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W.-M. Zheng
distribution of the system. According to the non-negativeness of relative entropy, we
have
i
Q i log(Q i /P i ) = log Z + β
i
E i Q i +
i
Q i log Q i = log Z + βE Q − S Q ≥ 0, (2.13)
where X Q is the average of random variable X under distribution Q, and S Q the
entropy associated with Q. In the case of identical P and Q, the equality implies
log Z = −βE P + S[P] = −βU + S.
Thus,
F(β, N , V ) ≡ F P ≡ −β
−1 log Z(β, N , V ) = U − TS,
(2.14)
which is just the Helmholtz free energy of thermodynamics. Accordingly the
Helmholtz free energy associated with any distribution Q may be defined as
F Q ≡ ≡E Q + T
i
Q i log Q i = U Q − TS Q .
(2.15)
Using (2.13), we have
F Q ≥ F(β, N , V ).
(2.16)
That is, the equilibrium state has the minimal free energy. Incidentally, the inequality
here is consistent with the principle of maximum entropy according to which the
canonical ensemble has maximal entropy. Note that the temperature T present in the
definition of free energy F Q is the same as that in the equilibrium distribution P,
or is the temperature attributed to the heatbath. That is to say, the free energy of an
arbitrary state in statistical mechanics is defined only for a system in a heatbath.
By further using (2.12), we have
dF = −SdT − pdV + μdN .
(2.17)
which is the change in free energy in an equilibrium or reversible process, and may be
denoted specially by (dF) rev . Consider an isothermal process at constant volume from
non-equilibrium to equilibrium. We then have (dF) irrev < (dF) rev = 0. This may
further be written as dF ≤ −SdT − pdV + μdN . We have thus derived the definition
of free energy for a non-equilibrium state or any general state in statistical mechanics.
However, it is impossible to deduce from equilibrium distributions the approach
to equilibrium, which is a problem concerning the direction of thermodynamical
processes.
W.-M. Zheng
distribution of the system. According to the non-negativeness of relative entropy, we
have
i
Q i log(Q i /P i ) = log Z + β
i
E i Q i +
i
Q i log Q i = log Z + βE Q − S Q ≥ 0, (2.13)
where X Q is the average of random variable X under distribution Q, and S Q the
entropy associated with Q. In the case of identical P and Q, the equality implies
log Z = −βE P + S[P] = −βU + S.
Thus,
F(β, N , V ) ≡ F P ≡ −β
−1 log Z(β, N , V ) = U − TS,
(2.14)
which is just the Helmholtz free energy of thermodynamics. Accordingly the
Helmholtz free energy associated with any distribution Q may be defined as
F Q ≡ ≡E Q + T
i
Q i log Q i = U Q − TS Q .
(2.15)
Using (2.13), we have
F Q ≥ F(β, N , V ).
(2.16)
That is, the equilibrium state has the minimal free energy. Incidentally, the inequality
here is consistent with the principle of maximum entropy according to which the
canonical ensemble has maximal entropy. Note that the temperature T present in the
definition of free energy F Q is the same as that in the equilibrium distribution P,
or is the temperature attributed to the heatbath. That is to say, the free energy of an
arbitrary state in statistical mechanics is defined only for a system in a heatbath.
By further using (2.12), we have
dF = −SdT − pdV + μdN .
(2.17)
which is the change in free energy in an equilibrium or reversible process, and may be
denoted specially by (dF) rev . Consider an isothermal process at constant volume from
non-equilibrium to equilibrium. We then have (dF) irrev < (dF) rev = 0. This may
further be written as dF ≤ −SdT − pdV + μdN . We have thus derived the definition
of free energy for a non-equilibrium state or any general state in statistical mechanics.
However, it is impossible to deduce from equilibrium distributions the approach
to equilibrium, which is a problem concerning the direction of thermodynamical
processes.