4 Contact Hamiltonian Systems for Probability Distribution Functions …
75
The set of initial values {ψ j (0)} is fixed with Proposition 7 or (4.16) as
ψ j (0) =
1
p
j
eq
p( j, 0),
j = 1, . . . , |Γ |,
where the right hand side above has been assumed to exist (see (4.13)). Then a pair
(C
Γ
f , λ
Γ
f ) is a (2|Γ | + 1)-dimensional contact manifold, where
C
Γ
f := R
|Γ |
× R
|Γ |
>0 × R,
and
λ
Γ
f = dI f −
j∈Γ
ψ j d p
f j
eq .
(4.18)
Its Darboux coordinates are ( p
f
eq , ψ, I f ) with ψ = {ψ 1 , . . . , ψ |Γ | }.
An example of f is
f ( p
j
eq ) = − ln p
j
eq ,
j = 1, . . . , |Γ |,
and in this case I
eq
f is the entropy of the equilibrium distribution function.
The function I
eq
f generates the Legendre submanifold A I
eq
f
⊂ C
Γ
f as in (4.7) with
= I
eq
f , which is explicitly written as
A I
eq
f
=
( p
f
eq , ψ, I f ) ∈ C Γ
f
ψ j =
∂ I
eq
f
∂ p
f j
eq
, I f = I
eq
f ( p
f
eq ), j = 1, . . . , |Γ |
,
which reduces to
A I
eq
f
=
( p
f
eq , ψ, I f ) ∈ C
Γ
f
ψ j = 1, I f = I
eq
f ( p
f
eq ), j = 1, . . . , |Γ |
,
due to
∂ I
eq
f
∂ p
f j
eq
= 1,
j = 1, . . . , |Γ |.
Notice that the state where ψ j = 1 for all j ∈ Γ is the equilibrium state due to (4.16).
Then combining the discussions so far and a part of Proposition 1, one has the
following.
Proposition 10 The equilibrium state of the solvable master equations is expressed
as the Legendre submanifold A I
eq
f
of the contact manifold (C
Γ
f , λ
Γ
f ).
Remark 3 Since
∂
2 I
eq
f
∂ p
f j
eq ∂ p
f k
eq
= 0,
j, k = 1, . . . , |Γ |,