2 On the Foundational Principles of Statistical Mechanics
93
The change in Hamiltonian with volume resulting from the confinement of the container wall is dE/dV = −p, where p is independent of microscopic conformations,
hence dE = −pdV . Thus, the first term inside the square bracket is the term describing work. From the expression of probability density we have
βE(r
N
, p
N
) = − log Z(β, N , V ) − log P(r
N
, p
N
).
(2.9)
Thus, the second term inside the square bracket is
− β
−1
d r
N d p
N log P(r
N , p
N )dP(r
N , p
N ) = −β
−1
d r
N d p
N d [P(r
N , p
N ) log P(r
N , p
N )] = TdS.
(2.10)
This is just the heat absorbtion. (Here the fact that log Z is a constant and probability
P is normalized has been taken into account. As a result of this the term concerning
log Z makes no contribution.) We then arrive at the first law of thermodynamics:
dU = −pdV + TdS.
(2.11)
It is straightforward to extend this to the general form of work. If the number of
particles is allowed to change, by introducing μ = (∂U/∂N ) β,V we have
dU = −pdV + TdS + μdN ,
(2.12)
where μ is the chemical potential measuring the increment in internal energy by
adding a particle. In summary, the change in internal energy due to the change
in Hamiltonian is work, while the change in internal energy due to the change in
distribution is heat.
The second law of thermodynamics is a fundamental postulate applicable in any
phenomena involving heat, and is used to explain irreversible processes in nature.
There are several equivalent formulations of the law. Clausius states that heat cannot
spontaneously flow from cold regions to hot regions. In the language of statistical
mechanics, the law may be retold as a minimum principle: the free energy of a
system in a heat bath never increases in any spontaneous process. Here the reduction
indicates the direction of process, and implies approaching equilibrium, which is
beyond the extent reachable by the equilibrium statistical mechanics. The minimum
principle of free energy claims only that amongst all states the equilibrium state has
minimal free energy, with no reference to the direction of any process.
To simplify notation, we use summation instead of integration. The relative
entropy of any two distributions {Q i } and {P i } is defined as
D(Q, P) =
i
Q i log(Q i /P i ).
By means of log x ≤ 1 − x, and setting x = P i /Q i , it is easy to prove that D(Q, P) ≥
0, and the equality is valid only when {P i } and {Q i } are identical. Now denote the
canonical distribution by P i = Z
−1 exp(−βE i ), and assume {Q i } to be any other
93
The change in Hamiltonian with volume resulting from the confinement of the container wall is dE/dV = −p, where p is independent of microscopic conformations,
hence dE = −pdV . Thus, the first term inside the square bracket is the term describing work. From the expression of probability density we have
βE(r
N
, p
N
) = − log Z(β, N , V ) − log P(r
N
, p
N
).
(2.9)
Thus, the second term inside the square bracket is
− β
−1
d r
N d p
N log P(r
N , p
N )dP(r
N , p
N ) = −β
−1
d r
N d p
N d [P(r
N , p
N ) log P(r
N , p
N )] = TdS.
(2.10)
This is just the heat absorbtion. (Here the fact that log Z is a constant and probability
P is normalized has been taken into account. As a result of this the term concerning
log Z makes no contribution.) We then arrive at the first law of thermodynamics:
dU = −pdV + TdS.
(2.11)
It is straightforward to extend this to the general form of work. If the number of
particles is allowed to change, by introducing μ = (∂U/∂N ) β,V we have
dU = −pdV + TdS + μdN ,
(2.12)
where μ is the chemical potential measuring the increment in internal energy by
adding a particle. In summary, the change in internal energy due to the change
in Hamiltonian is work, while the change in internal energy due to the change in
distribution is heat.
The second law of thermodynamics is a fundamental postulate applicable in any
phenomena involving heat, and is used to explain irreversible processes in nature.
There are several equivalent formulations of the law. Clausius states that heat cannot
spontaneously flow from cold regions to hot regions. In the language of statistical
mechanics, the law may be retold as a minimum principle: the free energy of a
system in a heat bath never increases in any spontaneous process. Here the reduction
indicates the direction of process, and implies approaching equilibrium, which is
beyond the extent reachable by the equilibrium statistical mechanics. The minimum
principle of free energy claims only that amongst all states the equilibrium state has
minimal free energy, with no reference to the direction of any process.
To simplify notation, we use summation instead of integration. The relative
entropy of any two distributions {Q i } and {P i } is defined as
D(Q, P) =
i
Q i log(Q i /P i ).
By means of log x ≤ 1 − x, and setting x = P i /Q i , it is easy to prove that D(Q, P) ≥
0, and the equality is valid only when {P i } and {Q i } are identical. Now denote the
canonical distribution by P i = Z
−1 exp(−βE i ), and assume {Q i } to be any other