Exactly Solvable Reaction Diffusion Models on the Bethe Lattice
through the empty interval method (EIM) with Analytical and Numerical approaches
Laleh.Farhang Matin
Batool.Ghalandary
Dept.of Physics Azad University,North
branch,Tehran
Dept.of Physics Azad University,North
branch,Tehran
Tehran,Iran
Tehran.Iran
b.ghalandary@gmail.com
Laleh.matin@gmail.com
Abstract—the most general reaction- diffusion model on the Bethe lattice with nearest-neighbor interactions is introduced. The
evolution equation of the system can be solved exactly through the empty interval method and comparing Analytical approach with
Numerical .the stationary and the dynamics solution of such models are discussed.
Keywords: Reaction-Diffusion, Bethe lattice, empty-interval method
I.
INTRODUCTION
Reaction-Diffusion systems have been studied using
various methods, including analytical techniques,
approximate methods and simulation. Approximate methods
are generally different in different dimensions, as for
example the mean field techniques, working well for high
dimensions, generally do not give correct results for lowdimensional systems. A large fraction of analytical studies
belong to low-dimensional (especially one-dimensional)
systems, as solving low-dimensional systems should in
principle be easier. [1-11].
The Bethe lattice of coordination number ξ≥2 can be
viewed as an interior part of the infinite Cayley tree: each
site is connected by bounds to ξ nearest neighbor sites, and
there are no closed loops formed by bounds. In fact the
details of the lattice connectivity are not important for our
considerations. However, we disregard any end-effects.
Consider a connected collection of n site on the Bethe
lattice. One interesting feature of a loop less lattice, shared
with the d=1 lattice for which ξ=2, is that in such a cluster
the n sites are connected by exactly n-1 internal bonds. This
statement is well known and easily established by induction:
each new site can only be connected to one existing cluster
site, by one bond, because loops are not possible. Another
useful conclusion is that the number of bonds shared by the
cluster sites and the nearest neighbor sites immediately
outside
the
cluster
under
consideration
is
.
In [12],the most general single-species reaction-diffusion
model with nearest-neighbor interactions on the Bethe
lattice has been investigated which could dissolved exactly
through the empty interval method .the stationary solution
of such models ,as well as their dynamics, have been
discussed .Here solvability means that evolution equation
for E n ( the probability that n consecutive sites be empty)
is closed . it turned out there, that certain relations between
the reaction rates are needed , so that the system is solvable
via EIM. The evolution equation of en is a recursive
equation in terms of n , and is linear . I n this article the
most general single-species reaction-diffusion model with
nearest-neighbor interactions on the Bethe lattice is
investigated, which can be solved exactly through the empty
interval method. The scheme of the paper is as follows. In
section 2 the most general reaction-diffusion which nearest
neighbor interaction on the Bethe lattice is studied, which
can be solved exactly through EIM.
In sections 3 and 4 Analytical and Numerical schemes of
such models are discussed. In sections 5 the comparison of
produce diagrams of Analytical and Numerical approaches
is studies. Finally, section 6 is devoted to concluding
remarks.
2-Models solvable through the empty interral
method on the Bethe lattice
The cayley tree is a tree (a lattice without loops) where
each site is connected to ξ sites (Fig.1).
There is no loop in a cayley tree.For ξ ≥ 3 the closeness
of the evolution equation for E n requires that the rate of
creating an empty site be zero. The reason is that if is not
the case, then an empty n-cluster can be created from two
disjoint empty cluster joined by a single occupied site
[13]. This shows that if the evolution of the empty
clusters is to be closed, then the only possible reactions
are the following, with the rates indicated.
   , r1 .
   , r2 .
   , r3 .
(2)
(There is no distinction between left and right, of course)
using these, one arrives at the following time evolution
for En :
Fig.1.the cayley tree with ξ=3
Two sites are called neighbors if they are connected
through a link..the Bethe lattice of coordination number
  2 can be viewed as an interior part of the infinite
cayley tree. Consider a system of particles on the Bethe
lattice. Each site is either empty or occupied by one
particle.
The interaction (of particles and vacancies) is nearest
neighbor. The probalitiy that a connecthed collection of n
sites be empty is denoted by E n . It is assumed that this
quantity does not depend on the choice of collection. An
example is a tree where the probability that a site is
occupied is
and is independent of the states of other
site.Then
(1)
E n  (1 -  ) n .
The following graphical representations help express
various relation in a more compact form..an empty
(occupied) site is denote by ○ (●).a connected collection
of n empty sites is denoted by Ο n . for example, for ξ=3
attend follow to the figure 2:
dEn
  Rn r1 p(  On )  Rn (r2  r3 ) p(o  On )
dt
 (n  1)( 2r2  r3 ) p(On ),
one has
p(  On )  p(o  On )  p(On )
from which
p(  On )  En  En 1
(4)
(5)
Using this, one arrives at
dEn
 Rn [r1 ( En  En1 )  (r2  r3 ) En1 ]
dt
 (n  1)( 2r2  r3 ) En
Throughout the paper, it is assumed that r1 ,
(6)
r2 and r3
are all nonzero.
3- The exact analytical Solution:
The stationary solution of the system (Es, for which the
time derivative vanishes), satisfies
Rn [r1 ( Ens  Ens1 )  (r2  r3 ) Ens1 ]
 (n  1)( 2r2  r3 ) Ens  0
As
(7)
En 's are nonnegative and no nincreasing in n, it is
easy to see that the only solution to (7) is :
Ens  o
Fig.2.an empty cluster on a cayley tree with ξ=3
(8)
This means that in the stationary configuration, all of the
sites are occupied, which is not a surprise since in all
reactions particles are created. Regarding dynamics, one
question is to obtain the spectrum of the evolution
Hamiltonian. This is equivalent to finding solutions with
exponential time dependence : ε
(9)
En (t )  En exp( t )
We obtain with results that study from article [14]explain
this subject, completely:
(10)
 k   r1  (k  1) , k  1
Where
(11)
  (  2)r1  2r2  r3
This spectrum is discrate , and there is a gap between .the
Largest eigenvalue and zero, which means that the
system evolves towards its stationary configuration with
a relaxation time. This relaxation time is

1
r1
(12)
The general solution to (6) is then

E n (t )   c k E n
k
k 1
,
k
Where C s
exp(  k t ),
are to be determined from the initial
condition.a special solution to (6) is of the form
n1
(14)
(15)
(16)

  2
1


1  b(o)[1  exp(  t )] 
defined in terms of E
t  dt
n
,E
is
t
n 1
Ent dt  Ent [1  t (r1 Rn  (n  1)( 2r2  r3 )]
 Ent 1 .t.Rn [r1  (r2  r3 )]
(24)
Stability condition for solving equation [23] is:
2- t 
1
r1 Rn  (n  1)( 2r2  r3 )
(25)
t  dt
In this solution En
convergence to zero
(17)
Using these , one obtains
En (t )  En (o) exp[ r1t  (n  1)  ] 


  2  n1
1


1  b(o)[1  exp(  t ) 
(18)
A special case where the ansataz (14) works is the case of
initially uncorrelated – sites , so that each site is occupied
with probability  regardless of other sites. One has
then
n
One can rearrange the equation (23), and then En
Will govern the stability condition of numerical
Simulation and the convergence of numerical scheme is
satisfied.
(  2)( r2  r3  r1 )
(  2)r1  2r2  r3
En(0)=(1-  ) ,
So that
E1 (0)=1-  ,
b(0)=1-  ,
By using forward FD scheme, the above equation (22)
will be appeared in the algebraic format as following :
t  dt
En  Ent
 Rn [r1 ( Ent  E t n1 )  (r2  r3 ) E t n1 ]
(23)
t
 (n  1)( 2r2  r3 ) E t n
3- E nt 1  E nt


(22)
1  r1  r2  r3
E1 (t )  E1 (o) exp( r1t )
Where :
dEn (t )
 Rn [r1 ( En  En1 )  (r2  r3 ) En1 ]
dt
 (n  1)( 2r2  r3 ) En
t  dt
(13)
En (t )  E1 (t )b(t )
Putting this in (6) obtains
b(o) exp(  t )
b(t ) 
1  b(o)[1  exp(   )]
4- Numerical solution with explicit finite differences
method
In this section study Numerical solution of evolution
equation of system; with explicit finite differences
method
(19)
(20)
(21)
5- comparing the results of Analytical and Numerical
approaches
After introduction of Analytical and Numerical methods
of evolution equation we compare all diagrams that we
obtain from these methods.
First in numerical solution number of sites on the Bethe
lattice must be limited.
We assume that numbers of sites are called n max .it
means, the Bethe has ceiled .then we compare results of
numerical and analytical approaches.
Because of in anastaz, we calculate E1 , E2 , E 3 in this
article first we compare analytical solutions of n=1,2,3
with numerical solution.
Step1:n max =3then E 1 , E 2 and E 3 in all solution are plotted in fig 3.
Figure3:=3then E 1 , E 2 and E 3 are plotted in n max =3.
Step2:n max =10then E 1 , E 2 and E 3 in all solution are plotted in fig 4.
Figure4:=3then E 1 , E 2 and E 3 are plotted in n max =10.
Step3:n max =30then E 1 ,E 2 and E 3 in all solution are plotted in fig 5
Figure5:= E 1 , E 2 and E 3 are plotted in n max =30.
We find that if n max be larger , solutions of numerical
approach; will be more compatible with analytical
solution.
6- concluding remark
The most general single – species exclusion model on the
Bethe lattice was considered, for which the evolution of
the empty-intervals is close and compared Analytical
approach with Numerical . the stationary and dynamics
solution of such models were discussd and was shown
that these to ideas (dynamics solution and stationary
solution were compatible . evolution equation of empty .
interval method was closed . it was shown that in the in
the stationary configuration of such models all sites are
occupied.
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