Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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7.5. IDEAL GAS IN CLASSICAL STATISTICAL MECHANICS. 149<br />
This example is studied in detail in a course on non-linear dynamics. An<br />
important question, studied actively in th<strong>at</strong> field, is the following. Wh<strong>at</strong> are the<br />
conditions for the Hamiltonian so th<strong>at</strong> most of phase space for most energies<br />
is covered by a chaotic orbital? In our words, can we construct a non-trivial<br />
many body Hamiltonian for which st<strong>at</strong>istical mechanics does not work? Note<br />
th<strong>at</strong> every Hamiltonian will have regular orbits, but the ergodic theorem is valid<br />
only if these orbits have a measure zero.<br />
7.5 Ideal gas in classical st<strong>at</strong>istical mechanics.<br />
The classical Hamiltonian for a system of N independent particles of mass m is<br />
given by<br />
Collisions are needed.<br />
H =<br />
N<br />
i=1<br />
p 2 i<br />
2m<br />
(7.32)<br />
Such a system is clearly not ergodic. The solutions for the trajectories are<br />
xi(t) = xi(0) + vit and sample only a small fraction of phase space. We have to<br />
perform a sp<strong>at</strong>ial average in order to get some meaningful answers. In reality,<br />
of course, there are collisions. These collisions are elastic and are assumed to be<br />
instantaneous and they randomly redistribute the energy between all particles.<br />
In this case, we can perform a time average to get a meaningful answer. The<br />
l<strong>at</strong>ter procedure is more s<strong>at</strong>isfying from an experimental point of view, and<br />
allows us to compare experimental results with ensemble averages derived from<br />
the Hamiltonian above.<br />
If we assume th<strong>at</strong> the <strong>at</strong>oms are hard spheres, instantaneous collisions are<br />
the only possibility. But the interactions between <strong>at</strong>oms are more than th<strong>at</strong>,<br />
there are long range tails in the potential. This means th<strong>at</strong> when the density of<br />
the gas is too high, the interaction potential plays a role. Hence the ideal gas<br />
is a first approxim<strong>at</strong>ion of a description of a gas <strong>at</strong> low density, where we can<br />
ignore the interaction terms.<br />
Density of st<strong>at</strong>es.<br />
We will calcul<strong>at</strong>e the entropy of the ideal gas in the micro-canonical ensemble.<br />
We replace the delta functions by Gaussian exponentials, and use the limit<br />
ɛ → 0. We have to use this ɛ-procedure in order to avoid problems with the<br />
product of a delta-function and its logarithm. The entropy follows from<br />
1<br />
S(U, V, N) = −kB<br />
N!h3N <br />
d XC 1<br />
ɛ √ (U−H)2<br />
e− ɛ<br />
π 2 <br />
log C 1<br />
ɛ √ (U−H)2<br />
e− ɛ<br />
π 2<br />
<br />
(7.33)