Re: Physical example for Maxwell-Boltzmann statistics

Thomas Smid says...

If the energy of particles can take on continuous values, then I can'
t see how there could be two particles with exactly the same energy.
The probability for that would be zero.

Generally speaking, the notions of 'identical' or 'non-identical'
should imply that one has certain discrete values or classes to which
the objects can be assigned unambiguously.

So in this sense I don't actually quite understand Gibbs' notion of
"entirely alike molecules" for a classical gas. There shouldn't be any
two molecules entirely alike.

In statistical mechanics, an important quantity (from which everything
in equilibrium thermodynamics can be derived) is the entropy associated
with a macroscopic state. The entropy associated with a macroscopic
state is the log of the number of microscopic states. If you specify the
macroscopic quantities:
(1) Total energy, (2) Volume, (3) Number of particles (of each type),
(4) Total momentum and angular momentum, etc., then how many microstates
are consistent with those macroscopic attributes? (In classical
statistical mechanics, it is necessary to discretize phase space
in order to get a finite entropy.) To figure out the answer, you
have to know when two microstates are "different". In particular,
if you have N particles, is there a difference between the following
microstates?

1. Particle 1 has energy E1, while particle 2 has energy E2.
2. Particle 1 has energy E2, while particle 2 has energy E1.

If the particles are identical, then there is no difference
between states 1 and 2. If they are not identical, then they
are two different microstates.

So to say that the particles are distinguished by their
energy (or momentum, or position, or anything else that
is not an intrinsic property of the particle) doesn't
answer the question. The question is: does *swapping*
energy between two particles leave the microstate
unchanged, or does it change it?

At least in the way I learned statistical mechanics,
the assumption of identical particles made a difference
in the computed entropy.

========== Moderator's note ==============================

Indeed, that's precisely right. Boltzmann had to introduce a
famous (or infamous?) factor 1/N! to make the entropy an extensive
quantity. Of course this is related to the indistinguishability of
the microstate under exchange of identical particles, but in classical
mechanics that's just an ad-hoc assumption about the correct statistics,
because in principle, particles are always distinguishable once and for
all by their initial position in phase space. Nowadays this ad-hoc assumption
is of course well justified by the quantum-theoretical notion of
indistinguishability of particles leading to the Bose-Einstein and Ferm-Dirac
statistics in spaces with \geq 3 dimensions.

--
Daryl McCullough
Ithaca, NY

.

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