4 Contact Hamiltonian Systems for Probability Distribution Functions …
85
By assuming the existence of ℘
eq
θ and that of ℘, the normalization condition is
written with (4.27) as
℘
eq
θ (1) =
∞
j=0
℘
eq
θ ( j) = 1,
and
℘(1, τ ) =
∞
j=0
℘ ( j, τ ) = 0, τ ∈ R.
Then, the system (4.29) is immediately written in terms of
℘ as
d
dτ
℘(s, τ ) =
℘
eq
θ (s) −
℘(s, τ ),
0 ≤ s < 1.
(4.30)
Thus, one has arrived at the following:
Proposition 17 The system (4.30) that is obtained from (4.26) is formally same as
the solvable master equations (4.2), where the systems (4.30) and (4.26) are linked
by the probability generating function (4.27) and a change of variables.
Remark 4 Although (4.30) is formally same as (4.2), there is a significant difference. The totality of the label s for (4.30) is [0, 1), and that of j for (4.2) is the
discrete set Γ . This continuous label s does not allow us to discuss (4.30) in terms
of the standard contact geometry employed in the main text of this paper, however,
we could formally proceed with a careful analysis. Such an analysis is left for future
work.
For the sake of completeness, the solutions to (4.29), (4.30), and (4.26) are
expressed as follows. First, the solution to (4.29) is immediately obtained as
℘ θ (s, τ ) = e
− τ
℘ θ (s, 0),
or equivalently,
℘ θ (s, t) = e
− γ (1−s) t
℘ θ (s, 0).
Second, the solution to (4.30) is then
℘(s, τ ) = e
− τ
℘ θ (s, 0) + (1 − e
−τ
)
℘
eq
θ (s),
0 ≤ s < 1.
In what follows the initial conditions for ℘ θ are imposed as
℘ θ ( j, 0) = δ j,0 ,
j = 0, . . . , ∞,
from which one has the relation:
℘ θ (s, 0) =
∞
j=0
δ j,0 s
j
= 1.
85
By assuming the existence of ℘
eq
θ and that of ℘, the normalization condition is
written with (4.27) as
℘
eq
θ (1) =
∞
j=0
℘
eq
θ ( j) = 1,
and
℘(1, τ ) =
∞
j=0
℘ ( j, τ ) = 0, τ ∈ R.
Then, the system (4.29) is immediately written in terms of
℘ as
d
dτ
℘(s, τ ) =
℘
eq
θ (s) −
℘(s, τ ),
0 ≤ s < 1.
(4.30)
Thus, one has arrived at the following:
Proposition 17 The system (4.30) that is obtained from (4.26) is formally same as
the solvable master equations (4.2), where the systems (4.30) and (4.26) are linked
by the probability generating function (4.27) and a change of variables.
Remark 4 Although (4.30) is formally same as (4.2), there is a significant difference. The totality of the label s for (4.30) is [0, 1), and that of j for (4.2) is the
discrete set Γ . This continuous label s does not allow us to discuss (4.30) in terms
of the standard contact geometry employed in the main text of this paper, however,
we could formally proceed with a careful analysis. Such an analysis is left for future
work.
For the sake of completeness, the solutions to (4.29), (4.30), and (4.26) are
expressed as follows. First, the solution to (4.29) is immediately obtained as
℘ θ (s, τ ) = e
− τ
℘ θ (s, 0),
or equivalently,
℘ θ (s, t) = e
− γ (1−s) t
℘ θ (s, 0).
Second, the solution to (4.30) is then
℘(s, τ ) = e
− τ
℘ θ (s, 0) + (1 − e
−τ
)
℘
eq
θ (s),
0 ≤ s < 1.
In what follows the initial conditions for ℘ θ are imposed as
℘ θ ( j, 0) = δ j,0 ,
j = 0, . . . , ∞,
from which one has the relation:
℘ θ (s, 0) =
∞
j=0
δ j,0 s
j
= 1.