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The probability of reaction events in a complex chemical system
PHYSICS LETTERS A
Volume 116, number 9
14 July 1986
T H E P R O B A B I L I T Y O F R E A C T I O N E V E N T S IN A C O M P L E X C H E M I C A L S Y S T E M M.O. V L A D L Med. Bucure~ti, CSsu(a po~tal5 77-49, Bucarest, Romania
Received 19 July 1984; accepted in revised form 12 May 1986
A stochastic model corresponding to R linearly independent chemical reactions, at equilibrium, can be formulated in the system's own scale, by means of the numbers of reaction events. Within the framework of the van Kampen approximation, a general formula for the probability of reaction events may be easily derived.
The problem of determining the stochastic properties for the numbers of reaction events has been raised by Perrin [1]. In the case of m o n o molecular equilibrium A ~ B, an analytical solution of this problem was given by Solc [2]. Vlad [3] dealt with the linear reaction network A~ ~ - A j, n >/i > j > 1, through a chemical master equation. This latter approach can be extended to non-linear systems, too. The master equation that arises in this case has a rather simple appearance, but in general it is by no means easy to solve in any systematic way. However, at equilibrium, a general solution is possible, on the basis of the van K a m p e n approximation [4]. Let us consider an S - c o m p o n e n t reacting system involving R different reactions S
k7
Y'. v ~ A ; ~
Y'~ v;jA/, i=1
j = 1 . . . . . R,
(1)
where rig are stoichiometric coefficients and k ~ are the corresponding forward and backward rate constants. These reactions are linearly independent, that is, the rank of the stoichiometric matrix
II f,.j II, Lj = vi~ v,~, i = 1 . . . . . S, j = l ..... R,
F=
~),P(N, ¢ ±, t)
= E [w,.±(v, N-Vl,-,N) j;±
×P(N-T-~, , ( ..... ¢ 7 - 1 ..... ,~, t) -Wj±(V,N--*N+g)P(N,,±,t)],
(4)
where N, ~, and c ± are S- and R-dimensional vectors N=(N,.),
~=(f,j),
i = 1 . . . . . S,
j=I...R,
,-+ = ( , f ) ,
(5)
S>R,
(2)
(6)
respectively, V is the volume of the system and the transition probabilities Wj+-At, j = 1 . . . . . R are given by [5]
Wj±(V, N ~ N + ~ s
-
is equal to R: rank F = R.
as
S
i=1
k/
molecular species Ai, i = 1 . . . . . S and by c7 the numbers of reaction events Y~vi~Ai---) Y~vi~Ai, the master equation for the reactions (1), based entirely on combinatorial kinetics [5], m a y be written
)
N,(N, - 1) • • • (N, - v,~ + 1)
= vk? FI i=1
1~±
V '*
S
(3)
Denoting by N,, i = 1 . . . . . S the numbers of
= V k f I-I (N~/V) ~?~,
J = 1 ..... R.
(V)
i=1
The equilibrium conditions involve that the 453
Volume 116, number 9
PHYSICS LETTERS A
moments of N, are constant d(U,(t))/dt
~(X, n ±, t) dx d'o + dTI
d(aN,(t)aNj(t))/dt
(8)
= 0....
To describe the composition fluctuations around equilibrium, we introduce the reaction extents Zj [6]: N , ( t ) = (N,) + Y2fq~-i, )
i = 1 . . . . . S.
(9)
c + - = V ( e ± ) + V1/2~ +-, t) dxd'q + d'q , ,
= kj ±
H(
(ni) + gf,,(~,) I
i = 1 ..... S,
j=l
(10)
j=l
..... R,
(19)
a~?=~?-(~?>, (20,21)
we get:
d(x/)/dt=
d(e?)/dt
Eb/,(X,),
= ri +-,
d(~/)/dt=O,
(22,23) (24,25)
l
d(AxjAxI)/dt
Employing
j=l
..... R,
(11)
..... R.
,
i = l ..... S,
d(~'/)/dt = rj+ - 5 - '
from which, taking into account (3), we obtain
)"+'
(17)
(18)
n~ = N / V ,
ax,=x,-(x,>,
Through mediation, these equations lead to
(~.j)=O,
V ( ~ ) + V1/2X,
= V3R/2P(~=
= O,
E L i ( Z , ) = O, J
14 July 1986
= y' bj~.(AxkAxl)
as state variables ~j and ~± , the master equation (4) becomes
k
+ Eb,k(Axkaxj)
+ ajzB,,
k
3 , P = E [Wj±('~j-T- 1--+ Z j ) P ( - - , T- 1, %+
-
1)
(26)
j;_+
d(Axjdxn?)/dt
- <~(-'-J-O-T + 1 ) e ] ,
k
(27)
where d(an?a~;)/dt =
W j ± ( g T 1--+ < )
= G ~ ( ( N ) ; / , + FZ-+ (N) + FZ).
(13)
Applying to (12) the "van Kampen extensivity Ansatz" [4], i.e., assuming that the dependences = = F,j(V), c f = E l ( V ) are given by ~/
~j = ('~j) + V-a/2X 2,
e/ = (el)
6j,3.~G~,
,t, =
-
(28)
Eb,~(x3"} ~ O Y ; ' ' j ,X/ ' i
32Nit
B~aX2 ++~
T/-+ - -
ax~a~F
(14) + G/± - -
+ V I/2~y.-+ = V E ? ,
+ V-'/2rlf,
a,B=+,
and
zj = v(~+) + v ' / ~ x j ,
{? = V(¢)
+ 3i, Tt*,
= Ebjk(AxkA~F)j
(12)
(15)
where V(~';), V(ej-+) and V I / 2 x j , v l / 2 l l / are the deterministic and the fluctuating contributions to .~j and of, respectively, with
(29)
where
¢., = E ( + 1)(aC/o(~,)),
(30)
±
B t = ~ r l -+'
(31)
_+
( 5 ) , X j, ( e l ) ,
rl? - V °,
and introducing the notations 454
(16)
T/± = + r / , GI +- = rl +- ,
(32) (33)
Volume 116, number 9
PHYSICS LETTERS A
Therefore, it is easy to check that
and 8jl is the Kronecker symbol. Combining (11) and (14) it turns out that
(Xs) = 0,
(~j) = 0.
(34,35)
and then eqs. (20), (22), (30)-(33) yield AXj = Xs,
(36) s
r/ =
=
/U
= H
14 July 1986
(37,38)
i=l
Fj, = 85,,
(49)
(x,xv) = (a-'),o,
(50)
are solutions of (45) and (46). Eqs. (50) are consistent with the results of Kubo, Matsuo and Kitahara [7] or thermodynamic [8] approaches. On the other hand, the integration of (23), (25) and (27)-(29) with the initial conditions
OrJ+ f bit= E ( + 1 ) O(n,) " i;+
(51)
(xj(to)ATI~(to)) = (A*/f(t0)a~/t~(/0)) = 0 .... ~
f.=
-~I~jt,
(52)
(39)
B t = 27,,
(40)
T~± = +?,,
(41)
GI± = rt,
(42)
where
gives
( e l ( t ) ) = ~(t - to), (xj(t)A~/~(t))
Q l f ( t ) ) = 0,
= (Xj(t)~I~(t))
(53,54)
= ± d p j t ( t ) r I,
(55)
(Any(t)An~(t)) = (~y(t)~l~(t))
$2jl = E f u f i l ( n i ) i
(43)
-1 .
=
Inserting (9) and (14) into the equilibrium conditions (8) and making use of (34) gives
a,/3-
+, (56)
g'(X, r/±, t ) = [detM(t)]x/2 e x p ( - ½ X t M - ' X ) ,
d( A Nq A N~2)/ d t = E EL,jfi2, d( Z.iY't) / d t j
Jj, e, (t - to),
( 2 q r ) 3R/2
l
(57)
= vY'~ ~_~fiofi=t d ( x j x , ) / d t = O, j
I
(44)
1 (2,~) R (t - to) RI-I~,
from which, taking into account (3), we obtain
d(XjY.t)/dt = V d ( x j x , ) / d t = 0.
(44')
By means of (39), (40) and (44'), the relations turn into a simple form:
28j" t = ~.Fj, + ~.tro,
(45)
with
)
(58)
where
ll~.~2.oll] dr,
(59)
to
(46)
k
Assuming now that all (n~) and k~ are different from 0, (3), (19), (37) and (38) lead to rank 12 = R.
(--~)2
t;+ rt( t - t O ) , '
q,= IIq,jzll = f ' e x p [ - ( t - r )
F.o = y' 12~k(XkXv).
?j > 0,
× e x p ( - ½ ~~
(47,48)
X* = (X*TI+*~I-*),
M=
¢* _ fl-1 ~t
D 0~
(60) ,
(61)
-i 455
V o l u m e 116, n u m b e r 9
PHYSICS LETTERS A
D = Ila,z~l(t- to)tl,
(62)
= II ~j,6 II.
(63)
The p h y s i c a l m e a n i n g of these equations is s t r a i g h t f o r w a r d : T h e p r o b a b i l i t y d i s t r i b u t i o n for + the reaction events E v , / A i ~ Y , v ~ A t,
{i = {7 + % = V~kj~ + V1/2Bj,
(64)
14 J u l y 1986
W e m a y c o n c l u d e that the evolution of the system m a y be r e p r e s e n t e d as a s u p e r p o s i t i o n of R i n d e p e n d e n t r a n d o m flights [9]. This work has been s u p p o r t e d by C I M C Bucure~ti. The a u t h o r is grateful to a referee for p o i n t i n g out some m i s p r i n t s in the original manuscript.
is n o r m a l :
References ,@()), t ) = f ~ ( ' q
±, t ) 8 ( B - - T I + - - ) I
R
=FI
i=1
) d r / + dr/
1 V"4"rr(t- t0)F /
[1] [2] [3] [4] [5] [6]
Xexp
4(t-t
0)?t
' [7]
the variances of T increasing linearly in time:
[8]
('q~ = 2~(t-
[9]
456
to).
(66)
J. perrin, Les atomes (Alcan, Paris, 1920) p. 201. M. Sole, Z. Phys. Chem. Neue Folge 92 (1974) 1. M. Vlad, Rev. Roumaine Chim. 21 (1976) 677. N.G. van Kampen, Can. J. Phys. 39 (1961) 551; Adv. Chem. Phys. 34 (1976) 245. G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems (Wiley, New York, 1977) ch. 10. Th. de Donder. L'affinit~ (Gauthier Villars, Paris, 1936) ch. 1. M.O. Vlad, E. Segal and V.T. Popa, Physica A 127 (1984) 333. M. Vlad and M. Dragos, Rev. Roumaine Chim. 23 (1978) 1233. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.
Volume 116, number 9
14 July 1986
T H E P R O B A B I L I T Y O F R E A C T I O N E V E N T S IN A C O M P L E X C H E M I C A L S Y S T E M M.O. V L A D L Med. Bucure~ti, CSsu(a po~tal5 77-49, Bucarest, Romania
Received 19 July 1984; accepted in revised form 12 May 1986
A stochastic model corresponding to R linearly independent chemical reactions, at equilibrium, can be formulated in the system's own scale, by means of the numbers of reaction events. Within the framework of the van Kampen approximation, a general formula for the probability of reaction events may be easily derived.
The problem of determining the stochastic properties for the numbers of reaction events has been raised by Perrin [1]. In the case of m o n o molecular equilibrium A ~ B, an analytical solution of this problem was given by Solc [2]. Vlad [3] dealt with the linear reaction network A~ ~ - A j, n >/i > j > 1, through a chemical master equation. This latter approach can be extended to non-linear systems, too. The master equation that arises in this case has a rather simple appearance, but in general it is by no means easy to solve in any systematic way. However, at equilibrium, a general solution is possible, on the basis of the van K a m p e n approximation [4]. Let us consider an S - c o m p o n e n t reacting system involving R different reactions S
k7
Y'. v ~ A ; ~
Y'~ v;jA/, i=1
j = 1 . . . . . R,
(1)
where rig are stoichiometric coefficients and k ~ are the corresponding forward and backward rate constants. These reactions are linearly independent, that is, the rank of the stoichiometric matrix
II f,.j II, Lj = vi~ v,~, i = 1 . . . . . S, j = l ..... R,
F=
~),P(N, ¢ ±, t)
= E [w,.±(v, N-Vl,-,N) j;±
×P(N-T-~, , ( ..... ¢ 7 - 1 ..... ,~, t) -Wj±(V,N--*N+g)P(N,,±,t)],
(4)
where N, ~, and c ± are S- and R-dimensional vectors N=(N,.),
~=(f,j),
i = 1 . . . . . S,
j=I...R,
,-+ = ( , f ) ,
(5)
S>R,
(2)
(6)
respectively, V is the volume of the system and the transition probabilities Wj+-At, j = 1 . . . . . R are given by [5]
Wj±(V, N ~ N + ~ s
-
is equal to R: rank F = R.
as
S
i=1
k/
molecular species Ai, i = 1 . . . . . S and by c7 the numbers of reaction events Y~vi~Ai---) Y~vi~Ai, the master equation for the reactions (1), based entirely on combinatorial kinetics [5], m a y be written
)
N,(N, - 1) • • • (N, - v,~ + 1)
= vk? FI i=1
1~±
V '*
S
(3)
Denoting by N,, i = 1 . . . . . S the numbers of
= V k f I-I (N~/V) ~?~,
J = 1 ..... R.
(V)
i=1
The equilibrium conditions involve that the 453
Volume 116, number 9
PHYSICS LETTERS A
moments of N, are constant d(U,(t))/dt
~(X, n ±, t) dx d'o + dTI
d(aN,(t)aNj(t))/dt
(8)
= 0....
To describe the composition fluctuations around equilibrium, we introduce the reaction extents Zj [6]: N , ( t ) = (N,) + Y2fq~-i, )
i = 1 . . . . . S.
(9)
c + - = V ( e ± ) + V1/2~ +-, t) dxd'q + d'q , ,
= kj ±
H(
(ni) + gf,,(~,) I
i = 1 ..... S,
j=l
(10)
j=l
..... R,
(19)
a~?=~?-(~?>, (20,21)
we get:
d(x/)/dt=
d(e?)/dt
Eb/,(X,),
= ri +-,
d(~/)/dt=O,
(22,23) (24,25)
l
d(AxjAxI)/dt
Employing
j=l
..... R,
(11)
..... R.
,
i = l ..... S,
d(~'/)/dt = rj+ - 5 - '
from which, taking into account (3), we obtain
)"+'
(17)
(18)
n~ = N / V ,
ax,=x,-(x,>,
Through mediation, these equations lead to
(~.j)=O,
V ( ~ ) + V1/2X,
= V3R/2P(~=
= O,
E L i ( Z , ) = O, J
14 July 1986
= y' bj~.(AxkAxl)
as state variables ~j and ~± , the master equation (4) becomes
k
+ Eb,k(Axkaxj)
+ ajzB,,
k
3 , P = E [Wj±('~j-T- 1--+ Z j ) P ( - - , T- 1, %+
-
1)
(26)
j;_+
d(Axjdxn?)/dt
- <~(-'-J-O-T + 1 ) e ] ,
k
(27)
where d(an?a~;)/dt =
W j ± ( g T 1--+ < )
= G ~ ( ( N ) ; / , + FZ-+ (N) + FZ).
(13)
Applying to (12) the "van Kampen extensivity Ansatz" [4], i.e., assuming that the dependences = = F,j(V), c f = E l ( V ) are given by ~/
~j = ('~j) + V-a/2X 2,
e/ = (el)
6j,3.~G~,
,t, =
-
(28)
Eb,~(x3"} ~ O Y ; ' ' j ,X/ ' i
32Nit
B~aX2 ++~
T/-+ - -
ax~a~F
(14) + G/± - -
+ V I/2~y.-+ = V E ? ,
+ V-'/2rlf,
a,B=+,
and
zj = v(~+) + v ' / ~ x j ,
{? = V(¢)
+ 3i, Tt*,
= Ebjk(AxkA~F)j
(12)
(15)
where V(~';), V(ej-+) and V I / 2 x j , v l / 2 l l / are the deterministic and the fluctuating contributions to .~j and of, respectively, with
(29)
where
¢., = E ( + 1)(aC/o(~,)),
(30)
±
B t = ~ r l -+'
(31)
_+
( 5 ) , X j, ( e l ) ,
rl? - V °,
and introducing the notations 454
(16)
T/± = + r / , GI +- = rl +- ,
(32) (33)
Volume 116, number 9
PHYSICS LETTERS A
Therefore, it is easy to check that
and 8jl is the Kronecker symbol. Combining (11) and (14) it turns out that
(Xs) = 0,
(~j) = 0.
(34,35)
and then eqs. (20), (22), (30)-(33) yield AXj = Xs,
(36) s
r/ =
=
/U
= H
14 July 1986
(37,38)
i=l
Fj, = 85,,
(49)
(x,xv) = (a-'),o,
(50)
are solutions of (45) and (46). Eqs. (50) are consistent with the results of Kubo, Matsuo and Kitahara [7] or thermodynamic [8] approaches. On the other hand, the integration of (23), (25) and (27)-(29) with the initial conditions
OrJ+ f bit= E ( + 1 ) O(n,) " i;+
(51)
(xj(to)ATI~(to)) = (A*/f(t0)a~/t~(/0)) = 0 .... ~
f.=
-~I~jt,
(52)
(39)
B t = 27,,
(40)
T~± = +?,,
(41)
GI± = rt,
(42)
where
gives
( e l ( t ) ) = ~(t - to), (xj(t)A~/~(t))
Q l f ( t ) ) = 0,
= (Xj(t)~I~(t))
(53,54)
= ± d p j t ( t ) r I,
(55)
(Any(t)An~(t)) = (~y(t)~l~(t))
$2jl = E f u f i l ( n i ) i
(43)
-1 .
=
Inserting (9) and (14) into the equilibrium conditions (8) and making use of (34) gives
a,/3-
+, (56)
g'(X, r/±, t ) = [detM(t)]x/2 e x p ( - ½ X t M - ' X ) ,
d( A Nq A N~2)/ d t = E EL,jfi2, d( Z.iY't) / d t j
Jj, e, (t - to),
( 2 q r ) 3R/2
l
(57)
= vY'~ ~_~fiofi=t d ( x j x , ) / d t = O, j
I
(44)
1 (2,~) R (t - to) RI-I~,
from which, taking into account (3), we obtain
d(XjY.t)/dt = V d ( x j x , ) / d t = 0.
(44')
By means of (39), (40) and (44'), the relations turn into a simple form:
28j" t = ~.Fj, + ~.tro,
(45)
with
)
(58)
where
ll~.~2.oll] dr,
(59)
to
(46)
k
Assuming now that all (n~) and k~ are different from 0, (3), (19), (37) and (38) lead to rank 12 = R.
(--~)2
t;+ rt( t - t O ) , '
q,= IIq,jzll = f ' e x p [ - ( t - r )
F.o = y' 12~k(XkXv).
?j > 0,
× e x p ( - ½ ~~
(47,48)
X* = (X*TI+*~I-*),
M=
¢* _ fl-1 ~t
D 0~
(60) ,
(61)
-i 455
V o l u m e 116, n u m b e r 9
PHYSICS LETTERS A
D = Ila,z~l(t- to)tl,
(62)
= II ~j,6 II.
(63)
The p h y s i c a l m e a n i n g of these equations is s t r a i g h t f o r w a r d : T h e p r o b a b i l i t y d i s t r i b u t i o n for + the reaction events E v , / A i ~ Y , v ~ A t,
{i = {7 + % = V~kj~ + V1/2Bj,
(64)
14 J u l y 1986
W e m a y c o n c l u d e that the evolution of the system m a y be r e p r e s e n t e d as a s u p e r p o s i t i o n of R i n d e p e n d e n t r a n d o m flights [9]. This work has been s u p p o r t e d by C I M C Bucure~ti. The a u t h o r is grateful to a referee for p o i n t i n g out some m i s p r i n t s in the original manuscript.
is n o r m a l :
References ,@()), t ) = f ~ ( ' q
±, t ) 8 ( B - - T I + - - ) I
R
=FI
i=1
) d r / + dr/
1 V"4"rr(t- t0)F /
[1] [2] [3] [4] [5] [6]
Xexp
4(t-t
0)?t
' [7]
the variances of T increasing linearly in time:
[8]
('q~ = 2~(t-
[9]
456
to).
(66)
J. perrin, Les atomes (Alcan, Paris, 1920) p. 201. M. Sole, Z. Phys. Chem. Neue Folge 92 (1974) 1. M. Vlad, Rev. Roumaine Chim. 21 (1976) 677. N.G. van Kampen, Can. J. Phys. 39 (1961) 551; Adv. Chem. Phys. 34 (1976) 245. G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems (Wiley, New York, 1977) ch. 10. Th. de Donder. L'affinit~ (Gauthier Villars, Paris, 1936) ch. 1. M.O. Vlad, E. Segal and V.T. Popa, Physica A 127 (1984) 333. M. Vlad and M. Dragos, Rev. Roumaine Chim. 23 (1978) 1233. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.