70
S. Goto and H. Hino
4.3 Distribution Functions from Solvable Master Equations
In this section a set of master equations with a particular set of Markov kernels
(transition matrix elements) is introduced, and then its time-development is analyzed.
Let Γ be a set of finite discrete states, t ∈ R time, and p( j, t) dt a probability
distribution function that a state j ∈ Γ is found in between t and t + dt. The first
objective is to realize a given distribution function p
eq
θ that can be written as
p
eq
θ ( j) =
π θ ( j)
Z (θ )
,
Z (θ ) :=
j∈Γ
π θ ( j),
(4.12)
where θ ∈ ⊂ R
n is a parameter set with θ = {θ
1
, . . . , θ
n
}, and Z : → R the
so-called partition function so that p
eq
θ is normalized :
j∈Γ p
eq
θ ( j) = 1. Although
it is often the case that n |Γ | when thermodynamic systems are considered, we
do not assume this case, where |Γ | is the number of elements of Γ , |Γ | := #Γ .
In what follows, attention is focused on a class of master equations. Let p :
Γ × R → R ≥0 be a time-dependent probability distribution function, where t ∈ R
denotes time. Then, consider the set of master equations (4.1),
∂
∂t
p( j, t) =
j ∈Γ \{ j}
w( j| j
) p( j
, t) − w( j
| j) p( j, t)
,
where w : Γ × Γ → I , (I := [ 0, 1 ] ⊂ R) is such that w( j| j
) denotes a probability
that a state jumps from j
to j. This w is referred to as a transition matrix (and kernel).
With (4.1), and the assumptions
w θ ( j| j
) = p
eq
θ ( j),
together with
p
eq
θ ( j) = 0, ∀ j ∈ Γ,
(4.13)
one derives the solvable master equations, (4.2):
∂
∂t
p( j, t) = p
eq
θ ( j) − p( j, t).
Although the term “solvable master equation” is used in this paper, such a term can be
replaced with “simple master equation” for example. In addition the solvable master
equations are related to another class of master equations that have interaction terms.
Such a class is related to the Poisson distribution (see Appendix A).
An explicit form of p( j, t) is obtained by solving (4.2). Then the following proposition can easily be shown.
S. Goto and H. Hino
4.3 Distribution Functions from Solvable Master Equations
In this section a set of master equations with a particular set of Markov kernels
(transition matrix elements) is introduced, and then its time-development is analyzed.
Let Γ be a set of finite discrete states, t ∈ R time, and p( j, t) dt a probability
distribution function that a state j ∈ Γ is found in between t and t + dt. The first
objective is to realize a given distribution function p
eq
θ that can be written as
p
eq
θ ( j) =
π θ ( j)
Z (θ )
,
Z (θ ) :=
j∈Γ
π θ ( j),
(4.12)
where θ ∈ ⊂ R
n is a parameter set with θ = {θ
1
, . . . , θ
n
}, and Z : → R the
so-called partition function so that p
eq
θ is normalized :
j∈Γ p
eq
θ ( j) = 1. Although
it is often the case that n |Γ | when thermodynamic systems are considered, we
do not assume this case, where |Γ | is the number of elements of Γ , |Γ | := #Γ .
In what follows, attention is focused on a class of master equations. Let p :
Γ × R → R ≥0 be a time-dependent probability distribution function, where t ∈ R
denotes time. Then, consider the set of master equations (4.1),
∂
∂t
p( j, t) =
j ∈Γ \{ j}
w( j| j
) p( j
, t) − w( j
| j) p( j, t)
,
where w : Γ × Γ → I , (I := [ 0, 1 ] ⊂ R) is such that w( j| j
) denotes a probability
that a state jumps from j
to j. This w is referred to as a transition matrix (and kernel).
With (4.1), and the assumptions
w θ ( j| j
) = p
eq
θ ( j),
together with
p
eq
θ ( j) = 0, ∀ j ∈ Γ,
(4.13)
one derives the solvable master equations, (4.2):
∂
∂t
p( j, t) = p
eq
θ ( j) − p( j, t).
Although the term “solvable master equation” is used in this paper, such a term can be
replaced with “simple master equation” for example. In addition the solvable master
equations are related to another class of master equations that have interaction terms.
Such a class is related to the Poisson distribution (see Appendix A).
An explicit form of p( j, t) is obtained by solving (4.2). Then the following proposition can easily be shown.