58
S. Goto and H. Hino
Contact geometry has been known as an odd-dimensional counterpart of symplectic geometry [7–9], and this geometry is applicable to various mathematical
sciences. They include thermodynamics [10–12], fluid mechanics [13], optimization
problems [14], and dissipative mechanical systems [15, 16]. In developing contact
geometric thermodynamics [17, 18], links between contact geometry and information geometry have been discussed [3, 19]. In this context, when a metric tensor
field is introduced on a contact manifold, it was found that para-Sasakian geometry
is suitable for describing thermodynamics [20, 21], where para-Sasakian manifolds
are para-contact metric manifolds satisfying some additional condition. It is then of
interest to explore how para-contact metric manifolds describe dynamical systems
for probability distribution functions inducing thermodynamics.
Master equations are used for modeling Markov processes [22] in the form of
continuous-time first order differential equations and of discrete-time maps, and are
used for describing probability distribution functions by modeling so-called transition matrices or rates. Let Γ be a set of finite states, and p : Γ × R → R ≥0 a timedependent probability distribution function, where t ∈ R denotes time. Continuoustime master equations are then written as
∂
∂t
p( j, t) =
j ∈Γ \{ j}
w( j| j
) p( j
, t) − w( j
| j) p( j, t)
,
(4.1)
where w : Γ × Γ → I , (I := [ 0, 1 ] ⊂ R) is such that w( j| j
) denotes a probability that a state jumps from j
to j, and is referred to as a transition matrix. These
systems are used in various scientific disciplines including nonequilibrium statistical mechanics [22, 23]. Aside from interests in the natural sciences, its resolution
has wide implications in mathematical engineering. One of them is to apply such
processes to Monte-Carlo simulations, called Markov-Chain Monte-Carlo methods
[24]. Further progress in building mathematical foundation of these methods and
their applications continues to attract attention [25, 26].
Given master equations, one is interested in time-development of the expectation
variables. Such variables are defined as follows. Given a set of prescribed functions
O a : Γ → R, (a = 1, . . . , n) the expectation variables O a (t) are defined as
O a (t) =
j∈Γ
O a ( j) p( j, t),
a = 1, . . . , n,
where { p( j, t)} obey (4.1), and n is independent of |Γ | in general. These expectation
variables are expected to play roles of time-dependent thermodynamic variables, and
thus how information and contact geometric theories are applied to these variables
should be clarified.
S. Goto and H. Hino
Contact geometry has been known as an odd-dimensional counterpart of symplectic geometry [7–9], and this geometry is applicable to various mathematical
sciences. They include thermodynamics [10–12], fluid mechanics [13], optimization
problems [14], and dissipative mechanical systems [15, 16]. In developing contact
geometric thermodynamics [17, 18], links between contact geometry and information geometry have been discussed [3, 19]. In this context, when a metric tensor
field is introduced on a contact manifold, it was found that para-Sasakian geometry
is suitable for describing thermodynamics [20, 21], where para-Sasakian manifolds
are para-contact metric manifolds satisfying some additional condition. It is then of
interest to explore how para-contact metric manifolds describe dynamical systems
for probability distribution functions inducing thermodynamics.
Master equations are used for modeling Markov processes [22] in the form of
continuous-time first order differential equations and of discrete-time maps, and are
used for describing probability distribution functions by modeling so-called transition matrices or rates. Let Γ be a set of finite states, and p : Γ × R → R ≥0 a timedependent probability distribution function, where t ∈ R denotes time. Continuoustime master equations are then written as
∂
∂t
p( j, t) =
j ∈Γ \{ j}
w( j| j
) p( j
, t) − w( j
| j) p( j, t)
,
(4.1)
where w : Γ × Γ → I , (I := [ 0, 1 ] ⊂ R) is such that w( j| j
) denotes a probability that a state jumps from j
to j, and is referred to as a transition matrix. These
systems are used in various scientific disciplines including nonequilibrium statistical mechanics [22, 23]. Aside from interests in the natural sciences, its resolution
has wide implications in mathematical engineering. One of them is to apply such
processes to Monte-Carlo simulations, called Markov-Chain Monte-Carlo methods
[24]. Further progress in building mathematical foundation of these methods and
their applications continues to attract attention [25, 26].
Given master equations, one is interested in time-development of the expectation
variables. Such variables are defined as follows. Given a set of prescribed functions
O a : Γ → R, (a = 1, . . . , n) the expectation variables O a (t) are defined as
O a (t) =
j∈Γ
O a ( j) p( j, t),
a = 1, . . . , n,
where { p( j, t)} obey (4.1), and n is independent of |Γ | in general. These expectation
variables are expected to play roles of time-dependent thermodynamic variables, and
thus how information and contact geometric theories are applied to these variables
should be clarified.