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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2.7. ENERGY FLUCTUATIONS. 35<br />
The form of the rel<strong>at</strong>ion between the partition function and the Helmholtz free<br />
energy is easily memorized by writing the equ<strong>at</strong>ion in the following form,<br />
which has Boltzmann like factors on each side.<br />
The entropy follows from<br />
F (T,V,N) ɛs<br />
− k e B T − k = e B T (2.50)<br />
S = 1<br />
<br />
ɛs<br />
<br />
<br />
ɛs<br />
(U−F ) = kB P (s) + log(Z) = kB<br />
T kBT kBT<br />
s<br />
s<br />
which gives<br />
s<br />
<br />
+ log(Z) P (s)<br />
(2.51)<br />
<br />
S = −kB P (s) log(P (s)) (2.52)<br />
s<br />
as before.This is just an easier way to derive the entropy as a function of volume<br />
and temper<strong>at</strong>ure in terms of the elementary probabilities.<br />
2.7 Energy fluctu<strong>at</strong>ions.<br />
A system in thermal contact with a large reservoir will have the temper<strong>at</strong>ure of<br />
this reservoir when equilibrium is reached. In thermal contact, energy is allowed<br />
to flow back and forth between the system and the reservoir and the energy of<br />
the system will not be equal to U <strong>at</strong> all times, only as an average. Thermal<br />
equilibrium implies only th<strong>at</strong> there is no net flow of energy between the system<br />
and the reservoir.<br />
Are the fluctu<strong>at</strong>ions real?<br />
So far we have made the connection between thermodynamics and st<strong>at</strong>istical<br />
mechanics in the following sense. Thermodynamics is a macroscopic theory<br />
which is experimentally correct. <strong>St<strong>at</strong>istical</strong> mechanics is a microscopic theory,<br />
and the averages calcul<strong>at</strong>ed in this theory seem to be the same as the corresponding<br />
quantities in thermodynamics. We have not addressed the concept<br />
of fluctu<strong>at</strong>ions, however. Fluctu<strong>at</strong>ions are much more difficult to describe in<br />
thermodynamics, although we have given some prescription how to do this.<br />
Fluctu<strong>at</strong>ions are easy to describe in st<strong>at</strong>istical mechanics. But are the fluctu<strong>at</strong>ions<br />
described in st<strong>at</strong>istical mechanics really wh<strong>at</strong> we observe experimentally?<br />
How does the thermodynamic limit play a role here? These are very difficult<br />
questions, but here we simply assume th<strong>at</strong> the answer is th<strong>at</strong> we have the correct<br />
description of fluctu<strong>at</strong>ions in thermodynamics, since the theory looks like<br />
the thermodynamic form. The fact th<strong>at</strong> in the thermodynamic limit all distributions<br />
are Gaussian will have to play an important role.
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