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Q. Wang
1. the entropy is maximum at the equilibrium, i.e., the entropy production rate is
nonnegative, which is ensured by the nonnegative definiteness of L;
2. dynamics near equilibrium is given by a linear response, i.e., kinetic equation
(3.4);
3. the mobility coefficient matrix satisfies the reciprocal relation L = L
T .
We note that mobility in the original Onsager principle is assumed independent of
x. In modern applications, this has been generalized to allow L to be functional of x.
We will discuss this when we generalize the Onsager principle in the next chapter.
3.2.2 Onsager Principle Accounting for Inertia
The Onsager principle alluded to earlier applies to purely dissipative systems, in
which inertia effect is not accounted for. We now turn to the system with nonnegligible inertia and discuss how to incorporate the inertia effect into the Onsager
principle. We consider a thermodynamical system in contact with its surrounding.
Let U be its internal energy, E k the kinetic energy corresponding to the inertia effect,
Q its heat, and W the work the system does to the surrounding, the first law of
thermodynamics states that
E k + U = Q − W.
(3.17)
We note that the kinetic energy is a part of the total free energy which is independent
of entropy. Let S be its entropy and T the absolute temperature defined from the
second and the third law of thermodynamics, respectively. According to the second
law, for a reversible process,
d S =
d Q
T
;
(3.18)
for an irreversible process on the other hand,
d S =
d Q
T
+
d Q
T
,
(3.19)
where d Q
is the energy (in the form of heat) lost internally during the irreversible
process. Recall that the Helmholtz free energy of the matter system is defined by
F = U − T S.
(3.20)
Hence,
d(E k + F) = d(E k + U ) − T d S − SdT = −dW − d Q
− SdT. (3.21)
Q. Wang
1. the entropy is maximum at the equilibrium, i.e., the entropy production rate is
nonnegative, which is ensured by the nonnegative definiteness of L;
2. dynamics near equilibrium is given by a linear response, i.e., kinetic equation
(3.4);
3. the mobility coefficient matrix satisfies the reciprocal relation L = L
T .
We note that mobility in the original Onsager principle is assumed independent of
x. In modern applications, this has been generalized to allow L to be functional of x.
We will discuss this when we generalize the Onsager principle in the next chapter.
3.2.2 Onsager Principle Accounting for Inertia
The Onsager principle alluded to earlier applies to purely dissipative systems, in
which inertia effect is not accounted for. We now turn to the system with nonnegligible inertia and discuss how to incorporate the inertia effect into the Onsager
principle. We consider a thermodynamical system in contact with its surrounding.
Let U be its internal energy, E k the kinetic energy corresponding to the inertia effect,
Q its heat, and W the work the system does to the surrounding, the first law of
thermodynamics states that
E k + U = Q − W.
(3.17)
We note that the kinetic energy is a part of the total free energy which is independent
of entropy. Let S be its entropy and T the absolute temperature defined from the
second and the third law of thermodynamics, respectively. According to the second
law, for a reversible process,
d S =
d Q
T
;
(3.18)
for an irreversible process on the other hand,
d S =
d Q
T
+
d Q
T
,
(3.19)
where d Q
is the energy (in the form of heat) lost internally during the irreversible
process. Recall that the Helmholtz free energy of the matter system is defined by
F = U − T S.
(3.20)
Hence,
d(E k + F) = d(E k + U ) − T d S − SdT = −dW − d Q
− SdT. (3.21)