Chapter 4
Contact Hamiltonian Systems for
Probability Distribution Functions
and Expectation Variables: A Study
Based on a Class of Master Equations
Shin-itiro Goto and Hideitsu Hino
Abstract Based on information and para-contact metric geometries, in this paper
two classes of dynamical systems are formulated. One is for describing the timedevelopment of probability distribution functions following continuous-time master
equations, and the other one for describing expectation variables with respect to
the distribution functions. These two classes of systems are described by contact
Hamiltonian systems on contact metric manifolds. Here the systems for expectation
variables are exactly derived from the master equations describing nonequilibrium
processes. Their relaxation processes are characterized by the lengths of integral
curves of contact Hamiltonian vector fields and the values of the Ricci tensor fields.
At equilibrium states realized as the time-asymptotic limit of the contact Hamiltonian
systems, possible information geometric structures are given.
Keywords Master equations · Para-contact metric manifolds · nonequilibrium
statistical mechanics · Information geometry
4.1 Introduction
Information geometry is a geometrization of mathematical statistics, and its differential geometric aspects and applications to statistics have been investigated [1,
2]. Examples of such applications include statistical inference, quantum information, thermodynamics. In addition, links between equilibrium thermodynamics and
information geometry have been studied [3]. Moreover, dynamical systems for probability distribution functions have been investigated from viewpoints of information
geometry [4–6].
S. Goto (B) · H. Hino
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
e-mail: sgoto@ism.ac.jp
H. Hino
e-mail: hino@ism.ac.jp
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_4
57
Contact Hamiltonian Systems for
Probability Distribution Functions
and Expectation Variables: A Study
Based on a Class of Master Equations
Shin-itiro Goto and Hideitsu Hino
Abstract Based on information and para-contact metric geometries, in this paper
two classes of dynamical systems are formulated. One is for describing the timedevelopment of probability distribution functions following continuous-time master
equations, and the other one for describing expectation variables with respect to
the distribution functions. These two classes of systems are described by contact
Hamiltonian systems on contact metric manifolds. Here the systems for expectation
variables are exactly derived from the master equations describing nonequilibrium
processes. Their relaxation processes are characterized by the lengths of integral
curves of contact Hamiltonian vector fields and the values of the Ricci tensor fields.
At equilibrium states realized as the time-asymptotic limit of the contact Hamiltonian
systems, possible information geometric structures are given.
Keywords Master equations · Para-contact metric manifolds · nonequilibrium
statistical mechanics · Information geometry
4.1 Introduction
Information geometry is a geometrization of mathematical statistics, and its differential geometric aspects and applications to statistics have been investigated [1,
2]. Examples of such applications include statistical inference, quantum information, thermodynamics. In addition, links between equilibrium thermodynamics and
information geometry have been studied [3]. Moreover, dynamical systems for probability distribution functions have been investigated from viewpoints of information
geometry [4–6].
S. Goto (B) · H. Hino
The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
e-mail: sgoto@ism.ac.jp
H. Hino
e-mail: hino@ism.ac.jp
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_4
57