4 Contact Hamiltonian Systems for Probability Distribution Functions …
77
dynamical systems expressing nonequilibrium process of thermodynamic variables
can be identified with a class of contact Hamiltonian vector fields on a contact
manifold. The above claim also holds on para-contact metric manifolds.
To clarify how time-development of probability distribution functions, that are
not thermodynamic variables, can be described on contact and para-contact metric
manifolds, in the following a geometric description of the time-development of the
master equations is shown. Our basic strategy is the same as that of thermodynamic
variables as explained below. As shown in Proposition 7, initial states approach to the
equilibrium state as time develops. This time-development can be reformulated on
contact manifolds and para-contact metric manifolds with Proposition 2. The details
are as follows.
Given a function f : R >0 → R, the functions { p
f j
eq }, {ψ j }, and I f have been
defined as (4.15), (4.16), and (4.17), respectively. Then the dynamic equations for
{ p
j
eq }, {ψ j } and I f are obtained from (4.2) as
d
dt
p
f j
eq = 0,
d
dt
ψ j = 1 − ψ j ,
d
dt
I f = I
eq
f − I f , j = 1, . . . , |Γ |. (4.19)
The dynamical system (4.19) is a contact Hamiltonian system as stated below.
Proposition 11 (Master equations as contact Hamiltonian system). Let (C
Γ
f , λ
Γ
f )
be a (2|Γ | + 1)-dimensional contact manifold, and ( p
f
eq , ψ, I f ) the Darboux coordinates so that λ
Γ
f is given by (4.18).
Then (4.19) can be written as a contact Hamiltonian system with the contact
Hamiltonian
h
Γ
I
eq
f
( p
f
eq , ψ, I f ) = I
eq
f ( p eq ) − I f .
(4.20)
In addition, it follows that
lim
t→∞
( p
f
eq , ψ(t), I f (t) ) ∈ A I
eq
f
.
(4.21)
Proof Identify p
f
eq , ψ, I f with x, y, z in Proposition 2, respectively. In addition,
identify I
eq
f with . Then it follows that (4.9) is equivalent to (4.19). Thus (4.19)
is the contact Hamiltonian system with the contact Hamiltonian (4.20). From the
general discussion stated as (4.10), the property (4.21) holds.
For the sake of completeness, the solution to (4.19) is given. This is immediately
obtained as
p
j
eq (t) f ( p
j
eq (t)) = p
j
eq (0) f ( p
j
eq (0)),
ψ j (t) = e
−t
ψ j (0) + (1 − e
−t
),
I f (t) = e
−t I f (0) + (1 − e
−t
)I
eq
f .
77
dynamical systems expressing nonequilibrium process of thermodynamic variables
can be identified with a class of contact Hamiltonian vector fields on a contact
manifold. The above claim also holds on para-contact metric manifolds.
To clarify how time-development of probability distribution functions, that are
not thermodynamic variables, can be described on contact and para-contact metric
manifolds, in the following a geometric description of the time-development of the
master equations is shown. Our basic strategy is the same as that of thermodynamic
variables as explained below. As shown in Proposition 7, initial states approach to the
equilibrium state as time develops. This time-development can be reformulated on
contact manifolds and para-contact metric manifolds with Proposition 2. The details
are as follows.
Given a function f : R >0 → R, the functions { p
f j
eq }, {ψ j }, and I f have been
defined as (4.15), (4.16), and (4.17), respectively. Then the dynamic equations for
{ p
j
eq }, {ψ j } and I f are obtained from (4.2) as
d
dt
p
f j
eq = 0,
d
dt
ψ j = 1 − ψ j ,
d
dt
I f = I
eq
f − I f , j = 1, . . . , |Γ |. (4.19)
The dynamical system (4.19) is a contact Hamiltonian system as stated below.
Proposition 11 (Master equations as contact Hamiltonian system). Let (C
Γ
f , λ
Γ
f )
be a (2|Γ | + 1)-dimensional contact manifold, and ( p
f
eq , ψ, I f ) the Darboux coordinates so that λ
Γ
f is given by (4.18).
Then (4.19) can be written as a contact Hamiltonian system with the contact
Hamiltonian
h
Γ
I
eq
f
( p
f
eq , ψ, I f ) = I
eq
f ( p eq ) − I f .
(4.20)
In addition, it follows that
lim
t→∞
( p
f
eq , ψ(t), I f (t) ) ∈ A I
eq
f
.
(4.21)
Proof Identify p
f
eq , ψ, I f with x, y, z in Proposition 2, respectively. In addition,
identify I
eq
f with . Then it follows that (4.9) is equivalent to (4.19). Thus (4.19)
is the contact Hamiltonian system with the contact Hamiltonian (4.20). From the
general discussion stated as (4.10), the property (4.21) holds.
For the sake of completeness, the solution to (4.19) is given. This is immediately
obtained as
p
j
eq (t) f ( p
j
eq (t)) = p
j
eq (0) f ( p
j
eq (0)),
ψ j (t) = e
−t
ψ j (0) + (1 − e
−t
),
I f (t) = e
−t I f (0) + (1 − e
−t
)I
eq
f .