Solvable reaction-diffusion processes without exclusion

Abstract

For reaction-diffusion processes without exclusion, in which the particles can exist in the same site of a one-dimensional lattice, we study all the integrable models which can be obtained by imposing a boundary condition on the master equation of the asymmetric diffusion process. The annihilation process is also added. The Bethe ansatz solution and the exact N-particle conditional probabilities are obtained. Comment: 13 pages, accepted for publication in Jour. Math. Phys

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arXiv:cond-mat/0601155v1 [cond-mat.stat-mech] 9 Jan 2006
Solvable reaction-diffusion processes without
exclusion
Masoud Alimohammadi∗
School of Physics, University of Tehran,
North Karegar Ave., Tehran, Iran.
February 6, 2008
Abstract
For reaction-diffusion processes without exclusion, in which the particles
can exist in the same site of a one-dimensional lattice, we study all the
integrable models which can be obtained by imposing a boundary con-
dition on the master equation of the asymmetric diffusion process. The
annihilation process is also added. The Bethe ansatz solution and the
exact N-particle conditional probabilities are obtained.
1Introduction
There are a variety of phenomena that can be explained by stochastic mod-
els, and their non-equilibrium behaviors can be understood by rather simple
rules [1–3]. One of the important examples of these models are the reaction-
diffusion processes on a one-dimensional lattice for which their dynamics are
fully specified by their master equation [4,5]. A simple example of reaction-
diffusion process is asymmetric simple exclusion process (ASEP) [2,6,7], which
is known to be relevant to various fields of science like the kinetics of biopolymer-
ization [8,9], traffic models [10], polymers in random media, dynamical models
of interface growth [11,12], noisy Burgers equation [13], study of shocks [14,15],
sequence alignment [16], and molecular motors [17]. For recent reviews, see for
instance [18–20].
The ASEP is a lattice model in which each particle hops to its right (left)
nearest-neighboring site with a probability DRdt(DLdt) in an infinitesimal in-
terval dt. In addition, particles are subject to hard-core exclusion: each site is
either occupied only by one particle or empty. ASEP has been studied in [21]
by introducing a master equation which describes the time evolution of proba-
bilities P(x1,··· ,xN;t), when the particles are not in neighboring sites, and a
so-called boundary condition, which specifies the situations in which the prob-
abilities go outside the physical region x1 < x2 < ··· < xN. These happen
when some of the particles are in adjacent sites and the master equation can
not be applied to them. It has been shown that the model is integrable in the
sense that the N-particle S-matrix is factorized into a product of two-particle
∗alimohmd@ut.ac.ir
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S-matrices. The coordinate Bethe ansatz has been used in this proof. Note that
the S-matrix can completely determine the dynamics of a Markovian process,
i.e. the N-particle probabilities of a model.
By choosing other suitable boundary conditions, without changing the mas-
ter equation, one may study the more complicated reaction-diffusion processes,
even with long-range interaction. In ref. [22], the so-called drop-push model has
been studied by this method. In this model the particle hops to the next right
site even it is occupied. The particle hops to this site by pushing all the neigh-
boring particles to their next right sites, with a rate depending on the number
of right neighboring particles. The generalization of this model, by consider-
ing both the right and left hopping, has been done in ref. [23]. This method
has been also applied to more-than-one-species situations, which become more
complicated. The complexity arises from the above mentioned factorization
of N-particle scattering matrix. In these cases, the factorization demands the
two-particle S-matrices to satisfy the spectral Yang-Baxter equation. Various
solvable multi-species models have been studied in this way, of which the most
recent general cases have been discussed in [24] and [25].
All of the above studies have been restricted to interactions which include
the hard-core exclusion. This made some simplification. In ref. [26, 27], an
asymmetric diffusion model without exclusion has been shown to be integrable
and to have the same R-matrix as that of the ASEP. In [28], it has been shown
that the processes:
mn→m − l,n + lwith rate
DR
[l],
DRγl
[l]
mn→m + l,n − lwith rate
, (1)
is integrable, in the sense of the above mentioned two-particle factorization. The
numbers ”m”,”n”,··· indicate the particle numbers on a site, γ = DL/DR, and
[l] =1 − γl
1 − γ.
(2)
Note that in [28], the reaction rates of eq.(1) are scaled by DR. These processes
are obtained by imposing the boundary condition
PN(··· ,xj,xj−1,··· ;t) = DRPN(··· ,xj−1,xj−1,··· ;t)+DLPN(··· ,xj,xj,··· ;t) (j = 1,··· ,N−1),
(3)
on master equation of ASEP:
∂
∂tP(x1,··· ,xj,··· ,xN;t)=
N
?
j=1
DRP(x1,··· ,xj−1,xj− 1,xj+1··· ,xN;t)
+
N
?
j=1
DLP(x1,··· ,xj−1,xj+ 1,xj+1··· ,xN;t)
−NP(x1,··· ,xj,··· ,xN;t),(4)
in which we have used a time scale so that DR+ DL= 1. In above equation,
P(x1,··· ,xN;t) is the probability for finding at time t the particles at sites
x1,··· ,xN. We take these functions to define probabilities only in the physical
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region x1 ≤ x2 ≤ ··· ≤ xN. In fact, in the domain ΩN = x1 ≤ x2 ≤ ··· ≤
xN⊂ ZN, the function P is the probability defined above, whereas in ZN/ΩN
it is defined by the master equation (4), but it is not a probability. The master
equation (4) is only valid for
xi< xi+1, (5)
since for xi = xi+1, there will be terms with xi= xi+1+ 1 in the right-hand
side of eq.(4), which is out of the physical region. One can, however, assume
that (4) is valid for all physical region xi≤ xi+1, and impose certain boundary
condition for xi= xi+1. The boundary condition (3) leads to interactions (1).
In this paper we want to study all possible boundary conditions for single-
species systems and derive all the integrable one-dimensional reaction-diffusion
processes without exclusion which can be obtained by this method. The scheme
of the paper is the following. In section 2, we show that there are two types
of boundary conditions when DR?= 0 and DL?= 0. The first is one considered
in [28], i.e. eq.(3), which we call it the type 1 model, and the second one is:
DRP(x,x − 1) + DLP(x + 1,x) = P(x,x).(6)
Here we have suppressed all the other coordinates for simplicity. We show that
the interactions of this type 2 model are
mn
mn
→
→
m − 1,n + 1
m + 1,n − 1
with rate DR,
with rate DL.(7)
This is the reactions (1), restricted to l = 1. In other words, to have an integrable
model, it is not necessary to have the simultaneous hoppings of any number of
particles from a common site to the neighboring site ( as indicated in (1)), but
the one particle hoppings can also lead to integrable models. Note that the
reactions (7) is not a subset of processes (1).
In the totally asymmetric case with DL = 0, we show in section 3 that
there is a new boundary condition which is a linear combination of boundary
conditions (3) and (6), i.e.
P(x,x − 1) = lP(x,x) + µP(x − 1,x − 1).(8)
It is shown that this type 3 model contains the reactions
mn → m − l,n + l,(9)
with rates
rl=
1
1 +l
µ+ ··· +
?
l
µ
?l−1. (10)
This is an interesting one-parameter family of interactions.
In section 4, we generalize the boundary conditions (3), (6) and (8) to include
annihilation to the processes (1), (7) and (9), respectively. The resulting models
are rather involved. To be specific, we consider DL= 0 case of (3), and show
that the boundary condition
P(x,x − 1) = µP(x − 1,x − 1)(µ < 1)(11)
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describes the reactions
mn→m − l,n + l



with rate µl−1,
n→


n − 1
n − 2
...
with total rates n −
n−1
?
l=0
µl.(12)
The second reactions are the annihilation processes. We call this model as the
type 4 model. It must be mentioned that we can not extend our investigation
to include the creation processes. The main reason is that if we do so, the
evolution equation of n-particle sector will not become closed and will depend
on the more-than-n-particle configurations.
The Bethe ansatz solution for different models is discussed in section 5 and
the exact N-point conditional probabilities of type 4 model is obtained in section
6. Some interesting physical quantities are also obtained. Finally we discuss
the multi-species extension of these processes in the last section and show that
this generalization is not possible for the reactions without exclusion.
2 Reactions with DL?= 0 and DR?= 0
Consider the master equation (4) for two particle sector, when the particles are
at site x1= x2= x:
˙P(x,x) = DRP(x−1,x)+DRP(x,x−1)+DLP(x+1,x)+DLP(x,x+1)−2P(x,x).
(13)
The second and third probabilities in the right-hand side of the above equation
are out of the physical region, and must be defined through some boundary
conditions. There are only two possibilities which are consistent in more-than-
two-particle sectors. The first one is
µ′P(x,x − 1) = DRP(x − 1,x − 1) + DLP(x,x), (14)
and the second one is
DRP(x,x − 1) + DLP(x + 1,x) = l′P(x,x).(15)
In the first choice, we take any unphysical terms of eq.(13) as a linear combina-
tion of physical functions, and in the second choice, we take the whole unphysical
terms as a linear combination of the physical probabilities. The right-hand sides
of eqs.(14) and (15) are the only allowed combinations which one can write. This
was discussed in [25] for ASEP cases. In fact, one can use any other parameters
α and β instead of DRand DLin eq.(14), with condition α + β = 1. Also one
can add two other terms µP(x−1,x−1) and νP(x+1,x+1) to the right-hand
side of eq.(15), but it can be shown that for obtaining a consistent description in
more-than-two-particle sectors, one must take µ = ν = 0 ( see section 3 of [25]
for more details in ASEP case).
To obtain the range of parameters µ′and l′, one can use eq.(4) to show that
∂ø∂t
?
x2
?
x1≤x2
P(x1,x2;t)=−
?
x
P(x,x;t) + DR
?
x
P(x,x − 1;t) + DL
?
x
P(x + 1,x;t)
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=−
?
x
P(x,x;t) +
?
x
P(x,x − 1;t).(16)
For boundary condition (14), eq.(16) results
∂ø∂t
?
x2
?
x1≤x2
P(x1,x2;t) = (1øµ′− 1)
?
x
P(x,x;t),(17)
and for boundary condition (15), it results
∂ø∂t
?
x2
?
x1≤x2
P(x1,x2;t) = (l′− 1)
?
x
P(x,x;t).(18)
In the first step, let us exclude the annihilation processes and therefore it is
assumed that the number of particles is constant in time. In this way, the
eqs.(17) and (18) lead to
µ′= 1
l′= 1
(for boundary condition (14)),
(for boundary condition (15)).(19)
Eq.(14) exactly becomes the one studied in [28], i.e. eq.(3), and it induces
the reactions (1), but the second one, eq.(15), is new. To obtain the reaction
introduced by this boundary condition, with l′= 1, we first consider˙P(x,x) in
eq.(13). Using eqs.(15) and (19), one finds
˙P(x,x) = DRP(x − 1,x) + DLP(x,x + 1) − P(x,x), (20)
which is the evolution equation of the following two-particle reactions
10
01
→
→
01
10
with rate DR,
with rate DL. (21)
Remember DR+ DL= 1. Generally, for n particles existing at a common site
x, one finds from (4):
∂
∂tP(x,··· ,x
? ?? ?
n
)=DRP(x − 1,x,··· ,x
? ?? ?
n−1
) + DLP(x,··· ,x
? ?? ?
n−1
,x + 1)
+
n
?
j=2
n
?
j=2
DRP(x1= x,··· ,xj−1= x,xj= x − 1,xj+1= x··· ,xn= x)
+
DLP(x1= x,··· ,xj−2= x,xj−1= x + 1,xj= x,··· ,xn= x)
−nP(x,··· ,x
? ?? ?
DRP(x − 1,x,··· ,x
n
)
=
? ?? ?
n−1
) + DLP(x,··· ,x
? ?? ?
n−1
,x + 1) − P(x,··· ,x
? ?? ?
n
),(22)
which is obviously the evolution equation of reactions (7).
3Totally asymmetric diffusion with DL= 0
In DL= 0, the boundary condition (3) and (6) becomes
P(x,x − 1)=P(x − 1,x − 1),
5