102
Q. Wang
serve as corner stones for the development of many nonequilibrium theories for such
systems [12–15]. In this chapter, we review the Onsager linear response theory for
nonequilibrium systems, known as the Onsager principle [5], and make a contact
with the second law of thermodynamics. The Onsager principle implies the second
law of thermodynamics in the form of the Clausius-Duhem inequality. It is in fact
more general in that it defines the dissipation mechanism that characterizes the constitutive relation for any matter systems. In the maximum Onsager-Machlup action
potential principle, it provides a way to compute the dissipative force and match
it up with the other non-dissipative forces. We then discuss how to generalize the
Onsager principle to not only allow quasi-linear dependence of the mobility operator
on thermodynamical variables in a functional way, but also to allow certain reversible
processes directly related to the elastic force to be included in the model [21]. This
allows one to apply the generalized Onsager Principle (GOP) to many complex systems beyond what the original Onager principle was intended for. The organization
of the chapter is as follows. In Sect. 3.2, we review the Onsager principle for dissipative systems in two equivalent yet distinct forms. In Sect. 3.3, we discuss the
generalized Onsager principle. In Sect. 3.4, we derive a host of thermodynamical
models including generalized hydrodynamical theories to show their variational and
dissipative structures.
3.2 Onsager Principle for Dissipative Systems
We discuss the Onsager linear response theory for a purely dissipative system in
details in this section. We proceed with it in a closed system first and then discuss it
in an open system. We give two distinct formulations of the Onsager linear response
theory together with the Onsager reciprocal relation in the name of the Onsager
principle. The first formulation is constructive and the second is variational. Various
applications of the Onsager principle in derivation of nonequilibrium thermodynamical models will be discussed as examples at the end of the section.
3.2.1 Constructive Onsager Principle
We consider a closed thermodynamical system not far from equilibrium x = 0,
whose state is described by a set of coarse-grained, thermodynamical variables
x(t) = (x 1 , · · · , x n )
T or fluctuations measured relative to equilibrium value x = 0.
Onsager states that entropy of the system S, which is assumed a function of x, reaches
its maximum value S e at the equilibrium [13–15]. We expand the entropy function
in its Taylor series at the equilibrium to arrive at the following approximation up to
the quadratic term
S = S e + S(x) + o(x
2
), ,S = −
1
2
x
T
· H · x,
(3.1)
Q. Wang
serve as corner stones for the development of many nonequilibrium theories for such
systems [12–15]. In this chapter, we review the Onsager linear response theory for
nonequilibrium systems, known as the Onsager principle [5], and make a contact
with the second law of thermodynamics. The Onsager principle implies the second
law of thermodynamics in the form of the Clausius-Duhem inequality. It is in fact
more general in that it defines the dissipation mechanism that characterizes the constitutive relation for any matter systems. In the maximum Onsager-Machlup action
potential principle, it provides a way to compute the dissipative force and match
it up with the other non-dissipative forces. We then discuss how to generalize the
Onsager principle to not only allow quasi-linear dependence of the mobility operator
on thermodynamical variables in a functional way, but also to allow certain reversible
processes directly related to the elastic force to be included in the model [21]. This
allows one to apply the generalized Onsager Principle (GOP) to many complex systems beyond what the original Onager principle was intended for. The organization
of the chapter is as follows. In Sect. 3.2, we review the Onsager principle for dissipative systems in two equivalent yet distinct forms. In Sect. 3.3, we discuss the
generalized Onsager principle. In Sect. 3.4, we derive a host of thermodynamical
models including generalized hydrodynamical theories to show their variational and
dissipative structures.
3.2 Onsager Principle for Dissipative Systems
We discuss the Onsager linear response theory for a purely dissipative system in
details in this section. We proceed with it in a closed system first and then discuss it
in an open system. We give two distinct formulations of the Onsager linear response
theory together with the Onsager reciprocal relation in the name of the Onsager
principle. The first formulation is constructive and the second is variational. Various
applications of the Onsager principle in derivation of nonequilibrium thermodynamical models will be discussed as examples at the end of the section.
3.2.1 Constructive Onsager Principle
We consider a closed thermodynamical system not far from equilibrium x = 0,
whose state is described by a set of coarse-grained, thermodynamical variables
x(t) = (x 1 , · · · , x n )
T or fluctuations measured relative to equilibrium value x = 0.
Onsager states that entropy of the system S, which is assumed a function of x, reaches
its maximum value S e at the equilibrium [13–15]. We expand the entropy function
in its Taylor series at the equilibrium to arrive at the following approximation up to
the quadratic term
S = S e + S(x) + o(x
2
), ,S = −
1
2
x
T
· H · x,
(3.1)