114
Q. Wang
3.2.6 Extension of Onsager Principle to Spatially
Inhomogeneous Systems
In the presentation of the Onsager principle so far, we have assumed the entropy and
the entropy production rate (analogously free energy and the energy dissipation rate)
are functions of the thermodynamical variables. This applies to spatially homogeneous systems only. Realistically, most matter systems are spatially inhomogeneous.
We can extend the Onsager principle to spatially inhomogeneous systems where
potentials and mobility are functionals of the thermodynamical variables. Specifically, we assume the entropy, free energy and internal energy are functionals of the
thermodynamical variables and the entropy production rate (or the energy dissipation rate) are functionals of the thermodynamical variables and quadratic functions
of their corresponding invariant time derivatives. Under these assumptions, the thermodynamical principles mentioned above are still valid.
We generalize the dissipation functional to cases where the coefficients of the
quadratic form in ˙
x depend on the thermodynamical variables and their spatial derivatives and/or integrals. Then, the kinetic equation derived from the Rayleighian is given
by
T X =
∂∂ F
∂ ˙
x
= −
δ F
δx
+ G −
d
dt
∂ E k
∂ ˙
x
,
(3.59)
where the free energy functional, kinetic energy, and the dissipation functionals are
given by
F[x] =
f (x, ∇x, · · · , )dr, E k =
ρ
2
˙ x
2 dr, , S =
1
2
˙
x · R · ˙
xdr,
(3.60)
where f is the free energy density, ρ is the density of the matter system, r is the
spatial variable, R is the friction coefficient which can be functions of x and its spatial
derivatives. This yields all the transport equations for the system. Notice that this is
a force balance equation that includes all the forces in the system: dissipative force
−
δδ F
δ ˙
x
, elastic force −
δ F
δx
, external force G, and the inertia force −
d
dt
δ E k
δ ˙
x
.
In practice, we either apply the Onsager principle directly to arrive at the constitutive relation by constructing the mobility matrix L to define the kinetic equation:
X = L · ˙
x,
(3.61)
or maximize the Onsager-Machlup action potential (or minimize the time integral of
the Rayleighian) to arrive at the same relation when the dissipation functional S is
available. In the latter case, after we obtain the energy dissipation functional S and
the Helmholtz free energy F, the Rayleighian is defined by
R =
˙
E k + F − G · ˙
x + F , φ F = −T S .
(3.62)
Q. Wang
3.2.6 Extension of Onsager Principle to Spatially
Inhomogeneous Systems
In the presentation of the Onsager principle so far, we have assumed the entropy and
the entropy production rate (analogously free energy and the energy dissipation rate)
are functions of the thermodynamical variables. This applies to spatially homogeneous systems only. Realistically, most matter systems are spatially inhomogeneous.
We can extend the Onsager principle to spatially inhomogeneous systems where
potentials and mobility are functionals of the thermodynamical variables. Specifically, we assume the entropy, free energy and internal energy are functionals of the
thermodynamical variables and the entropy production rate (or the energy dissipation rate) are functionals of the thermodynamical variables and quadratic functions
of their corresponding invariant time derivatives. Under these assumptions, the thermodynamical principles mentioned above are still valid.
We generalize the dissipation functional to cases where the coefficients of the
quadratic form in ˙
x depend on the thermodynamical variables and their spatial derivatives and/or integrals. Then, the kinetic equation derived from the Rayleighian is given
by
T X =
∂∂ F
∂ ˙
x
= −
δ F
δx
+ G −
d
dt
∂ E k
∂ ˙
x
,
(3.59)
where the free energy functional, kinetic energy, and the dissipation functionals are
given by
F[x] =
f (x, ∇x, · · · , )dr, E k =
ρ
2
˙ x
2 dr, , S =
1
2
˙
x · R · ˙
xdr,
(3.60)
where f is the free energy density, ρ is the density of the matter system, r is the
spatial variable, R is the friction coefficient which can be functions of x and its spatial
derivatives. This yields all the transport equations for the system. Notice that this is
a force balance equation that includes all the forces in the system: dissipative force
−
δδ F
δ ˙
x
, elastic force −
δ F
δx
, external force G, and the inertia force −
d
dt
δ E k
δ ˙
x
.
In practice, we either apply the Onsager principle directly to arrive at the constitutive relation by constructing the mobility matrix L to define the kinetic equation:
X = L · ˙
x,
(3.61)
or maximize the Onsager-Machlup action potential (or minimize the time integral of
the Rayleighian) to arrive at the same relation when the dissipation functional S is
available. In the latter case, after we obtain the energy dissipation functional S and
the Helmholtz free energy F, the Rayleighian is defined by
R =
˙
E k + F − G · ˙
x + F , φ F = −T S .
(3.62)