PHYSICA
ELSEVIER
Physica A 237 (1997) 205-219
Fluctuating hydrodynamics approach to chemical reactions
I. P a g o n a b a r r a g a * , A. P 6 r e z - M a d r i d , J.M. R u b i
Department de Fisica Fonamental, Faeultat de Fisica, Universitat de Barcelona, Diagonal 647,
08028 Barcelona, Spain
Received 27 August 1996
Abstract
We have used the thermodynamical description of a chemical reaction as a diffusion process
along an internal coordinate to analyze fluctuations in the density of the constituents, which are
treated under the framework of fluctuating hydrodynamics. We then obtain a Langevin
equation for the density, as a function of the internal coordinate, whose stochastic source
statisfies a fluctuation-dissipation theorem. After contraction of the description, by means of
integration in the internal coordinate, we derive the Langevin equation for the concentration of
reactants and products as well as the statistical properties of the random source which agree
with the corresponding results obtained by means of Keizer's theory. Application of the
formalism is illustrated by considering particular cases. An extension to coupled chemical
reactions is also discussed.
1. Introduction
Non-equilibrium thermodynamics has been successfully applied to the study of
transport phenomena in different physical systems and for diverse physical situations.
The linear laws relating fluxes and forces are derived by the theory in a systematic
way, which, when employed in the balance equations permit us to formulate the
differential equations describing the evolution of the fields, characterizing the local
state of the system [1].
The study of relaxation phenomena reveals, however, the necessity to propose
a different scheme from the one used for transport processes. In fact, when considering, for example, chemical reactions, the linear law coupling the reaction rate with the
corresponding affinity implicitly restricts its validity in the case when the system is
close to the equilibrium state, since the general relationship between both quantities,
the law of mass action, is a non-linear function. One then concludes that the
* Corresponding author.
0378-4371/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved
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L Pagonabarraga et aL /Physica A 237 (1997) 205-219
206
procedure employed to analyze transport phenomena, which is satisfactorily described by linear laws, does not completely account for the kinetics of chemical
reactions for which the constitutive equations are intrinsically non-linear.
To reestablish generality in the application of the theory, one must adopt the proper
description of the kinetic process. The reaction is then viewed as a diffusion process
through a potential barrier whose minima are related to the initial and final states of
the reaction, which correspond to reactants and products, respectively [2]. One then
assigns a coordinate (referred to as the internal coordinate or degree of freedom)
whose values, ranging from the initial (reactants) to the final (products) states characterize the diferent steps of the process. In this description and following Arrhenius, one
implicitly assumes the existence of an activated intermediate state between reactants
and products [3]. From the entropy production related to the diffusion process, one
then derives the law of mass action. This picture has been applied to diverse physical
situations for which a diffusion process along an internal coordinate may be defined,
and in a general way enables one to obtain kinetic equations of the Fokker-Planck
type for the density in the space of the internal coordinate [1,4].
Our purpose in this paper is to show that, as for transport phenomena, fluctuating
hydrodynamics provides a solid scheme to describe fluctuation dynamics in systems in
which the relaxation phenomena take place. We will show that by using fluctuating
hydrodynamics in the space of the internal coordinate, it is possible to arrive at the
proper formulation of the properties of the random part of the reaction rate, from
which one analyzes the fluctuation dynamics.
To this end, we have distributed the paper in the following way. In Section 2 we
briefly review the approach of non-equilibrium thermodynamics to study chemical
reactions, contrasting the procedure used to analyse transport phenomena with the
theory of internal degrees of freedom. Section 3 is devoted to the fluctuation analysis.
In Section 4 we extend our previous results to the case of an arbitrary number of
coupled chemical reactions. Finally, in the discussion section we emphasize our main
results.
2. Non-equilibrium thermodynamics and the law of mass action
We consider the chemical reaction in which substance 1 transforms into (1 ~ 2 ) .
This reaction will be characterized by the reaction rate J and the affinity A defined
through the expression
2
A = ~
vlPi,
(1)
i=1
where #i are the chemical potentials of the different species and vl are the stoichiometric coefficients, with sign ' - ' if they correspond to reactants, and ' +' for products.
L Pagonabarraga et al. /Physica A 237 (1997) 205-219
207
Following the scheme of non-equilibrium thermodynamics, one may obtain the
entropy production [1]
1
(2)
a = - -- JA,
T
which, as usual, admits an interpretation in terms of flux-force pairs. From this
quantity one may derive the phenomenological equation
-A
J = - 1 T'
(3)
where i is a phenomenological coefficient, thus establishing a linear relationship
between the reaction rate and affinity. This linear law is only valid near equilibrium
and one then concludes that non-equilibrium thermodynamics in its standard version
is not able to reproduce the law of mass-action, basic to formulate the kinetic
equations for the different constituents. As shown in Ref. [5], the form of the entropy
production given through Eq. (2) is still valid when the system is far from equilibrium.
To avoid this limitation, and maintaining the essentials of the non-equilibrium
thermo-dynamics line of reasoning, it is possible to adopt the following scenario. The
transformation of 1 into 2 or viceversa may be viewed as a diffusion process along an
internal coordinate 7 ranging from an initial state 71 (reactants) to a final state )~2
(products), in which a particle of the activated complex, originating from the reactants
[6], crosses the potential barrier between both states. One then assumes local
equilibrium in the internal space which enables one to formulate the Gibbs equation
[1]
?2
6s-
Pt°t1 T f /2(7,t ) b n ( ? , t) d~ ,
(4)
?l
where n(~,, t) is the number density in the internal space,/2(% t) its conjugated chemical
potential, T the temperature, and
?2
Pro, = m f n( 7, t) dr,
(5)
?1
the total density, with m the mass of a molecule of the activated complex.
Additionally, the evolution of the number density is governed by the continuity
equation
On
&
~J
QT'
(6)
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L Pagonabarraga et al. /Physica A 237 (1997) 205-219
which introduces the diffusion current in the internal space J(7, t). The entropy
production related to the diffusion process then follows from Eq. (4). One obtains
a=--~
1 f J ( j - ~0#(7)
?
dT,
(7)
This expression also has the form of products of currents and thermodynamic forces
which now are related to each value of the internal coordinate. The resulting linear
laws are then given by
J(j
-
L(7) OU(7)
- - ,
T
07
(8)
where locality in the internal space has been assumed for which only fluxes and forces
corresponding to the same value of 7 are coupled. In this last expression L ( J is
a phenomenological coefficient.
Upon substitution of(8) in (6) and by using the expression for the chemical potential
valid for ideal systems
I~ = k B T In n(7, t) + ~(7),
(9)
one then obtains the diffusion equation in the internal space
~1- 07 \
~7 + nb-~7 "
(10)
Here 4~ is the potential through which the diffusion process takes place. Although in
principle we do not need to give its specific form, it is characterized by two minima at
71 and 72 related to the equilibrium states of reactants and products, respectively,
separated by an energy maximum which will be assumed large compared to k ~ T [6].
Furthermore, we have also introduced the corresponding diffusion coefficient
D = kBTb, with b being the mobility in the internal space given by b = L / n T , and
kB the Boltzmann's constant. In our subsequent analysis, the mobility and consequently the diffusion coefficient will be considered constant as a first approximation.
The diffusion current given through Eq. (8) can be alternatively written as
J ( 7 ) = _ D e _ • / k . r __c? eu/ker .
(11)
07
When the height of the barrier is large compared to thermal energy, the system
achieves a quasi-stationary state in which
J ( 7 , t ) = J(t){O(7 - 71) - 0(7 - 72)},
(12)
where J is the quasi-stationary current, assumed uniform, and O(x) is the unit step
function. Additionally, due to the height of the barrier we can also consider that
equilibrium in each potential well is reached independently; thus, the chemical
I. Pagonabarraga et al. / Physica A 237 (1997) 205-219
209
potential is uniform in these domains, and is given by
//(7) = / / ( 7 1 ) 0 ( 7 o - 7) + / / ( 7 2 ) 0 ( ~
- 7o),
(13)
where 7o is the coordinate at the maximum of the potential. By using Eq. (9) in (13),
this last relation yields
n(7, t) = na(t)e-r~)-~(~l)l/k"rO(7o -- 7) + n2(t)e-F~')-~el211/k"rO(7 -- 7o),
(14)
where nl - n(7 = 71) and n2 - n(7 = 72) are the number densities of the activated
complex at the initial and final states, respectively.
Equating (11) and (12) and integrating over 7 one then obtains the law of massaction [1]
d = kBl(1
-- eA/ksT),
(15)
where 1 is related to the diffusion coefficient D through the expression
D exp (//1/knT)
l = k , S~ e x p ( ~ ( 7 ) / k s V ) d T '
(16)
with//1 =//(71) being the chemical potential of the first species (reactant). Moreover,
/ / ( 7 2 ) - / / ( 7 1 ) = / / 2 - //1 = A, with //2 = / / ( 7 2 ) being the chemical potential of the
product, what has been used to obtain Eq. (15). Near equilibrium, l becomes time
independent
1 --,
Dn~q
=-- l,
kB S~ exp[~(7) -- q~(71)]/kBTd7
(17)
where n~q corresponds to the number density of reactant at equilibrium. Thus, and due
to the small size of A / k ~ T , expression (15) reduces to the linear law (3). It is worth
pointing out that by substituting Eqs. (12) and (13) into (7) we obtain the expression of
the entropy production given in (2) although the law of mass action (15) is satisfied in
agreement with the corresponding result in Ref. [5].
Thus, thinking of the chemical reaction as a diffusion process through an energy
barrier has allowed us to write the linear law (8) relating the local reaction rate and the
"gradient" of the chemical potential, the thermodynamic force in the internal space.
Then, integration of this equation over the internal coordinate has enabled us to
obtain a nonlinear relationship between the stationary current and the affinity even
though a linear relation between the flux and the thermodynamic force is satisfied in
the internal space, something that can not be achieved in the framework of the classic
nonequilibrium thermodynamics.
To obtain the kinetic equations for the reactant and product concentrations, we will
start from the balance equation for the number density (6). By substituting in this
equation the stationary current given in Eq. (12) and employing (14) one has
d
d
d~ rt(71) = -- ~ n(72) = -- J '
(18)
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I. Pagonabarraga et al. /Physica A 237 (1997) 205-219
where we have assumed that the exponential appearing in (14) approximates to delta
functions. Thus, using (15), (16) and (18) one can derive the rate equations for the two
components
dnl
dt
dn2
--dt
k-n2 - k+nl,
(19)
where the rate constants for the forward ( + ) and reverse ( - ) reactions are defined by
k+
=
D exp(~(71)/kBT )
kBl
= -I~ exp(CI)(7)/k,T)dy
nl
(20)
and
k_
D exp(~(72)/knT)
f ~ exp(O(),)/kaT)d7
kBl
n2
(21)
respectively. The ratio between both constants defines the equilibrium constant of the
reaction
K - ~-_ = exp
(4(72) - ~(71)) ,
(22)
which in the ideal case we have studied depends only on temperature through the
quantity k~T.
The law of mass action Eq. (15) and Eqs. (16)-(22), can also be obtained for the more
"
- " 2i=,+
"
general chemical reaction Yi=
1 ( - vi)Bi.--1 viBi, but the affinity should be
deduced in this case, since now the concentration of the activated complex is not equal
to that of the reactant and the product. According to this stoichiometric equation,
because the activated complex is formed when the appropriate number of molecules
join, the densities of the activated complex at the reactant and product states are
(-v,,
n(T1) = f i nB,
and
fi
n(72) =
i=1
v,
nB,.
(23)
i=n+l
Therefore, in this general case, we can still relate the affinity of the reaction to the
difference of the chemical potential of the activated complex at the reactant and
product states. Hence,
m+n
/~(72) -/~(71) = ~ villi = A .
(24)
i=1
According to the stoichiometric equation corresponding to the chemical reaction
the number density of the ith species is nn, = vin(ya), for i ~< n, and nn, = vin(72), if
n < i ~< m. Thus, by using Eq. (19) the rate equations are written in this case as
dtnnj=vJ
k+
n~] v ' ) - k _
i=1
n,, ; 1 < ~ j ~ n + m .
i=n+l
(25)
Z Pagonabarraga et al./Physica A 237 (1997) 205-219
211
Finally, it is interesting pointing out that our analysis can be extended to the case of
a non-ideal unimolecular chemical reaction. In this case, the chemical potential
contains an activity coefficient f(7), and is given by
i~ = knTlnfn + q~.
(26)
The diffusion current is then written as
I/ O
J(y,t)=-D~-~nne"
/K.TX~ - 1 O
)
~ e "/K"r,
(27)
where we have defined the diffusion coefficient
L c31~_ kBT ( L ) (
D = T c3n
1+
Ol n f ' ]
c31nn }
(28)
which reduces to our previous definition when f = 1. After integration in the internal
coordinate assuming constancy of D an equation similar to (15) is obtained but now
the phenomenological coefficient is modified to account for the non-ideality of the
system
D exp (I~x/kRT)
I = kB ~ (O/On)exp(l~(y)/kBT)dT"
(29)
We can also derive the corresponding kinetic equations for the reactant and product
densities, which are similar to their corresponding related equations given through
Eqs. (18) and (19). The rate constants are in this case
Df2 exp (cI).~/kBT)
k_ = ~ (8/8n)exp(l~(7)/k.r)d7
(30)
and
De1 exp(~rffkBT)
0n
S,~1(8/)exp(~(7)/knT)d 7'
k+ = ~2
(31)
and the equilibrium constant is given by
flK,
K . . . . ideal = f 2
(32)
where fl - f ( 7 =- 71) and f2 = f ( 7 = 72). The previous results for the ideal case are
then recovered by setting f~ =f2 = 1.
3. Fluctuating hydrodynamics in the internal space
In this section, we will study the dynamics of the fluctuations in the number density
of reactants and products around a given reference state. In order to take into account
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I. Pagonabarraga et al. / Physica A 237 (1997) 205-219
the presence of fluctuations in the diffusion process along the internal coordinate, we
will adopt the framework of fluctuating hydrodynamics which has been fruitfully
applied to the analysis of fluctuations in the case of the transport phenomena [7]. We
then consider that the diffusion current in the internal space splits up into systematic
(s) and random (r) contributions in the form
J(7, t) = JS(7, t) + Jr(7, t).
(33)
The systematic part is given by an expression equivalent to Eq. (11), while the random
part constitutes a gaussian stochastic process with zero mean and fluctuation-dissipation theorem given by [7]
<jr(y, t)J'(7', t')) = 2kBL6(7 - 7')b(t - t') = 2Dn6(7 - 7')6(t - t'),
(34)
with D the diffusion coefficient introduced in the previous section, and n(~,, t) the
average number density in the quasi-stationary state given by Eq. (14). In this state,
the time scale related to changes in density is larger than the one for fluctuations of
such a quantity. Then, for all practical purposes we can assume that the process is
quasi-stationary [8].
When employing the decomposition (33) in the continuity equation (6), we obtain
the stochastic differential equation describing the evolution of the fluctuations of the
number density, 6n, around the average number density
~--
if7 JS(7, t) -
J'(Y,t)=-~TLDe
-~7(6ne*/k"T)---~TJr(y,t).
(35)
Note at this point that since the diffusion coefficient has been assumed constant
Eq. (35) is a linear stochastic differential equation. Here, and for simplicity's sake, we
will consider the unimolecular reaction, although the generalization of the theroy to
several reactions is straight forward.
We will now assume that, due to the height of the energy barrier, the quasistationary regime applies to the whole mass flux, thus, we can use Eq. (12) for the total
diffusion flux given in Eq. (33). We can then proceed along the same lines of the
previous section. In particular, if we integrate along the internal coordinate, we can
obtain the corresponding expression for the total homogeneous flux. After using
Eqs. (12), (14) and (33) we achieve
J(t) = k i l o
-
-
e A/k"T) -k- Jr(t),
(36)
where we have defined the homogeneous fluctuating current as
)'2
1
Jr(t) -
e /kBT d y
.t e*/kBT jr (7' t) dT .
"/1
(37)
L Pagonabarraga et aL/Physica A 237 (1997) 205-219
213
This current J'(t) can be interpreted as the weighted average of the diffusion current in
the internal space which takes into account that in the stationary regime the flux is
controlled by the behaviour of the system around the maximum.
The stochastic properties of Jr(t) then follow from those of J'(7, t). Moreover, in
view of Eqs. (14), (34) and (37) one obtains the second moment of the fluctuating
current,
(J~(t)J'(t')}
7o
_ 2
~2
(k+nl f e~/knT dT + k_n2 f e~/kBT dT)6(t _ t,)
)~1
(38)
YO
Since J"(7, t) is assumed gaussian, this equation together with the result ( j r } = 0
completely determines the properties of the homogeneous random flux. Since the
energy barrier is very high, the dominant contribution to the integrals in Eq. (38)
occurs at to. Therefore, Eq. (38) leads to
( J r ( t ) J r ( t ' ) } = (k+nl + k_nz)b(t - t'),
(39)
which coincides with the expression obtained by Keizer using a different approach
[9]. For the particular case of equilibrium, according to Eqs. (16), (19) and (20) one
has, k_ ne2q = k +ne~q = kB l, the first equality corresponding to detailed balance. Consequently, (39) becomes, in this case,
{ Jr(t)Jr(t') ) = 2 k j 6 ( t - t')
(40)
which constitutes the formulation of the fluctuation-dissipation theorem consistent
with the linear law (3) which was already obtained by Zwanzig [10]. This equation
also leads to the expression of the reaction constants in terms of density correlation
functions (see in this context Ref. [11] and references quoted therein).
We can now derive the corresponding Langewin equation for the number density of
reactants and products. From Eqs. (18) and (36), we may write
d6nl
- k-~n2 - k+3nl - j r
dt
(41)
d6n2
- k+Jnl - k-•n2 + J ' ,
dt
(42)
and
where we have used Eqs. (16), (20) and (2l).
As an example of the application of the formalism we have developed, we will
proceed to use expression (39) for different cases:
(i) For the dimerization reaction 2B,~-B2, the corresponding expression for the
second moment of the random current is
( J'(t)J~(t') } = (k+n~ + k_nB~)J(t -- t').
(43)
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I. Pagonabarraga et al. /Physica A 237 (1997) 205-219
Note that now Eq. (23) has been used according to which n~ = n2 since, as it is written
in the stoichiometric equation of the reaction, two molecules of B are necessary to
form one of the activated complex.
(ii) For the multicomponent chemical reactions, y,7= 1 (-vi)Bi ~ Y~i~, + 1 viBi discussed previously, one obtains by substitution of Eq. (23) in (39)
( J ' ( t ) J ' ( t ' ) ) = ( k+ f l
(-v~) +
riB,
~l
k_
i=l
~)6(t_t,)
riB,
.
(44)
i=n+l
(iii) For non-ideal systems we obtain an equation similar to (39) but containing the
appropriate expression for the reaction constants, as given in Eqs. (30) and (31).
4. Extension to coupled chemical reactions
In this section we will proceed to extend the formalism we have developed to the
case of coupled chemical reactions, usually found in many situations of interest. Let us
now consider all the possible unimolecular chemical reactions occurring among the
n species Bi (i = 1. . . . . n). From the total number of elementary reactions ½n(n - 1),
(n - 1) are independent and will be taken as B1 , ~ B k ( k = 2 . . . . . n) [1].
For each elementary reaction, we can then define its corresponding reaction rate
J, and affinity A, with e = 1, ..., ½n(n - 1), in terms of which the entropy production
reads
1
T
a-
n(n- 1)/2
~
J,A, .
(45)
~=1
These currents and affinities are related through the law of mass action for each
a-reaction (see the appendix for details about its derivation)
J, = kBl,(1
-
e'4"/knT),
1 <~ a <~ ½n(n -
1),
(46)
where
A: -=/t=(7~2) -- P,(7=1),
(47)
D~ exp(#~(~A1)/kBT)
l~ -- kn f~2 exp(q) (7~)/kBT)d;~ .
(48)
and
JT~t
Here, 7,1 and 3)~2represent the position of the minima for the ~th reaction, and D, is
the corresponding diffusion coefficient. The rate equations are now written as
d
dt
ni~ =
vi~J~,
1 <. i <. n, and 1 ~< a ~< ½n(n - 1),
(49)
where ni, is the number density of the ith species participating in the ~th reaction, and
v~, the corresponding stoichiometric coefficient.
L Pagonabarraga et al./Physica A 237 (1997) 205 219
215
From Eq. (49) we obtain the rate of change of the number density of the ith species
ni = ~nt_-n I 1)/2 niT, that reads
dni
dt
1)/2
= n(n
~
n(n-1)/2
viTS7 =
~
7=1
kBvi~l~(1 -- ea'/k"T),
i = 1. . . . . n.
(50)
~=1
This equation can be rewritten as follows
d n i _ 1 ~ .~n 11/2
dt
T ""
~,
viTl°(A,)v~,Ftj,
j=l
i = 1..... n,
(51)
~=1
where the expression for the affinity An = Y~= 1 v~,p~ has been used, /~i being the
chemical potential of the ith species, and
=
(.1 - eA /k"T
L/--~B
T J'
(52)
By taking into account the mass conservation law in each reaction, ~ = 1 v i~ = O,
Eq. (51) leads to the set of phenomenological equations
n
dni_ 1 ~=zLij({A,})(pj_pl),
dt
T j=
i = 2 ..... n,
(53)
in which the terms dni/dt denote a set of (n - 1) independent fluxes and (pj - p l ) / T
the corresponding thermodynamic forces, while the phenomenological coefficients are
defined through
n ( n - 1)/2
L,j({A,}) =
~
o
vi~l~(A,)vj~,
i,j = 2 . . . . . n.
(54)
7=1
These coefficients, in number of (n - 1)2, satisfy the Onsager relations
L~j({A~})=Lj,({A~}),
i,j=Z,...,n.
(55)
In the representation of the independent fluxes and forces, the entropy production (45)
can be written as
1 ~
dni
a = ~ ,~z ~ - (#' - #1).
(56)
Near equalibrium, A , / k B T ~ 1, for ~-- 1, .... ~1 n ( n - 1), l°(A,) ~ r 7 and
Lij({AT}) ~ L ~ j , which is a constant coefficient. Thus Eq. (53) becomes a linear
relationship between fluxes and forces.
By using the following expression for the chemical potentials near equilibrium
I~i = k n T ln(nl/n eq) + #~q,
i = 1. . . . . n,
(57)
it is possible to write Eq. (53) as a master equation
dt
/'JinJ -j~l
)~ijrli'
j=l
j ¢ i,
(58)
I. Pagonabarraga et al. / Physica A 237 (1997) 205-219
216
where the rate constants are 21j = Lij/n~ q and 2j~ = Lji/ny q. The first term on the
right-hand side in (58) expresses the gain in the amount of the ith component at the
expense of the rest of species and the second the loss in this component. It is obvious,
by the definition of the rate constants and due to the Onsager relations, that the
principle of detailed balance is satisfied, 2jin~q = 2ijn eq.
Fluctuations are again incorporated by introducing for each reaction random
currents, J~(?,, t), in the internal space. These random currents have a zero mean and
second moment given by
(J~(?~, t)J~(?¢, t') ) = 2D,n,6(7, - 7a)3(t - t').
(59)
In the same way as we did previously, we then proceed to define the random current
7~2
S::: e¢~,/kBT
via d? ~ f e~'/k"Tj[,(7,,t)d,~
J~(t) = ~ vi~J r =
(60)
ct
which also has zero mean and a second moment given by
2:~2
(Jr(t)Ji(t')) =
~
• /k.T
~i,2
I,':; e"
d?otl]~flle ~'/a"T dyp
eG/knTd7~
x f e*~/k"Td7a(Y~(?~,t)Yj(?p,t')),
i,j = 1 , . . . , n .
(61)
When employing Eq. (59) this expression transforms into
(d~(t)Jff(t')) = ~ (v,~k~vi~) [exp { #~(y~)/kBT }
+ exp{l~(7~z)/knT}] 6(t - t'),
(62)
with
k,
L.'2
D,
(63)
The result (62) is also equivalent to a corresponding expression given in Ref. [9].
5. Discussion
In this paper we have presented a formalism, based on the introduction of internal
degrees of freedom [1], which permits us to analyze the kinetics of single and coupled
chemical reactions in their deterministic and stochastic versions and for ideal and
non-ideal systems. In agreement with Kramers' picture [2], the kinetic mechanism for
the reaction is described by means of a diffusion process along an internal coordinate
I. Pagonabarraga et al. / Physica A 237 (1997) 205-219
217
whose initial and final values correspond to the minima of a potential separated by an
energy barrier, and related to the reactants and products.
Non-equilibrium thermodynamics can then be applied to describe the diffusion
process in the internal space. After postulating a Gibbs equation, in which one
includes the number density in the internal space as an independent variable, one
obtains the entropy production and from it the corresponding phenomenological
equation. When the energy barrier is high compared to thermal energy, a quasistationary regime is achieved. Then integrating the relationship between current and
thermodynamic forces along the internal coordinate, we obtain the law of mass action
governing chemical kinetics. It must be emphasized that the formulation of nonequilibrium thermodynamics in its classic version (as for the case of transport
phenomena) is not able to derive the non-linear relationship between reaction rate
and affinity.
We have also derived the law of mass action for a general multicomponent chemical
reaction and for a non-ideal mixture, generalizing previous derivations of this law by
using the theory of internal degrees of freedom [1].
Our main objective in this paper has been to show that the fluctuation analysis can
be performed by using fluctuating hydrodynamics in the space of the internal variable.
Accordingly, one decomposes the corresponding diffusion current into systematic and
random contributions and formulates a fluctuation-dissipation theorem in which the
noise strength is proportional to the phenomenological coefficient coupling the
current and its conjugated force in the internal space. Integration of the random
current correlation function in the internal coordinate yields the same expression for
the random current correlation function as the one found by Keizer [9] on the basis of
a statistical mechanical approach.
Our derivation of the law of mass action is in some sense equivalent to the
procedure used in obtaining reaction rate equations by means of addiabatic elimination of fast variables [11,12]. In fact, if we define
6N(7, t) -
f dT' 6n(7', t)
(64)
and integrate Eq. (35) we obtain
0
tc~ ON(?) = -- JS(7 ) -- jr(y),
(65)
where JS(7 ) is given by Eq. (15). Now we define the integral operator ~(7) through
~2
1
f
(66)
I. Pagonabarraga et al. / P h y s i c a A 237 (1997) 2 0 5 - 2 1 9
218
with ~k(y)a function in the internal space. This operator has the property ~2 = 1, and
is consequently a projector. By applying this operator to both sides of Eq. (65) one
obtains
d6nl
- dt
--
JS(t)
--
(67)
J'(t),
where JS(t) and J'(t) are given by Eqs. (15) and (37), respectively. Moreover, in the
derivation of (67) we have used the fluctuating version of Eq. (14).
We have indicated how to proceed in order to extend the theory to the case of an
arbitrary number of coupled chemical reactions. Finally, it is worth pointing out that
our analysis in general could be applied to the case of thermally activated processes.
Acknowledgements
It is a pleasure to acknowledge Prof. P. Mazur and G. Gomila for fruitful discussions. This work has been supported by DGICYT of the Spanish Government
under grant PB92-0859.
Appendix A
Our purpose in this appendix is to derive the phenomenological equations and the
expression of the kinetic coefficients in the case of coupled chemical reactions following the procedure indicated in Section 2 [13]. To this end, by using the n-fluid model
[14], we consider the whole set of chemical reactions as if it were constituted by
½n(n - 1) different "fluids". The total change in the entropy is then given by the sum of
the corresponding changes related to each reaction and is expressed through the
Gibbs equation
1)/2 1
1 n(n-
68-
T
/~
~
- - J l~(7~)fn,(7,)dT,,
=1
Ptot
(68)
where
n(n
1)/2
Ptot =
m,
f
n~(7~)dT~.
(69)
~=i
As before, one then proposes the continuity equations for the evolution of the
number density of each reaction
&n,(7,)-
a7 J,;
~=l,...,½n(n--1).
(70)
1. Pagonabarraga et al. / Physica A 237 (1997) 205 219
219
F r o m Eqs. (68) and (70) one obtains the entropy p r o d u c t i o n
a -
l nCn~)/2f
0
T ~:,
J,(7,) ~7~ #,(7,)dY~,
(71)
from which we infer the linear laws
J~(7,) = - n~ ~
P,(7,) •
(72)
By using the expression of the chemical potential for the ~th reaction,
P,(7,) = k B T In n,(7~) + 4~,(7,),
(73)
in Eq. (72) one obtains
J~(7~) = -
D, e-¢'/k"T 0 eu,/k. T
07~
(74)
As before, quasi-stationarity in each kinetic process leads to
J(7~, t) = J,(t){O(7 , - 7,,) -- 0(7, -- 7,2)},
(75)
/~(7~) = #,(7,,)0(7,o - 7,) + #,(7,2)0(7~ - 7,o),
(76)
and
where 7,o is the coordinate at the position of the m a x i m u m of the ~th reaction. Thus
by equating Eqs. (74) and (75), and after integrating in the internal coordinate we
obtain the mass action law (46). O n the other hand, by substituting Eqs. (75) and (76)
in (71) one obtains Eq. (45).
References
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