A.1 Master Equation
151
This probability distribution is called Poisson Distribution, with the following
properties
M(t) = λt,
(A.9)
σ
2
M = λt
(A.10)
A.1.5 Detailed Balance
We consider the steady sate of a master equation that satisfies:
J i→j = 0, ∀i, ∀j (detailed balance)
(A.11)
or, equivalently,
P i ω i→j = P j ω j →i , ∀i, ∀j (detailed balance)
(A.12)
We call such steady state the state of detailed balance.
Detailed balance allows us to formulate relations among rates parameters. This
may be easily seen in the isomerization reactions shown in Fig. A.1. At equilibrium,
detailed balance means that
k 1 [A] = k 2 [B] ,
k 3 [B] = k 4 [C] ,
k 5 [C] = k 6 [A] ,
(A.13)
from which
k 1 k 3 k 5 = k 2 k 4 k 6
(A.14)
More general relations may be developed for complex reaction networks. From
Chap. 2, Eq. 2.4 we obtain the overdamped limit of the Langevin equation,
˙
x (t) = −
dV (x)
dx
+ (2D)
1/2 ξ(t)
(A.15)
Fig. A.1 Reaction sequence
of three isomeric compounds,
A, B, and C
151
This probability distribution is called Poisson Distribution, with the following
properties
M(t) = λt,
(A.9)
σ
2
M = λt
(A.10)
A.1.5 Detailed Balance
We consider the steady sate of a master equation that satisfies:
J i→j = 0, ∀i, ∀j (detailed balance)
(A.11)
or, equivalently,
P i ω i→j = P j ω j →i , ∀i, ∀j (detailed balance)
(A.12)
We call such steady state the state of detailed balance.
Detailed balance allows us to formulate relations among rates parameters. This
may be easily seen in the isomerization reactions shown in Fig. A.1. At equilibrium,
detailed balance means that
k 1 [A] = k 2 [B] ,
k 3 [B] = k 4 [C] ,
k 5 [C] = k 6 [A] ,
(A.13)
from which
k 1 k 3 k 5 = k 2 k 4 k 6
(A.14)
More general relations may be developed for complex reaction networks. From
Chap. 2, Eq. 2.4 we obtain the overdamped limit of the Langevin equation,
˙
x (t) = −
dV (x)
dx
+ (2D)
1/2 ξ(t)
(A.15)
Fig. A.1 Reaction sequence
of three isomeric compounds,
A, B, and C