Class of solvable reaction-diffusion processes on Cayley tree

0.19 MB·English

by  M. Alimohammadi

Class of solvable reaction-diffusion processes on
Cayley tree
0
1 M. Alimohammadi∗ and N. Olanj
0
2 Department of Physics, University of Tehran,

n North Karegar Ave., Tehran, Iran.
a
J

9
Abstract

]
h Considering the most general one-species reaction-diffusion processes
c
onaCayleytree,ithasbeenshownthatthereexisttwointegrablemodels.
e
In the first model, the reactions are the various creation processes, i.e.
m
◦◦ → •◦, ◦◦ → •• and ◦• → ••, and in the second model, only the
- diffusion process •◦ → ◦• exists. For the first model, the probabilities
t
a Pl(m;t), of finding m particles on l-th shell of Cayley tree, have been
t
s found exactly, and for the second model, thefunctions Pl(1;t) havebeen
. calculated. It has been shown that these are the only integrable models,
t
a ifonerestricts himself toL+1-shell probabilities P(m0,m1,··· ,mL;t)s.
m
-
d 1 Introduction
n
o
The integrable reaction-diffusion processes have been investigated by various
c
[ methods on one-dimensional lattice. A class of these models are characterized

by a master equation with appropriate boundary conditions. These boundary
2
terms,whichdeterminetheprobabilitiesattheboundaryofthespaceofthepa-
v
7 rameters,arechosensuchthatthestudyingofthevariousreactionsbecomespos-
4 sibleviaasimplemasterequation. Thebasicquantityinthesemodelsisthecon-
8 ditionalprobabilityP(α ,··· ,α ,x ,··· ,x ;t|β ,··· ,β ,y ,··· ,y ;0),which
1 N 1 N 1 N 1 N
0 is the probability of finding particles α ,··· ,α at time t at sites x ,··· ,x ,
1 N 1 N
.
4 respectively,ifatt=0we haveparticles β ,··· ,β atsites y ,··· ,y ,respec-
1 N 1 N
0 tively [1–6].
9
Another class of models are those which are solvable through the empty
0
interval method. In these models, the main quantity, in the most general case,
:
v isE (t)whichistheprobabilitythatnconsecutivesites,startingfromthesite
k,n
i
X k, are empty. Several examples have been investigated by this method [7–13].
The crucial property which causes the above mentioned cases to be analyt-
r
a ically solvable, is the dimension of the lattice which particles move on it. One
dimensional lattice is clearly the simplest case.
Consideringthehigher-dimensionallatticeinstudyingtheintegrablemodels
is obviously a great improvement in theoretical physics. One of the important
exampleinthisareaisCayleytree. ACayleytreeisaconnectedcycle-freegraph
where each site is connected to z neighbor sites. z is called the coordination
∗[email protected]
1
Figure 1: Cayley tree with coordination number z =3.
number. The Cayley tree of coordination number z may be constructed by
starting from a single centralnode (called the rootof the lattice) at shell l=0,
whichisconnectedtoz neighborsatshelll=1. Eachofthenodesinshelll>0
is connected to (z−1) nodes in shell l+1. This constructionmay stop at shell
l=L, or continue indefinitely (Fig.1). The interior of an infinite Cayley tree is
calledthe Bethe lattice. Due to distinctive topologicalstructureof Cayleytree,
the reaction-diffusion processes on this graph may be integrable.
Thediffusion-controlledprocessofclustergrowth,introducedbyWittenand
Sander[14],hasbeenstudiedonCayleytreein[15],andin[16],thetwo-particle
annihilation reaction for immobile reactant has been studied on Bethe lattice.
Diffusion-limited coalescence, A+A → A, and annihilation, A+A → 0 [17],
and random sequential adsorption [18] have also been studied on Cayley tree.
Alsothereaction-diffusionprocessesonCayleytree,solvablethroughtheempty
interval method, have been studied in [19].
The present paper is devoted to the study of some integrable reaction-
diffusionprocessesonCayleytree. Denotingthenumberofparticlesonl-thshell
bym ,weseekthesituationsinwhichtheprobabilityP(m ,m ,m ,··· ,m ;t)
l 0 1 2 L
can be calculatedexactly. l =L is the last shell of the Cayley tree. By con-
max
sidering the most general reactions on Cayley tree in section 2, it is shown
that we must restrict ourselves to three distinct creation reactions so that
a subclass of L + 1-shell probabilities P(m ,m ··· ,m ;t), i.e. P (m;t) ≡
0 1 L l
P(0,0,··· ,0,m = m,0,··· ,0;t), becomes solvable. The exact solution of
l
P (m;t) is obtained in section 3. Finally in conclusion section it is shown that
l
the diffusionprocesscanalsoleadto anintegrablemodel,ifonerestricthimself
tooneparticleprobabililitiesP (1;t),andifanextratrappingreactionexistsat
l
the central node. An ideal spherical trap, surrounded by a swarm of Brownian
particles,is afundamentalmodelwhichhadbeenfirstpresentedby vonSmolu-
chowski[20], andhas been generalizedto a mobile trapin[21]. Also the energy
transfer process in dendrimer supermolecule on the Cayley tree with a central
trap has been discussed in [22]. These two models are only integrable models
in this context.
It must be added that the procedure introduced in this paper can be easily
extendedtothesituationsinwhichthenumberofoccupiedshellsisgreaterthan
one, but there must exist at least one empty shell between any two of them.
2
Figure 2: The sourceterms of configuration(a) of(0,1,2)occupationnumbers.
Figure3: The sourcetermsofconfiguration(b) of(0,1,2)occupationnumbers.
2 The integrable models
The most general reactions of single species models with nearest-neighbor in-
teractions are
1:◦◦ → •◦, r 4:•◦→◦•, r
1 4
2:◦◦ → ••, r 5:•◦→◦◦, r
2 5
3:◦• → ••, r 6:••→◦◦, r
3 6
7:••→•◦, r (1)
7
where an empty (occupied) site is denoted by ◦(•). The r s are reaction rates
i
and there is no distinction between left and right. Our main goal is to find
the situations which lead to integrable models. By integrability, we mean the
possibility of the exact calculation of probabilities P(m ,m ,··· ;t).
0 1
The necessary condition for achieving this goal is that the occupation num-
bers of shells (the number of occupied sites of every shell) must be the only
parameters needed to characterize the configurations. Clearly this is not the
case if two adjacent shells are occupied. For example consider P(0,1,2;t) and
the reaction (1) of eq.(1). In Figs.(2) and (3), the source terms of two different
configurations (a) and (b), with same occupation numbers (0,1,2), are shown.
As it is seen, the configuration (a) has two source terms, while the configura-
tion (b) has three. The number behind each configuration is its multiplicity,
i.e. the number of ways which this configuration can lead to the desired state.
This shows that if two adjacent shells are occupied, it is not possible to write
the same evolution equation for different configurations of a given occupation
numbers. Therefore, we only consider the situations in which there is only one
non-zerooccupationnumber,i.e. m =mδ . Asmentionedearlier,itispossible
i il
to extend our procedure to the situations in which there is at least one empty
shell between any two occupied shells.
Considering the probabilities P (m;t) = P(0,0,··· ,0,m = m,0,··· ,0;t),
l l
we must yet check whether all the reactions of eq.(1) are acceptable or not. By
the term acceptable, we mean to have two properties. First, the determination
3
of occupation number m is sufficient for having a unique evolution equation,
irrespectiveofparticles’distribution. Second,allthetermsofevolutionequation
areexpressibleintermsofP (n;t)s. Nowitcanbeseenthatthisisnotthecase
k
forreactions(4)-(7)ofeq.(1). Thisisbecauseinallthesecases,thesourceterms
ofPl(m;t)sare2-shellprobabilitiesP(ml−1,ml;t)andP(ml,ml+1;t),whichcan
notbeexpressedintermsofP (n;t)s. Forthesereactions,themasterequations
k
are:
∂
∂tPl(m;t) = a4P(ml−1 =1,ml =m−1;t)+b4P(ml =m−1,ml+1 =1;t)
−s P (m;t) ( for r =r ), (2)
4 l 4
∂
∂tPl(m;t) = a5P(ml−1 =1,ml =m;t)+b5P(ml =m,ml+1 =1;t)
+c P (m+1;t)−s P (m;t) ( for r=r ), (3)
5 l 5 l 5
∂
∂tPl(m;t) = a6P(ml−1 =1,ml =m+1;t)
+b P(m =m+1,m =1;t) ( for r =r ), (4)
6 l l+1 6
and
∂
∂tPl(m;t) = a7P(ml−1 =1,ml =m;t)
+b P(m =m,m =1;t) ( for r=r ), (5)
7 l l+1 7
respectively. The parameters a ,b ,c and s are some constants. The case
i i i i
m=1 is an exception which will be discussed in section 4.
So the integrable model, through the P (m;t) probabilities, is a model in
l
which the allowed reactions are reactions (1)-(3) of eq.(1). The evolutionequa-
tion of P (m;t) on a Cayley tree with coordination number z and L+1 shells
l
then becomes (for l=0,1,2,··· ,L−1)
∂
P(m;t) = (n −m+1)zr P(m−1;t)
∂t l l 1 l
L L
−[2( nl′ −mz)r1+( nl′ −mz)r2+mzr3]Pl(m;t).(6)
lX′=1 lX′=1
In above equation
z(z−1)l−1 for l≥1
n = (7)
l (cid:26) 1 for l=0
is the number of sites of the l-th shell. The first term in the right-hand site of
eq.(6) is the source term of reaction (1), and 2( Ll′=1nl′ −mz)r1Pl(m;t) is its
sink term. This is because Ll′=1nl′ −mz is thPe number of empty pairs (◦ ◦)
of a Cayley tree with m parPticles on shell l, which each of them can change to
either • ◦ or ◦ • with rate r . The reaction(2) has no source term for P (m;t),
1 l
and its sink term is clearly ( Ll′=1nl′ −mz)r2Pl(m;t). Finally the reaction 3
doesnothavesourceterm,anPditssinktermismzr3Pl(m;t)becauseeachofthe
neighbors of each particles of l-th shell can be changed via ◦•→•• reaction.
For l = L, since there is no upper shell, the number of neighboring sites of
each occupied site is one, instead of z. Therefore for P (m;t), the evolution
L
equation is same as eq.(6), when z is replaced by 1.
4
3 Solutions of the master equation
In this section we try to solve the master equation
∂
P (m;t) = (n −m+1)zr P (m−1;t)
∂t l l 1 l
−[(r −r −2r )mz+(2r +r )N]P (m;t) (l <L), (8)
3 2 1 1 2 l
with
L
N ≡ nl′, (9)
lX′=1
in two cases r −r −2r 6= 0 and r −r −2r = 0. For l = L, we must set
3 2 1 3 2 1
z =1 in eq.(8).
3.1 The case r3 6= r2 +2r1
To solve eq.(8), we use a recursive method. We first consider the case m = 0,
where eq.(8) is reduced to
∂
P (0;t)=−N(2r +r )P (0;t), (10)
∂t l 1 2 l
with solution
P (0;t)=P (0)e−N(2r1+r2)t. (11)
l l
P (0) ≡ P (0;t = 0) is the probability of finding no particle in shell l at t = 0.
l l
Inserting eq.(11) in m=1 case of eq.(8), results in P (1;t) as follows
l
n r P (0)
P (1;t) = P (1)− l 1 l e−[z(r3−r2−2r1)+N(2r1+r2)]t
l (cid:20) l r −r −2r (cid:21)
3 2 1
n r P(0)
+ l 1 l e−N(2r1+r2)t. (12)
r −r −2r
3 2 1
Continuing this method, one can deduce the following general expression for
P (m;t):
l
m k (n −j)! r m−j
P (m;t) = l 1 (−1)k−j
l (n −m)!(m−k)!(k−j)!(cid:18)r −r −2r (cid:19)
kX=0Xj=0 l 3 2 1
×P (j)e−[jz(r3−r2−2r1)+N(2r1+r2)]t (l <L), (13)
l
in which, as we will show, P (j)≡P (j;t=0).
l l
It canbe seen that for m=0 and m=1, eq.(13) leads to eqs.(11) and (12),
respectively. Toprovethateq.(13)isthesolutionofeq.(8),weinsertitineq.(8).
The summations in the left-hand side and in the second term of the right-hand
side of eq.(8) are both m k . If we write them as m + m−1 k ,
k=0 j=0 j=0 k=0 j=0
thenitcanbeseenthatPthe Pm termsinbothsidesarethPesame,PandthePrefore
j=0
theyarecancelled. SoinallPthreetermsofeq.(8),thesummationshavethesame
form m−1 k , which after some simple calculations, it can be shown that
k=0 j=0
the rePmaininPg terms are cancelled, which proves the expression (13) satisfies
5
the master equation (8). Note that because of the (n − m)! factor in the
l
denominator of eq.(13), P (m;t) satisfies
l
P (m>n ;t)=0, (14)
l l
which shows the correct behavior of eq.(13).
It is also necessary to prove the physical interpretation of P (j) as P (j;t=
l l
0). To see this, we consider eq.(13) at t=0:
m m r m−j (n −j)!
P (m;t=0) = 1 l P(j)
l (cid:18)r −r −2r (cid:19) (n −m)! l
Xj=0Xk=j 3 2 1 l
(−1)k−j
× , (15)
(m−k)!(k−j)!
in which we use the equality:
m k m m
A(j,k)= A(j,k). (16)
kX=0Xj=0 Xj=0Xk=j
Using k′ =k−j, eq.(15) becomes
m r m−j (n −j)!
P (m;t=0) = 1 l P(j)
l (cid:18)r −r −2r (cid:19) (n −m)! l
Xj=0 3 2 1 l
m−j (−1)k′
× . (17)
k′!(m−j−k′)!
kX′=0
From binomial expansion
(a−b)n = n (−1)k′n! an−k′bk′, (18)
k′!(n−k′)!
kX′=0
one finds for a=b
n (−1)k′
0=n!an . (19)
k′!(n−k′)!
kX′=0
Sothelastsummationofeq.(17)iszeroforallm−j 6=0,orj =0,1,··· ,m−1.
The only remaining term is j =m, which results in
P (m;t=0)=P(m). (20)
l l
Thiscompletesthe proofofeq.(13)asthe exactsolutionofthemasterequation
(8), with P (j)s as the initial values of probabilities. The probability P (m;t)
l L
is found from (13), by taking l =L and z =1.
Eq.(13)givestheprobabilityoffindingmparticlesattimetonshelll,when
allother shells are empty, if we begin with any number of particles on shell l at
t=0. Of coursefor a specific initial condition,i.e. P (j;t=0)=δ , only one
l j,j0
term of summation k survives.
j=0
P
6
3.2 The case r = r +2r
3 2 1
For the case r =r +2r , the master equation (8) becomes
3 2 1
∂
P (m;t)=(n −m+1)zr P (m−1;t)−Nr P (m;t) (l <L). (21)
∂t l l 1 l 3 l
Using the method of the previous case, the solution of above equation is found
to be
m
(n −j)!
P(m;t)=e−Nr3t l P (j)(r zt)m−j (l <L). (22)
l (m−j)!(n −m)! l 1
Xj=0 l
It can be easily shown that the solution (22) satisfies (21) and has the desired
properties P (m >n ;t)=0 and P (m;t=0)=P(m). For l =L, the solution
l l l l
is found from eq.(22), by taking l =L and z =1.
4 Conclusion
Tostudytheintegrablereaction-diffusionprocessesonamore-than-one-dimensional
lattices,themostgeneralinteractions(1)havebeenconsideredonaCayleytree
with coordination number z. It has been shown that among the probability
functions, the probabilities P (m;t)s are independent of distribution of parti-
l
cles on various shells, and may lead to integrable models if we consider two
situations. The first one is the creation-reactions (1)-(3) of eq.(1), with master
equation (6). The P (m;t)s for two cases r 6= r + 2r and r = r + 2r
l 3 2 1 3 2 1
have been considered in section 3, with the final exact results (13) and (22),
respectively.
As is clear from eqs.(2)-(5), in the case m=1, the master equations (3)-(5)
still contain the 2-shell probabilities, but for diffusion process r = r , only the
4
one-shell probabilities (i.e. one point functions) appear in eq.(2), which may
lead to an integrable model.
ThemasterequationofP (1;t),withl>1,fordiffusionprocessonaCayley
l
tree with coordination number z is:
∂
∂tPl(1;t)=(z−1)Pl−1(1;t)+Pl+1(1;t)−zPl(1;t), (23)
in which the diffusion rate r has been absorbedin the rescaling of time t. The
4
evolution equations of P (1;t) and P (1;t) are
0 1
∂
P (1;t)=P (1;t)−zP (1;t), (24)
∂t 0 1 0
and
∂
P (1;t)=zP (1;t)+P (1;t)−zP (1;t), (25)
∂t 1 0 2 1
respectively. Theseequationstakeaformsimilartoeq.(23),providedonedefines
P−1(1;t):=0, (26)
P (1;t):=0. (27)
0
7
The first boundary condition is trivially satisfied, since P (m;t)s have been
l
definedforl ≥0. Butthe secondoneindicatesa trappedreactionatthe origin.
So eq.(23), with arbitrary l, defines an integrable model on a Cayley tree if
the particles have diffusion process, and if there exists a trap at the root of
the lattice. When the reactions are coalescence A+A → A and annihilation
A+A → 0, the density of particles in shell l, i.e. ρ (t), has been calculated
l
in [17] for the same situation, that is a trap at the root of a Cayley tree.
It can be shown that the solution of master equation (23) with boundary
condition (27) is
Pl(1;t)=(z−1)(l−l0)/2 Il−l0(2(z−1)1/2t)−Il+l0(2(z−1)1/2t) e−zt, (28)
h i
in which I is the n-th order modified Bessel function.
n
Finallyitmustbeaddedthatitremainsonemorecasewhichmayleadtoan
integrablemodel. Lookingateqs.(2)-(5),itisclearthatifonetakesm=0,only
the master equation (3) leads to an equation which only consists of one-point
functions, i.e.
∂
∂tPl(0;t)=a5Pl−1(1;t)+b5Pl+1(1;t)+c5Pl(1;t)−s5Pl(0;t). (29)
Itcanbeshownthats =0. ButthepointisthatthedeterminationofP (0;t)s
5 l
depends on the evaluation of P(1;t)s, which can not be calculated. This is
l
because the master equation of P (1;t)s contains the two-point probabilities
l
P(1,1;t) and therefore is not closed. Therefore the two considered models are
the only cases which can be exactly solved.
Acknowledgement: Thisworkwaspartiallysupportedbyagrantfromthe
”EliteNationalFoundationofIran”andagrantfromthe UniversityofTehran.
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