Statistical Mechanics - Physics at Oregon State University

1.7. LAWS OF THERMODYNAMICS. 19
1.7 Laws of thermodynamics.
Thermodynamics is based upon four laws or postulates. In statistical mechanics
these laws follow immediately from our previous discussion.
Zeroth Law.
If two systems are in thermal equilibrium with a third system, they are in
thermal equilibrium with each other. Trivial from TA = TB ∪ TB = TC ⇒ TA =
TC.
First Law.
Heat is a form of energy. Trivial, since we defined thermal contact by an exchange
of energy and used this exchange of energy to define entropy.
Second Law.
Entropy always increases. Also trivial, since if the combined system starts with
energies U 0 A and U 0 B we automatically have gA(U 0 A )gB(U 0 B ) gA( ÛA)gB( ÛB),
+ Sinit Sfinal .
or after taking the logarithms S init
A
B
Third Law.
A
+ S final
B
The entropy per particle is zero at zero temperature in the thermodynamic limit.
This is more a statement about quantum mechanics. In general, the degeneracy
of the ground state is small. Suppose the degeneracy of the ground state is
N p , which is unusually large. Then the entropy per particle in the ground state
(and hence at T = 0 ) is S(0) = kBp log(N)
N , which is zero when N → ∞. Only
when the degeneracy of the ground state is on the order of eN or N! does one
see deviations from this law. Such systems are possible, however, when we have
very large molecules!
We also see that the local stability requirements derived in thermodynamics
are fulfilled. Since the entropy is related to the maximal term in a product, we
recover expressions like
0.
∂ 2 S
∂U 2
N,V
In summary, the first law is an assumption made in both thermodynamics
and statistical mechanics. The only difference is that the introduction of heat as
a form of energy is more acceptable when based on a microscopic picture. The
other three laws all follow from the definition of entropy in statistical mechanics.
Entropy is a more tangible quantity in a microscopic theory. It is still useful,
however, to be able to define entropy from a macroscopic point of view only. This
approach leads to generalizations in systems theory. For any complicated system
for which the state only depends on a small number of macroscopic variables
one can define a quantity like entropy to represent all other variables which are
not of interest. Such a system might not have a clear microscopic foundation
and therefore a generalization of the microscopic definition of entropy in terms
of states might be impossible.