4 Contact Hamiltonian Systems for Probability Distribution Functions …
81
The relation between G
O and g
eq is
g
eq = ι
O ∗ G
O
,
where ι
O
: A eq → C
O is the embedding.
Then from discussions in Sect. 4.2.4, one has the following.
Proposition 15 The length between a state and the equilibrium state for the moment
dynamical system calculated with (4.4) is
l[ G
O
, X eq ]
∞
t =
∞
t
G O (X eq , X eq ) dt
= | h
O
eq (θ, O θ , ,) |, (4.24)
where O θ = { O 1 θ , . . . , O n θ }, and X eq the contact Hamiltonian vector
field associated with h
O
eq . Then the convergence rate for (4.24) is exponential. In
addition, it follows that
Ric
G
O ( X eq , X eq ) = −2n
h
O
eq
2 ,
and
Ric
G
O
X eq ,
∂
∂∂
= −2n h
O
eq .
Combining Propositions 10–15, one arrives at the main theorem in this paper.
Theorem 1 (Geometric descriptions of the solvable master equations and moment
dynamical systems). The solvable master equations and moment dynamical system
derived from the solvable master equations are described on para-contact metric
manifolds, and its convergence to the equilibrium states are characterized by the
Mrugala metric fields and the Ricci tensor fields associated with the Levi-Civita
connections.
4.7 Beyond the Toy Model
So far as a toy model, called the solvable master equations in this paper, the master
equations with a particular choice of transition matrix have mainly been discussed.
Although various mathematical structures have been clarified due to its simplicity,
this toy model lacks generality. In what follows, how much the present contact
geometric approach and its variant can possibly be applied to the case with general
transition matrices is briefly discussed.
81
The relation between G
O and g
eq is
g
eq = ι
O ∗ G
O
,
where ι
O
: A eq → C
O is the embedding.
Then from discussions in Sect. 4.2.4, one has the following.
Proposition 15 The length between a state and the equilibrium state for the moment
dynamical system calculated with (4.4) is
l[ G
O
, X eq ]
∞
t =
∞
t
G O (X eq , X eq ) dt
= | h
O
eq (θ, O θ , ,) |, (4.24)
where O θ = { O 1 θ , . . . , O n θ }, and X eq the contact Hamiltonian vector
field associated with h
O
eq . Then the convergence rate for (4.24) is exponential. In
addition, it follows that
Ric
G
O ( X eq , X eq ) = −2n
h
O
eq
2 ,
and
Ric
G
O
X eq ,
∂
∂∂
= −2n h
O
eq .
Combining Propositions 10–15, one arrives at the main theorem in this paper.
Theorem 1 (Geometric descriptions of the solvable master equations and moment
dynamical systems). The solvable master equations and moment dynamical system
derived from the solvable master equations are described on para-contact metric
manifolds, and its convergence to the equilibrium states are characterized by the
Mrugala metric fields and the Ricci tensor fields associated with the Levi-Civita
connections.
4.7 Beyond the Toy Model
So far as a toy model, called the solvable master equations in this paper, the master
equations with a particular choice of transition matrix have mainly been discussed.
Although various mathematical structures have been clarified due to its simplicity,
this toy model lacks generality. In what follows, how much the present contact
geometric approach and its variant can possibly be applied to the case with general
transition matrices is briefly discussed.