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PHYSICAL REVIEW A VOLUME 43, NUMBER 10 15 MAY 1991
Symmetry between diffusion-limited coagulation and annihilation
Jian-Cheng LinDepartment ofPhysics, Clarkson Uniuersity, Potsdam, ¹wYork 13676
(Received 10 December 1990)
In homogeneous systems, symmetry is found to exist between the hierarchies of kinetic equationsof multipoint joint probability density functions of two diffusion-limited reactions A + A ~A(coagulation) and A + A ~inert (annihilation) by rescaling their multipoint joint density functions.Given one set of the joint density functions, this symmetry allows the solution of the others. A gen-eral class of diffusion-limited reactions, which are incomplete reactions between coagulation (with
probability p) and annihilation (with probability 1 —p), are found to share the same scaling symme-
try.
Recently much interest has been focused on diffusion-lirnited reactions in physical and chemical systems. '
In diffusion-limited reactions, particles are transported bydiffusion. While diffusing, two of them may encountereach other by chance. Upon encountering, the two parti-cles take part in a reaction immediately. In A + A ~2,a one-species irreversible coagulation reaction, they fuseto one, while in 3 + 3 —+inert, a one-species irreversibleannihilation reaction, they annihilate to an inert sub-stance which is unable to react any further. The main in-teresting physical quantity in the study of the diffusion-limited reactions is the particle concentration. Moststudies have been concentrated on the asymptotic behav-ior of the time-dependent particle concentration n (r) atlong time periods. ' ' ' Some exact results for bothdiffusion-limited coagulation and annihilation reactionsinvolving a single species have been available over the lastfew years. ' ' ' Although approaches to find theparticle concentration and its asymptotic behavior atlong time periods for both reactions are quite differentand the approach in one reaction cannot be generalizedto the other (e.g. , Ising model mapping, ' many-fermionsystem mapping, ' and consideration of the probabilityfunction of empty intervals ), a remarkably simple rela-tion could be found between the asymptotic behavior ofthe particle concentrations of the two reactions. The re-lation is that asymptotically the particle concentration inthe annihilation reaction is one-half of that in the coagu-lation reaction. This relation is true in higher dimen-sions. ' ' ' It suggests that some symmetry might existbetween the two reactions. Indeed, dual relations havebeen observed within stochastic processes of a coagula-tion system, an annihilation system, and a voter-modelsystem in one dimension' and the simple relation hasalso been obtained via a random-walk consideration inhigher dimensions as well. Nevertheless, a straight andsimple account for this relation does not exist. In thisstudy, by writing out the hierarchies of equations for themultipoint joint probability-density functions of the tworeactions, a symmetry between the two reactions couldeasily be found by making a scaling transformation of thejoint density functions. This symmetry determines the
simple relation between the asymptotic behavior of thetwo particle concentrations. Given one set of the mul-tipoint joint density functions, the symmetry allows oneto obtain the other set. In addition, a general class ofdiffusion-limited reactions, which are partially coagula-tion with probability p and partially annihilation withprobability 1 —p, is found to share the same scaling sym-metry.
A standard and systematic way of attacking diffusion-reaction problems is to set up the full hierarchy of kineticequations for the joint probability-density functions ofthe positions of any given number of particles, accom-panied with their physical boundary conditions. ' Thisinfinite hierarchy is usually unsolvable and thus uselessunless some approximate truncating schemes are intro-duced to cut off the infinite hierarchy. ' ' ' Neverthe-less, the symmetry existing within hierarchies of differentreactions can be studied without solving the equations.For simplicity, consider the symmetry that exists betweenhierarchies of irreversible coagulation and irreversible an-nihilation reactions in one dimension. The same symme-try could be found in higher dimensions following paral-lel consideration in one dimension.
The model is first formed on a discrete lattice with reg-ular spatial spacing l. Each site of the lattice is eitherempty or occupied by at most one particle. Particles arethen diffused with a macroscopic diffusion constant Dalong the lattice at each discrete temporal spacing ~. Re-actions take place when two of the particles collide. Ki-netic equations for the multipoint joint density functionsare found on the discrete ground and then both temporaland spatial continuum limits are taken in an order that~~0 goes before l ~0. This continuum-limit order elim-inates events involving two or more events of diffusion orreaction happening simultaneously. However, symmetrycould be found both before and after the spatial continu-um limit. It is more general to consider the symmetry be-fore the spatial continuum limit because in higher dimen-sions such a continuum limit is not possible due to thesizelessness of the particles. Denote the n-point jointdensity function at time r by p"„(x,, . . . , x„;r ). Itrepresents the probability density for finding n particles
43 5714 1991 The American Physical Society
43 BRIEF REPORTS 5715
located at x &, . . . , x„,where x&(xz & . (x„. The su-
perscript p represents reaction type with p=1 corre-sponding to a coagulation reaction and p =0 an annihila-tion reaction. The particle concentration then corre-
I
sponds to p](x;t) In homogeneous systems, p/]'(x;t) doesnot depend on the spatial variable x and is represented byn/'(t) T. he kinetic equation for p„(xi, . . . , x„;t) is givenby
2D Dpn(x]) . . ~ )Xnjt) p ypn(x]) . ~ . )x/). . . )xn)t)+ ypn(x]) ~ . . )x/ ])x/ +al)x/ +]) . . ~ )Xn)t)
D y [pn+](x]) ~ ~ ~ )x/ ])x/ l)x/)x/ +]). . . )X,t)k
+Pn+1(x 1) ~ ' ) Xk —1)Xk)Xk +1)Xk+]) . ) Xn) t)]
vvh«e ~=+1, k =1, . . . , n, and C]=1 (coagulation reaction) and Co =2 (annihilation reaction). The boundary condi-tions for n-point joint density functions associated with the kinetic equations are
pn(x]» ~ ~ ~ )Xm —1)Xm)xm)xm+2) ' ' ) n)x t&=0
where m =1, . . . , n —I, and
P"„(x„.. . , xk „xk,xk+]=L)xk+2+L). . . )Xn+L)t)=C((x]). . . )xk , t)P„'k(xk+„xk+2). . . , x„;t) .
(2)
(3)
A detailed derivation of the kinetic equations (1) for the joint density functions and their corresponding boundary con-ditions (2) can be seen in Ref. 22. From Eqs. (1)—(3) it is seen that multipoint joint density functions in both the coagu-lation and annihilation reactions share the same boundary conditions. The difference between the two kinetic equationsfor n-point joint density functions appears only in the coefficient of (n+1)-point joint density function. The characterof this difference allows a rescaling of the multipoint joint density function to remove this difference. Consider a res-caled n-point joint density function p „(x1, . . . , x„;t) given by
p~(x„. . . , x„;t)=(C )"p/„'(X„. . . , X„;t) .
This rescaling does not change the boundary conditions given by Eqs. (2) and (3). Kinetic equations satisfied byp~(x„. . . , x„;t)are
Pn(x]). . . )Xn)t)= 2 XPn(xl) ' ')xk) ' ' ')xn) )+p XPn(X]) ' )xk —1)xk+~1»xk+1) ' ' ')xn)
l k l k
DX [P n+1 x]) ' ' )xk —1)xk )xk)xk+1) ' )xn)t)
k
+p„+](X],. . . , Xk ],Xk, Xk+l, Xk+], . . . , X„,t)]
These are actually equations satisfied by the joint densityfunctions in coagulation reactions. With the rescaledjoint density functions, the coefticient C, which describesthe difference between kinetic equations of the joint den-sity functions of the two reactions, is absorbed. Thus there- scaled joint density functions satisfy the same kineticequations, regardless of their reaction type. Now, consid-ering the homogeneous systems with initial concentra-tions such that no correlations existed between the parti-cles, the following initial conditions for the joint densityfunctions result:
p/„'(x „.. . , x„;0)= [n~(0) ]" .
If the initial concentration is chosen such that
n~(0) = n '(0)/C~,
nP{0)=n (0)/Cp
(x I x 0 ) I n ( 0 ) / Cp ]
for particle concentration, and
=n '(t)IC
„(x„.. . , x„,t) nP{0)= n (0)/Cp
p (x&, . . . , x;0)=[n (0)/C )"
I
The initial conditions of the rescaled joint density func-tions do not depend on the reaction type as well. Equa-tions (2)—(5) and (8) yield that the rescaled joint densityfunctions, P~ (x„.. . , x„;t) =p„'(x„.. . , x„;t), do notdepend on reaction type. This implies that bothcoagulation- and annihilation-reaction homogeneous sys-tems with special initial particle concentrations belong tothe same class under this scaling symmetry. From thissymmetry, for the homogeneous systems, in general,
p~(x „.. . , x„;0)= [n '(0)]" . (8)
p„'(x„.. . , x„;t)(C~ )"
5716 BRIEF REPORTS 43
for the n-point joint density functions. Since in homo-geneous systems the asymptotic behavior of particle con-centration in a coagulation reaction does not depend oninitial concentration [n (t)-1/&2nDt when t~ ao ],9 itfollows immediately that the asymptotic behavior of par-ticle concentration in an annihilation reaction is simplyone-half of that in a coagulation reaction[n (t) —I/t/8rrDt when t~ co] because C =2 for anannihilation reaction.
From the finding of the scaling symmetry, there is aconstant of freedom, C, which distinguishes the reactiontypes. C& =1 corresponds to a coagulation reaction andCO=2 corresponds to an annihilation reaction. However,the value p could be generalized to take a real number on[O, l], and thought of as a measure of probability. Thegeneralized constant could then be expressed asC =2—p. For a given p, a reaction could be found to as-sociate with the corresponding kinetic equation. This is areaction between coagulation and annihilation such thattwo encountering particles either fuse to one with proba-bility p, or annihilate and become inert with probability1 —p. The asymptotic behavior of its particle con-centration is given by n~(t) —1/(2 p)+2mDt—whent ~ ~. Thus reactions for di6'erent p values form a classunder this scaling symmetry. Given the solution of mul-tipoint joint density functions to an element in this reac-tion class, the solution of joint density functions for thewhole reaction class via the symmetry consideration can
easily be obtained. To generalize this symmetry con-sideration into higher dimensions, the constraintx, & x2 - « x„ for the positions of particles is
dropped, and the boundary conditions are modified.Boundary condition Eq. (2) is modified in such a way thatthe n-point joint density function is set to zero if any twoof the n positions of the particles are at the same site.Boundary condition Eq. (3) is modified in a way such thatif the n positions could be separated into two groups ofpositions separated by an infinite distance, then the n-
point joint density function breaks into a product of twolower joint density functions in the same way given in Eq.(3). With these modifications, the form of the kineticequations is unchanged with the positions [xk ] replacedby [xI, I. In conclusion, it is very useful to consider sym-metry hidden between the kinetic equations of diff'erentkinds of reactions. Symmetry would permit going from asolution of one reaction to another without solving theinfinite hierarchies of the kinetic equations.
The author is grateful to Professor C. R. Doering forfruitful discussions and for making me aware that Dr.Jun Zhou has found a similar result. He would also liketo thank Dr. M. T. Frawley, Dr. D. C. N. Robb, and Dr.C. Frankenberger for their careful reading of this paper.This work was supported by National Science Founda-tion Grants No. PHY-8958506 and No. PHY-8907755.
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