120
Q. Wang
In the generalization of the Onsager principle, we assume the existence of a kinetic
energy functional (E k ), an entropy functional or a free energy functional (F). Given
a nonconservative external forces G, the generalized force X is given by
T X = −
δ F
δx
+ G −
d
dt
∂ E k
∂ ˙
x
,
(3.91)
where T is the absolute temperature. The generalized Onsager principle states as
follows.
1. There exists a mobility operator L such that ˙
x = L[x] · X, where L is a functional
of x;
2. mobility operator L = M sym + M anti , where M sym = M
T
sym ≥ 0 and M
T
anti =
−M anti , may be functions of x;
3. the nonnegative definiteness of M sym ≥ 0 ensure the free energy is dissipative
absent of any external forces in the isothermal cases
d
dt
(F + E k ) = −T X
T
· M sym · X + G · ˙
x,
(3.92)
or the entropy production is increasing
d
dt
S total = X
T
· M sym · X −
1
T
G · ˙
x.
(3.93)
The antisymmetric part M anti corresponds to the reversible process that is nonenergetic and not built in the Onsager-Machlup maximum action potential nor the
Rayleighian so that the variational version of the Onsager principle can not be generalized. Thus, the generalized Onsager principle does not have an equivalent variational counterpart. By allowing antisymmetric mobility operators, we are able to
handle energy conservative systems, like Hamiltonian systems. In the generalized
Onsager principle, we basically say that the constitutive relation includes not only
the dissipative force but also the nondissipative force!
Remark 3.3.1 We have presented two distinct formulations of the Onsager principle for dissipative systems. One is in the form of the kinetic equation, maximum
entropy principle and the Onsager reciprocal relation; and the other is in the form
of the maximum Onsager-Matchup action potential or equivalently the minimum
Rayleighian. In the presentation, we have clearly identified the thermodynamic variables or fluctuations away from the equilibrium as x and assumed the entropy is given
by a functional of these variables. Moreover, we assume the dissipation functional
is given by a quadratic functional of flux variable ˙
x.
However, many dissipation functionals given for hydrodynamic theories in fluid
systems are not given as functionals of ˙
x, but rather functionals of their spatial gradients in the Eulerian coordinate, like in the viscous fluid flows. For these systems, the
corresponding generalized force is the stress tensor instead of the force. Normally,
Q. Wang
In the generalization of the Onsager principle, we assume the existence of a kinetic
energy functional (E k ), an entropy functional or a free energy functional (F). Given
a nonconservative external forces G, the generalized force X is given by
T X = −
δ F
δx
+ G −
d
dt
∂ E k
∂ ˙
x
,
(3.91)
where T is the absolute temperature. The generalized Onsager principle states as
follows.
1. There exists a mobility operator L such that ˙
x = L[x] · X, where L is a functional
of x;
2. mobility operator L = M sym + M anti , where M sym = M
T
sym ≥ 0 and M
T
anti =
−M anti , may be functions of x;
3. the nonnegative definiteness of M sym ≥ 0 ensure the free energy is dissipative
absent of any external forces in the isothermal cases
d
dt
(F + E k ) = −T X
T
· M sym · X + G · ˙
x,
(3.92)
or the entropy production is increasing
d
dt
S total = X
T
· M sym · X −
1
T
G · ˙
x.
(3.93)
The antisymmetric part M anti corresponds to the reversible process that is nonenergetic and not built in the Onsager-Machlup maximum action potential nor the
Rayleighian so that the variational version of the Onsager principle can not be generalized. Thus, the generalized Onsager principle does not have an equivalent variational counterpart. By allowing antisymmetric mobility operators, we are able to
handle energy conservative systems, like Hamiltonian systems. In the generalized
Onsager principle, we basically say that the constitutive relation includes not only
the dissipative force but also the nondissipative force!
Remark 3.3.1 We have presented two distinct formulations of the Onsager principle for dissipative systems. One is in the form of the kinetic equation, maximum
entropy principle and the Onsager reciprocal relation; and the other is in the form
of the maximum Onsager-Matchup action potential or equivalently the minimum
Rayleighian. In the presentation, we have clearly identified the thermodynamic variables or fluctuations away from the equilibrium as x and assumed the entropy is given
by a functional of these variables. Moreover, we assume the dissipation functional
is given by a quadratic functional of flux variable ˙
x.
However, many dissipation functionals given for hydrodynamic theories in fluid
systems are not given as functionals of ˙
x, but rather functionals of their spatial gradients in the Eulerian coordinate, like in the viscous fluid flows. For these systems, the
corresponding generalized force is the stress tensor instead of the force. Normally,