4 Contact Hamiltonian Systems for Probability Distribution Functions …
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Proposition 7 (Solutions of the master equations, [27]). The solution of (4.2) is
p( j, t) = e
−t p( j, 0) + (1 − e
−t
) p
eq
θ ( j), from which lim
t→∞
p( j, t) = p
eq
θ ( j).
With this proposition, one notices the following.
1. Every solution p depends on θ .
2. The equilibrium state is realized with (4.2) as the time-asymptotic limit.
3. If
j p( j, 0) = 1, then
j p( j, t) = 1 for any t > 0.
Taking into account 1, p( j, t) is also denoted p( j, t; θ) in this paper. From 3, { p( j, t)}
is an element of the probability simplex,
S
|Γ |−1
:=
⎧
⎨
⎩
{ p( j, t) } ∈ R
|Γ |
≥0
|Γ |
j=1
p( j, t) = 1
⎫
⎬
⎭
.
4.3.1 Denormalization
Denormalized distribution functions are often considered in the literature [33]. In
this paper they are denoted { p( j, t)}. For { p( j, t)}, the normalization condition 3
above is not imposed,
j p( j, 0) = 1. From the linearity of master equations and
(4.2), one has
∂
∂t
p( j, t) = p
eq
θ ( j) − p( j, t).
Discrete distribution functions belong to the exponential family, then one can
write
p( j, t) = exp( θ
a
O a ( j) − (θ) ).
One realization of denormalization is to introduce a non-zero constant w ∈ R so that
p( j, t) = w p( j, t).
For the exponential family, if
is introduced so that
p( j, t) = exp
θ
a
O a ( j) −
(θ)
,
then one has
(θ) = (θ) − ln w.