80
S. Goto and H. Hino
Proposition 13 The equilibrium state of the moment dynamical system is expressed
as the Legendre submanifold A eq of the contact manifold (C
O
, λ
O
). Then such
equilibrium state induces a dually flat space.
The induced dually flat space is denoted by (A eq , ∇
eq , g
eq ), where the Riemannian metric tensor field g
eq is
g
eq =
n
a=1
n
b=1
∂
2
eq
∂θ a ∂θ b dθ
a
⊗ dθ
b
.
The dual coordinates are
θ
a
and
η a =
∂∂
eq
∂θ a .
4.6.2 Geometry of Nonequilibrium States
So far geometry of equilibrium states has been discussed. In this section our interpretation of outside the Legendre submanifold is same as that in Sect. 4.5.2.
Proposition 8 is written in a contact geometric language here. In what follows
phase space is identified with a (2n + 1)-dimensional para-contact metric manifold
(C, φ, ξ, λ, G).
The moment dynamical system is a contact Hamiltonian system as stated below.
Proposition 14 (Moment dynamical system as a contact Hamiltonian system, [27]).
The dynamical system in Proposition 8 can be written as a contact Hamiltonian
system.
Proof Identify x, y, z and in Proposition 2 with (θ, O , ,
eq
, ,) in
Proposition 8 as
x
a
= θ
a
, y a = O a θ , ,(x) =
eq
(θ ), z = .
This set of identifications yields the proof.
In nonequilibrium statistical physics, attention is often concentrated on how far a
state is close to the equilibrium state. To characterize points on an integral curve of
the contact Hamiltonian vector field X eq associated with the contact Hamiltonian
h
O
eq (θ, O θ , ,) =
eq
(θ ) − ,
introduce the Mrugala metric tensor field adapted to the contact manifold (C
O
, λ
O
),
G
O
:=
1
2
n
a=1
d θ
a
⊗ d O a θ + d O a θ ⊗ d θ
a
+ λ O ⊗ λ O .
S. Goto and H. Hino
Proposition 13 The equilibrium state of the moment dynamical system is expressed
as the Legendre submanifold A eq of the contact manifold (C
O
, λ
O
). Then such
equilibrium state induces a dually flat space.
The induced dually flat space is denoted by (A eq , ∇
eq , g
eq ), where the Riemannian metric tensor field g
eq is
g
eq =
n
a=1
n
b=1
∂
2
eq
∂θ a ∂θ b dθ
a
⊗ dθ
b
.
The dual coordinates are
θ
a
and
η a =
∂∂
eq
∂θ a .
4.6.2 Geometry of Nonequilibrium States
So far geometry of equilibrium states has been discussed. In this section our interpretation of outside the Legendre submanifold is same as that in Sect. 4.5.2.
Proposition 8 is written in a contact geometric language here. In what follows
phase space is identified with a (2n + 1)-dimensional para-contact metric manifold
(C, φ, ξ, λ, G).
The moment dynamical system is a contact Hamiltonian system as stated below.
Proposition 14 (Moment dynamical system as a contact Hamiltonian system, [27]).
The dynamical system in Proposition 8 can be written as a contact Hamiltonian
system.
Proof Identify x, y, z and in Proposition 2 with (θ, O , ,
eq
, ,) in
Proposition 8 as
x
a
= θ
a
, y a = O a θ , ,(x) =
eq
(θ ), z = .
This set of identifications yields the proof.
In nonequilibrium statistical physics, attention is often concentrated on how far a
state is close to the equilibrium state. To characterize points on an integral curve of
the contact Hamiltonian vector field X eq associated with the contact Hamiltonian
h
O
eq (θ, O θ , ,) =
eq
(θ ) − ,
introduce the Mrugala metric tensor field adapted to the contact manifold (C
O
, λ
O
),
G
O
:=
1
2
n
a=1
d θ
a
⊗ d O a θ + d O a θ ⊗ d θ
a
+ λ O ⊗ λ O .