104
Q. Wang
of the motion in the dynamics is linearly proportional to the force acting on it. In the
framework of the classical mechanics (Newtonian mechanics), this is equivalent to
setting inertia to zero while retaining the friction force, which is assumed linear in
˙
x, and all the other external forces in the momentum balance equation. This system
subject to friction is also known as the over-damped system in mechanics. Hence,
the kinetic equation defines relaxation dynamics for how the nonequilibrium system
relaxes back to the equilibrium in the linear response theory. Different mobility matrices define different dynamics although the entropy function of the system remains
the same. It is therefore the crucial component in any nonequilibrium dynamical
model.
Now that mobility defines actual dynamics of the system while relaxing back to
equilibrium. We will relate the entropy production rate to the mobility coefficient
explicitly in the irreversible process so that which dynamics the system adopts is
fully determined by either specifying the mobility or the energy dissipation rate.
Remark 3.2.1 The linear response adopted in this formulation is an assumption
which dictates the development of the entire nonequilibrium theory. There is no
reason why this cannot be modified to arrive at a truly nonlinear response and thereby
yielding nonlinear, nonequilibrium response theories. However, this has not been
explored in depth so far. For instance, the friction force might well be a nonlinear
response function given by
X = h(˙ x),
(3.6)
where h(˙ x) is a nonlinear function of ˙
x. Assuming h is invertible (for example, h is
monotonic), we have
˙
x = h
−1
(X),
(3.7)
where h
−1 is the inverse of h. In order for the system to be dissipative, some constraint
must be imposed on the function H. To a large extent, how to proceed with this remains
an open problem.
On the other hand, the mobility can be a function of the thermodynamical variable
x. So, the kinetic equation for the linear response can in fact be a nonlinear equation of
x. However, we hope one should keep in mind that the quasi-linear relation between
the flux and the force in the linear response can deduce a nonlinear equation for the
thermodynamical variables, but a nonlinear relation between ˙
x and x may not always
be a consequence of the linear response theory.
Under the condition that S is quadratic in x, i.e., the sign of S remains invariant
under a time-reversal operation, Onsager derived the well-known reciprocal relation
L i j = L ji ,
(3.8)
and, consequently, R i j = R ji , from the microscopic reversibility, which states that
for any t > 0 and τ ,
Q. Wang
of the motion in the dynamics is linearly proportional to the force acting on it. In the
framework of the classical mechanics (Newtonian mechanics), this is equivalent to
setting inertia to zero while retaining the friction force, which is assumed linear in
˙
x, and all the other external forces in the momentum balance equation. This system
subject to friction is also known as the over-damped system in mechanics. Hence,
the kinetic equation defines relaxation dynamics for how the nonequilibrium system
relaxes back to the equilibrium in the linear response theory. Different mobility matrices define different dynamics although the entropy function of the system remains
the same. It is therefore the crucial component in any nonequilibrium dynamical
model.
Now that mobility defines actual dynamics of the system while relaxing back to
equilibrium. We will relate the entropy production rate to the mobility coefficient
explicitly in the irreversible process so that which dynamics the system adopts is
fully determined by either specifying the mobility or the energy dissipation rate.
Remark 3.2.1 The linear response adopted in this formulation is an assumption
which dictates the development of the entire nonequilibrium theory. There is no
reason why this cannot be modified to arrive at a truly nonlinear response and thereby
yielding nonlinear, nonequilibrium response theories. However, this has not been
explored in depth so far. For instance, the friction force might well be a nonlinear
response function given by
X = h(˙ x),
(3.6)
where h(˙ x) is a nonlinear function of ˙
x. Assuming h is invertible (for example, h is
monotonic), we have
˙
x = h
−1
(X),
(3.7)
where h
−1 is the inverse of h. In order for the system to be dissipative, some constraint
must be imposed on the function H. To a large extent, how to proceed with this remains
an open problem.
On the other hand, the mobility can be a function of the thermodynamical variable
x. So, the kinetic equation for the linear response can in fact be a nonlinear equation of
x. However, we hope one should keep in mind that the quasi-linear relation between
the flux and the force in the linear response can deduce a nonlinear equation for the
thermodynamical variables, but a nonlinear relation between ˙
x and x may not always
be a consequence of the linear response theory.
Under the condition that S is quadratic in x, i.e., the sign of S remains invariant
under a time-reversal operation, Onsager derived the well-known reciprocal relation
L i j = L ji ,
(3.8)
and, consequently, R i j = R ji , from the microscopic reversibility, which states that
for any t > 0 and τ ,