82
S. Goto and H. Hino
4.7.1 Equilibrium States
By recalling the claims in Sect. 4.5.1, one realizes that a peculiarity of the choice
of transition matrix for master equations has never been used in the discussion of
equilibrium states. Therefore claims on geometric description of equilibrium states
for master equations can be extended to general master equations. For example, from
Proposition 10, one immediately has the following.
Proposition 16 Given master equations, assume that there exists an equilibrium
state. The equilibrium state of the master equations is then expressed as the Legendre
submanifold of the contact manifold.
4.7.2 Nonequilibrium States
For the toy model studied in this paper there is a simple relaxation process, that is,
p
eq
θ ( j) is realized in the asymptotic limit, t → ∞. This model with the carefully
chosen transition matrix enables the process to be described in the contact geometric language. Since some systems with other complicated transition matrices show
relaxation process, one possible extension of the present toy model could be obtained
by choosing such a transition matrix carefully. In addition, as shown in Appendix A,
the use of the probability generating function simplifies a class of master equations.
However it is unclear which class of master equations can be described by contact
Hamiltonian systems, and such a direction should be pursued as a future work.
Another direction to study master equations in the contact geometric language is to
focus on the Hamilton-Jacobi equation obtained from master equations under some
approximation [41], since contact geometric descriptions of the Hamilton-Jacobi
equation have been well-recognized [42].
4.8 Conclusions
This paper has offered a viewpoint that the solvable master equations and expectation
variables of the moment dynamical system derived from the master equations can
be described on para-contact metric manifolds. To give a geometric description of
these, contact Hamiltonian vector fields have been introduced on para-contact metric
manifolds. Then relaxation processes have been characterized by the use of metric
and the Ricci tensor fields. Moreover possible information geometric structures for
moment dynamical systems have been clarified.
The significance of these descriptions given in this paper is mentioned here.
Although a number of pure mathematical studies of para-contact metric manifolds
exist in the literature, there were a few applications to geometric thermodynamics.
Together with this, Riemannian metric tensor fields are often considered in contact
S. Goto and H. Hino
4.7.1 Equilibrium States
By recalling the claims in Sect. 4.5.1, one realizes that a peculiarity of the choice
of transition matrix for master equations has never been used in the discussion of
equilibrium states. Therefore claims on geometric description of equilibrium states
for master equations can be extended to general master equations. For example, from
Proposition 10, one immediately has the following.
Proposition 16 Given master equations, assume that there exists an equilibrium
state. The equilibrium state of the master equations is then expressed as the Legendre
submanifold of the contact manifold.
4.7.2 Nonequilibrium States
For the toy model studied in this paper there is a simple relaxation process, that is,
p
eq
θ ( j) is realized in the asymptotic limit, t → ∞. This model with the carefully
chosen transition matrix enables the process to be described in the contact geometric language. Since some systems with other complicated transition matrices show
relaxation process, one possible extension of the present toy model could be obtained
by choosing such a transition matrix carefully. In addition, as shown in Appendix A,
the use of the probability generating function simplifies a class of master equations.
However it is unclear which class of master equations can be described by contact
Hamiltonian systems, and such a direction should be pursued as a future work.
Another direction to study master equations in the contact geometric language is to
focus on the Hamilton-Jacobi equation obtained from master equations under some
approximation [41], since contact geometric descriptions of the Hamilton-Jacobi
equation have been well-recognized [42].
4.8 Conclusions
This paper has offered a viewpoint that the solvable master equations and expectation
variables of the moment dynamical system derived from the master equations can
be described on para-contact metric manifolds. To give a geometric description of
these, contact Hamiltonian vector fields have been introduced on para-contact metric
manifolds. Then relaxation processes have been characterized by the use of metric
and the Ricci tensor fields. Moreover possible information geometric structures for
moment dynamical systems have been clarified.
The significance of these descriptions given in this paper is mentioned here.
Although a number of pure mathematical studies of para-contact metric manifolds
exist in the literature, there were a few applications to geometric thermodynamics.
Together with this, Riemannian metric tensor fields are often considered in contact