3 Generalized Onsager Principle and It Applications
107
For an isothermal system, dT = 0. Then,
d(E k + F) = d(E k + U ) − T d S = −dW − d Q
.
(3.22)
The term on the right hand side is the energy loss to the surrounding and internally
while the term on the left is the kinetic energy plus the Helmholtz free energy,
termed the generalized free energy. The right hand side is termed energy loss or
energy dissipation. It then follows that
d
dt
(E k + F) = −
d(W + Q
)
dt
,
(3.23)
called the energy dissipation rate. This is consistent with the second law of thermodynamics.
By definition,
d
dt
(E k + F) =
d
dt
(E k + U ) − T
d S
dt
=
d
dt
(E k + U ) − T X
T
· L · X. (3.24)
For a closed isothermal system, the total energy never changes, i.e.,
d
dt
(E k + U ) = 0.
Then,
d
dt
(E k + F) = −T
d S
dt
= −T X
T
· L · X.
(3.25)
The energy dissipation rate is the negative of the entropy production rate. Let X is
the total generalized force. We adopt the kinetic equation from the liner response
theory
˙
x = L · X.
(3.26)
From (3.61), we deduce
d
dt
(E k + F) = [
∂
∂x
F +
d
dt
∂ E k
∂ ˙
x
] · ˙
x = −T ˙
x
T
· X,
(3.27)
where E k = ρ˙ x
2
/2 is used. This implies
X = −
1
T
[
∂
∂x
F +
d
dt
∂ E k
∂ ˙
x
]
(3.28)
So, in the thermodynamical system with inertia, the Onsager principle can be modified to accommodate the inertia. This version of the kinetic equation incorporates the
force balance among the inertia, dissipative and the potential force, which generalizes
the classical Onsager principle alluded to earlier.