
Then try to explain the emergence of thoughts from a substrate of 100
billion neurons by statistical mechanics alone. Good luck.
There's a difference between "not being able to understand it now" and
"not ever being able to understand it". I'm not a fan of putting
arbitrary physical phenomena beyond the reach of science.
I generally agree with this, however, when it comes to minds, I would
not consider this an arbitrary case. I've read somewhat convincing
arguments that mental states cannot in principle be totally understood
in terms of physical processes.
"X believes Y" is problematic in that the phenomenological component
of a belief is not a property you can ascribe to a physical system. As a
bare minimum, you need to be able to translate mental-statements into
brain-statements, which means they are two logically different domains.
(Granted, this may be a linguistic problem: consciousness could be
a "global property" of a brain, but this seems to simply be a purge of
scare words.)
But the problem with your argument is that one must accept the existence
of a non-physical entity: the soul, for lack of a better word. And this
'soul' is by definition not detectable or measurable by physical
methods, and yet it (presumably) interacts with a physical object: the
body.
My argument actually refers to conscious experience. Chalmers'
"The Conscious Mind" makes the argument that consciousness is not
logically supervenient on the physical, so the issue comes down to
awareness.
Even if you believe that conscious mental states correspond to brain
states (the most reasonable assumption, IMO), there is the issue that when
seeing something blue, for instance, there is nothing blue in the physical
brain, but the activity of blue-caring neurons. So we would have no way
of determining what an ET would see when viewing the sky: it's locked
out of the physical system to all who don't have access to the experience.
So you would have to map "sees blue" to "V1B cluster neurons firing."
But there is nothing to logically relate those neurons firing to the color
blue, other than that's what you experience. (Not everyone sees this as
a problem. But to say noting the physical aspects of experience exhausts
all there is to say about the situation is a kind of behaviorism. Dennett
in particular is quite guilty of this.)
--
Craig Franck
craig.franck@verizon.net
Cortland, NY
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| User: "Andy Resnick" |
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| Title: Re: Entropy question |
25 Aug 2005 12:56:31 PM |
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Craig Franck wrote:
<snip>
My argument actually refers to conscious experience. Chalmers'
"The Conscious Mind" makes the argument that consciousness is not
logically supervenient on the physical, so the issue comes down to
awareness.
Even if you believe that conscious mental states correspond to brain
states (the most reasonable assumption, IMO), there is the issue that when
seeing something blue, for instance, there is nothing blue in the physical
brain, but the activity of blue-caring neurons. So we would have no way
of determining what an ET would see when viewing the sky: it's locked
out of the physical system to all who don't have access to the experience.
So you would have to map "sees blue" to "V1B cluster neurons firing."
But there is nothing to logically relate those neurons firing to the color
blue, other than that's what you experience. (Not everyone sees this as
a problem. But to say noting the physical aspects of experience exhausts
all there is to say about the situation is a kind of behaviorism. Dennett
in particular is quite guilty of this.)
Well, at this point, I think we can't really speak quantitatively about
consciousness. But I can address a few of the points you raised.
From my persepctive, a valid question regards the existence of
"blue-caring neurons". Are mine the same as yours? How similar or
dissimilar are our neural networks? And, since I'm color-blind, what I
call 'blue' is surely different than what you call 'blue'. To be sure,
you don't need an ET: I put it to you that you have no way of completely
determining what the sky looks like to me, either. My particular brain
is different from yours.
So, I claim that what's interesting isn't so much if the "blue-caring
neuron" fires or not, but how the neuron is *regulated* to fire: the
inhibition and excitation, which (crudely) act like AND or OR gates. My
understanding is that pattern recognition works like this- neurons are
ANDed and ORed, and those that fire AND and OR with others to create
another level of processing... I think the human brain has 6 layers.
The retina works like this, too- adjacent neurons are subtracted to
detect motion, like a gradient filter.
So, what logically relates a certain neural cluster firing to a given
stimulus is the connections that occur and are modified during growth.
Now, one could then ask why certain regions in the brain correspond to
particular gross functions, and these areas are conserved from
individual to individual.... I don't have the answer to that.
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
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| User: "Zigoteau" |
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| Title: Re: Entropy question |
22 Aug 2005 04:27:02 PM |
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Hi, Don,
Absolutely. Right on.
Cheers,
Zigoteau.
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