long-time tails in the kinetics of reversible bimolecular reactions

VOLUME 86, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 29 JANUARY 2001

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Long-Time Tails in the Kinetics of Reversible Bimolecular Reactions

Irina V. Gopich,1,* Alexander A. Ovchinnikov,2,† and Attila Szabo1

1Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases,National Institutes of Health, Bethesda, Maryland 20892

2Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany(Received 31 May 2000)

A new approach is developed to study the relaxation of concentrations to equilibrium in reversiblebimolecular reactions. For A 1 B % C, a general analytic expression is derived for the amplitude ofthe power law t2d2 asymptotics for arbitrary diffusion coefficients and concentrations of the reactants.Our formalism is based on the analysis of the time correlation functions describing the equilibriumfluctuations of the concentrations. This powerful and simple procedure can be readily used to studyother bimolecular reactions such as A 1 B % C 1 D.

DOI: 10.1103/PhysRevLett.86.922 PACS numbers: 05.70.Ln

To determine the kinetics of diffusion-influenced re-versible reactions, such as A 1 B % C, over the entiretime range, one must solve a many body problem. Thisproblem simplifies considerably (but not sufficiently toyield an exact solution) when the B’s are in great excessand when two of the species are static. When A and Care static, based on an approximate microscopic theoryand on the results of simulations, it has been conjec-tured [1] that the normalized deviations of the concentra-tions from their equilibrium values decay asymptoticallyas Ke1 1 KeBe24pDBt2d2, where d is the dimen-sion, Ke is the equilibrium constant, and DB and Be arethe diffusion constant and the concentration of the speciesin excess. This result has recently been rigorously provedfor d 1, 3 by using a sophisticated many body perturba-tion technique [2]. This technique was previously used totreat the case DB DC 0, where it was found that theconcentrations relax as Ke1 1 KeBe124pDAt232 inthree dimensions [3]. The structure of these results sug-gests that, at least for pseudo-first-order reactions, a gen-eral simple result for the amplitude of the power lawrelaxation might exist.

The fact that, in bimolecular reversible reactions, theconcentrations approach equilibrium not exponentially butrather as t2d2 was first discovered by using simple physi-cal arguments involving spatial concentration fluctuations[4]. The basic idea is that, since the initial and equi-librium spatial fluctuations are different, a redistributionof particles through diffusion must occur before equilib-rium can be reached. Since the relaxation due to reactionproceeds exponentially, the final approach to equilibriumis determined by the diffusion of the particles. The powerlaw decay was subsequently obtained in various ways[5,6]. However, it was only recently that fluctuation the-ory yielded rigorous expressions for the amplitude of therelaxation in a few special cases [7]. Unfortunately, thisapproach, based on field theory techniques, does not ap-pear to be applicable to static A or B particles [7]. Thusat present there is no overlap between results obtained

by microscopic and what may be called hydrodynamicapproaches.

The purpose of this paper is to present the complete so-lution of the asymptotics of A 1 B % C for uniform ran-dom initial concentrations. Our starting point is a set ofreaction-diffusion equations that describes concentrationfluctuations at equilibrium. These are obtained by addingdiffusive terms and random forces to the conventional non-linear rate equations of chemical kinetics [8]. We thencalculate equilibrium time correlation functions of the con-centrations in Fourier space. To get the relaxation of theconcentrations for the system removed from equilibrium,we exploit the Onsager regression hypothesis that equilib-rium concentration fluctuations and small deviations of thebulk concentrations from equilibrium have the same relax-ation law. In the standard theory of light scattering bychemically reacting systems, one linearizes the reaction-diffusion equations about equilibrium [8–10]. That is,one neglects the term that is quadratic in the deviationsof the concentrations from their equilibrium values. Herewe treat this quadratic nonlinearity as a small perturbationin the simplest possible way.

This nonlinearity gives rise to the t2d2 relaxation of theconcentrations to equilibrium just like it is the nonlinearityin the Navier-Stokes equation that is responsible for thelong-time tail of the velocity autocorrelation function [11].However, these problems are different and, in fact, theproblem considered below is a much simpler example ofhow a nonlinearity can alter the long-time dynamics of astochastic system.

Let Ar, t, Br, t, and Cr, t be the local random con-centrations of A, B, and C particles. When the particlesare weakly interacting (i.e., an ideal solution) the con-centration fluctuations due to diffusion in the presence ofchemical reaction are described by the following system ofequations:

≠

≠tAr, t DA=2A 2 kaAB 1 kdC 1 fAr, t , (1a)


≠

≠tBr, t DB=2B 2 kaAB 1 kdC 1 fBr, t , (1b)

≠

≠tCr, t DC=2C 1 kaAB 2 kdC 1 fCr, t . (1c)

The first terms on the right-hand side of Eqs. (1a)–(1c) de-scribe the diffusion of A, B, and C particles with diffusioncoefficients DA, DB, and DC . The next two terms describethe chemical reaction with the association ka and dissocia-tion kd rate constants. Finally, the last terms representnoise in the system. For the present purposes, we needonly to assume that they have zero mean and are statisti-cally uncorrelated with the initial concentrations. The ini-tial concentrations are the equilibrium ones and when thesystem is infinite the distribution of particles is Poissonian[12] with mean concentrations Ae, Be, and Ce. They obeythe mass action law KeAeBe Ce, where Ke kakd isthe equilibrium constant.

Let the Fourier transform of the concentration devia-tions from equilibrium, cq, t, be a vector with thecomponents c1q, t

Rdr expiqr Ar, t 2 Ae, etc.

Equations (1a)–(1c) can then be rewritten as≠

≠tcq, t 2Kqc 2 kay

Z dq0

2pd

3 c1q 2 q0, tc2q0, t 1 f q, t . (2)

Here y is a column vector with components 1, 1, 21,and f q, t is a vector with the random forces fA, fB, andfC in q space as components. Matrix Kq contains bothdiffusion and reaction terms:

Kq

0B@ kaBe kaAe 2kd

kaBe kaAe 2kd

2kaBe 2kaAe kd

1CA

1 q2

0B@ DA 0 0

0 DB 00 0 DC

1CA . (3)

We begin by calculating the 3 3 3 matrix of concentra-tion correlation functions with elements cjq, tc

l q0, 0,where · · · denotes the ensemble averaging over the noise

and equilibrium (Poissonian) initial conditions. It is con-venient to define the relaxation matrix Rq, t as

cq, tcyq0, 0 Rq, t cq, 0cyq0, 0 . (4)

The relaxation of the bulk concentrations At, Bt, andCt from a uniform (random) initial condition that is closeto equilibrium is then obtained as follows. Assuming thatdecay of the bulk concentrations to equilibrium is the sameas the decay of equilibrium concentration fluctuations,we have

CCC t Rq 0, tCCC 0 , (5)

where CCC t is a vector with the components C1t At 2 Ae, C2t Bt 2 Be, and C3t Ct 2 Ce.Since the total number of particles is conserved during thereaction, C1t C2t 2C3t and hence CCC t equalsy times some function of time. Consequently, the concen-tration relaxation function defined as Rt CitCi0is the same for all i and is the eigenvalue of the relaxationmatrix R at q 0 corresponding to eigenvector y:

R0, ty Rty . (6)

In the standard theory of light scattering [8–10] the non-linear term in Eq. (2) is neglected because it is quadratic inthe deviations of the concentrations from their equilibriumvalues. Rq, t is then obtained by first solving for cq, t,multiplying the result on the left by cyq0, 0, and finallytaking the equilibrium average by using the fact that thenoise and the initial concentration fluctuations are uncorre-lated. In this way, one finds that Rq, t exp2Kqt.Since K0y k0y where k0 kd 1 kaAe 1 Be, itfollows from Eq. (6) that Rt exp2k0t which is thestandard result of ordinary chemical kinetics.

To go beyond this theory, we treat the nonlinearity asa small perturbation and solve Eq. (2) iteratively in lowestorder. Specifically, we first formally solve this equation forcq, t, approximating c1 and c2 in the nonlinear term bythe solution of the linearized problem. We then multiplythe result by cyq0, 0 and take the equilibrium average.By using the statistical independence of the noise and theinitial conditions, we obtain

cq, tcyq0, 0 e2Kqtcq, 0cyq0, 0

2 e2Kqt kayZ dq00

2pde2Kq2q00tcq 2 q00, 01e2Kq00tcq00, 02cyq0, 0 , (7)

where “” denotes time convolution ft gt Rt

0 ft 2 t0gt0 dt0. Note that the random forces do not contributeto this order and thus the same result would be obtained if these forces were averaged out right from the beginning. Theproperties of the correlation functions for a uniform (Poisson) equilibrium distribution are

ciq1, 0cj q2, 0 dij2pddq1 2 q2Ce

i , (8a)ciq1, 0cjq2, 0c

l q3, 0 dijdil2pddq1 1 q2 2 q3Cei , (8b)

where Ce1 Ae, Ce

2 Be, and Ce3 Ce. Using these in Eq. (7), we find [see Eq. (4)]

Rq, t e2Kq,t 2 kae2Kq,ty uq, t , (9)

where row vector uq, t has the components

ujq, t Z dq0

2pde2Kq2q0t1je2Kq0t2j . (10)

The Fourier transform of Rq, t in the time domain determines the light scattering spectrum [8].

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The concentration relaxation function, which is the eigenvalue of R0, t corresponding to the eigenvector y, is

Rt e2k0t 2 ka

Z t

0dt e2k0t2t

3Xj1

yj

Z dq0

2pde2Kq0t1je2Kq0t2j . (11)

To find its asymptotic behavior, it is convenient to represent the exponentials in Eq. (11) as a sum over eigenvalues liqof matrix Kq: exp2Kqt

P3i1 Xi exp2liqt, where Xi are built from the appropriate eigenvectors. Such a

decomposition can be used to define the “normal modes” of the linearized reaction-diffusion system [8,10]. Two of threemodes live infinitely long as q ! 0 (hydrodynamic modes) while the third one has a finite lifetime. In the q ! 0 limit,the eigenvalues of Kq are l1,2 q2D1,2 and l3 k0 1 q2D3, where Di are given by

D1,2

∑XkaDb 1 Dg 6

qXk2

aDb 2 Dg2 1 2kakbDg 2 Da Dg 2 Db∏ ¡

2k0 , (12a)

D3 3X

a1

kaDak0 . (12b)

Here ka and Da are defined by the diagonal elements of Kq in Eq. (3) as Kaaq ka 1 Daq2. The sums in Eq. (12a)are taken over permutations.

At long times only small values of q contribute to the integral in Eq. (11). Using the spectral representation ofexp2Kqt and evaluating the integrals, we find that, as t ! `,

Rt Ke

1 1 KeAe 1 Be

Ωf2, 2 1 g21

8pD1td2 1f1, 1 1 g22

8pD2td2 2f1, 2 1 f2, 1 1 2g1g2

4pD1 1 D2td2

æ, (13)

where fi, j D1 2 D22 bDi 1 D3 2 2DA 1 DA 2 Di aDj 1 D3 2 2DB 1 DB 2 Dj, gi D1 2 D2 pab Di 2 DC, and a KeAe1 1 KeAe 1 Be21, b aBeAe. Note that the coefficients are determined solely

by the diffusion constants, the equilibrium constant, and the equilibrium (or, equivalently, the initial) concentrations. Theasymptotics consist of three terms that correspond to the two hydrodynamic modes and their cross term. Let us considersome special cases where this result simplifies.

(a) Pseudo-first-order reaction.—As Ae ! 0, the first two terms in Eq. (13) vanish, so the relaxation function becomes

Rt Ke

1 1 KeBe2 4pDB 1 D t2d2, (14)

where D DA 1 KeBeDC1 1 KeBe. This result is surprisingly simple: DB 1 D is just the average of the rela-tive diffusion coefficients, DA 1 DB and DC 1 DB, weighted by the fraction of the time the system is in the A [i.e.,1 1 KeBe21] and C [i.e., KeBe1 1 KeBe21] states. This relaxation function reduces to the known exact results forthe reversible target problem DA DC 0 [1,2] and to the reversible trapping problem DB DC 0 [3].

(b) Two equal diffusion coefficients: DB DC .—The relaxation function now contains two nonzero terms:

Rt 2K3

e AeBe

1 1 KeAe 1 Be3 8pD t2d2 1Ke

1 1 KeAe 1 Be2 4pDB 1 D t2d2, (15)

where now the effective diffusion coefficient is D 1 1 KeAeDA 1 KeBeDC1 1 KeAe 1 Be. The last termin Eq. (15) is the generalization of the previous case, reducing to it in the limit Ae ! 0 or Be ! 0. The first term,however, does not appear in the pseudo-first-order case since it vanishes as Ae ! 0. When DA DB DC , Eq. (15)reduces to the result first obtained by Rey and Cardy [7].

(c) Two equal diffusion coefficients: DA DB.—All three terms of the asymptotics are manifested in this case:

Rt 2KeAeBe

Ae 1 Be21 1 KeAe 1 Be8pDBt2d2 1

2KeAeBe

Ae 1 Be21 1 KeAe 1 Be3 8pD t2d2

1KeAe 2 Be2

Ae 1 Be21 1 KeAe 1 Be2 4pDB 1 D t2d2 (16)

with D DA 1 KeAe 1 BeDC1 1 KeAe 1 Be.This result coincides with the asymptotics obtained inRef. [7] only if we specialize to the case that Ae Be.

We have presented a simple unified derivation ofthe long-time asymptotics of the reversible reactionA 1 B % C in the framework of fluctuation theory. Thefact that our general expression [Eq. (13)] reduces tofew previously known exact results is of course only a

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necessary condition for it to be exact in general. However,because these exact results span different regions of pa-rameter space and were obtained by completely differentand nonoverlaping methods, we fully expect that our resultis exact for uniform random initial conditions in ideal(i.e., the particles are not strongly interacting) solutions.Our procedure can be readily used to treat more complex


bimolecular reactions. (Needless to say, this procedureworks for simpler reactions such as A 1 A % A2, wherewe obtain the result given in Ref. [7].) The asymptoticswill be a power law for any reversible chemical reactionfor which (i) the rate equations are nonlinear and (ii) atleast one linear combination of the concentrations doesnot change with time. This latter requirement meansthat the reaction matrix K0 must have at least one zeroeigenvalue. This is necessary for the occurrence of a slowhydrodynamic mode (i.e., an eigenvalue that vanishes asq2 as q ! 0) which, when coupled by the nonlinearity,gives rise to the long-time tail.

We conclude by presenting a few results forA 1 B % C 1 D. In this case, there are three long-livedmodes and, in general, the asymptotics consist of sixterms. In the pseudo-first-order limit (Ae ! 0, Ce ! 0),one can show that

Rt KeBe

KeBe 1 De2 4pDD 1 D t2d2

1KeDe

KeBe 1 De2 4pDB 1 D t2d2 (17)

with D DeDA 1 KeBeDCKeBe 1 De and Ke CeDeAeBe. When all of the diffusion coefficients arethe same,

Rt ∑

KeAe 1 Be 1 Ce 1 DeKeAe 1 Be 1 Ce 1 De2

12Ke 2 12KeAeBe

KeAe 1 Be 1 Ce 1 De3

∏8pDt2d2.

(18)

The general result is too complex to be presented here, butno new ideas or techniques are involved in its derivation.

We are grateful to A. Berezhkovskii for helpfuldiscussions.

*On leave from the Institute of Chemical Kinetics andCombustion SB RAS, Novosibirsk 630090, Russia.

†On leave from the Institute of Chemical Physics RAS,Moscow 117334, Russia.

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long-time tails in the kinetics of reversible bimolecular reactions

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