Re: Entropy question
- From: "Zigoteau" <zigoteau@xxxxxxxxx>
- Date: 20 Aug 2005 04:00:56 -0700
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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