4 Contact Hamiltonian Systems for Probability Distribution Functions …
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geometric thermodynamics, applications of facts found in pure mathematical studies are thus expected to yield beneficial statements in thermodynamics. The present
study has given some of such applications. In addition, although there have been a
variety of studies of master equations as dynamical systems, any contact geometric
approach to these equations had not been shown. Thus the present study is a first step
to build a complete contact geometric theory of master equations.
There remain unsolved problems that have not been addressed in this article.
These include how this analysis is applied to a wider class of master equations and
corresponding moment dynamical systems. In addition, how theorems found in the
study of para-contact metric geometry can be applied to the master equations more,
and how other geometric formulations of thermodynamics [43, 44] can be related to
the present study should be addressed. By addressing these questions together with
this study, it is expected that a relevant and sophisticated geometric methodology
will be established for dealing with master equations and their applications.
Acknowledgements The author S.G. is partially supported by JSPS (KAKENHI) grant number
JP19K03635. The other author H.H. is partially supported by JSPS (KAKENHI) grant number
JP17H01793. In addition, both of the authors are partially supported by JST CREST JPMJCR1761.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
A. Link Between the Solvable Model and One-Step Process
In this section it is shown that there is a link between the solvable model (4.2) and a
model described by master equations with non-trivial transition matrix elements.
The solvable model (or the toy model) (4.2) is rewritten by introducing the set of
the variables
q θ ( j, t) = p( j, t) − p
eq
θ ( j),
as
d
dt
q θ ( j, t) = − q θ ( j, t),
j = 1, . . . , |Γ |.
(4.25)
It is shown below that this set of equations, (4.25), is related to another class of
master equations.
First, consider the master equations with some parameter θ ∈ ⊂ R
n
d
dt
℘ θ ( j, t) = γ [ ℘ θ ( j − 1, t) − ℘ θ ( j, t) ],
j = 0, . . . , ∞
(4.26)
where γ > 0 is constant, and ℘ θ ( j, t) = 0 for all t and j < 0. This model, (4.26),
belongs to a class of one-step processes [22], and is obtained by choosing Γ =
{0, 1, . . .} and {w( j| j
)} appropriately in (4.1).
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geometric thermodynamics, applications of facts found in pure mathematical studies are thus expected to yield beneficial statements in thermodynamics. The present
study has given some of such applications. In addition, although there have been a
variety of studies of master equations as dynamical systems, any contact geometric
approach to these equations had not been shown. Thus the present study is a first step
to build a complete contact geometric theory of master equations.
There remain unsolved problems that have not been addressed in this article.
These include how this analysis is applied to a wider class of master equations and
corresponding moment dynamical systems. In addition, how theorems found in the
study of para-contact metric geometry can be applied to the master equations more,
and how other geometric formulations of thermodynamics [43, 44] can be related to
the present study should be addressed. By addressing these questions together with
this study, it is expected that a relevant and sophisticated geometric methodology
will be established for dealing with master equations and their applications.
Acknowledgements The author S.G. is partially supported by JSPS (KAKENHI) grant number
JP19K03635. The other author H.H. is partially supported by JSPS (KAKENHI) grant number
JP17H01793. In addition, both of the authors are partially supported by JST CREST JPMJCR1761.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
A. Link Between the Solvable Model and One-Step Process
In this section it is shown that there is a link between the solvable model (4.2) and a
model described by master equations with non-trivial transition matrix elements.
The solvable model (or the toy model) (4.2) is rewritten by introducing the set of
the variables
q θ ( j, t) = p( j, t) − p
eq
θ ( j),
as
d
dt
q θ ( j, t) = − q θ ( j, t),
j = 1, . . . , |Γ |.
(4.25)
It is shown below that this set of equations, (4.25), is related to another class of
master equations.
First, consider the master equations with some parameter θ ∈ ⊂ R
n
d
dt
℘ θ ( j, t) = γ [ ℘ θ ( j − 1, t) − ℘ θ ( j, t) ],
j = 0, . . . , ∞
(4.26)
where γ > 0 is constant, and ℘ θ ( j, t) = 0 for all t and j < 0. This model, (4.26),
belongs to a class of one-step processes [22], and is obtained by choosing Γ =
{0, 1, . . .} and {w( j| j
)} appropriately in (4.1).