1. Introduction
The multi-species asymmetric simple exclusion process on is a generalization of the asymmetric simple exclusion process (ASEP) on in the sense that each particle may belong to a different species labelled by an integer . Each particle jumps to the right by one step with the probability p or to the left by one step with the probability after a waiting time that is exponentially distributed with rate 1. If a particle belonging to l attempts to jump to the site occupied by a particle belonging to , the jump is prohibited; however, if a particle belonging to attempts to jump to the site occupied by a particle belonging to , then the jump occurs by interchanging positions.
The transition probabilities and some determinantal formulas for the multi-species ASEP and its special cases were found in [
1,
2,
3,
4,
5]. For certain special initial conditions with a single second class particle, some distributions and their asymptotics were studied in [
6,
7]. More recently, asymptotic behaviors of the second class particles were studied using the color-position symmetry, see [
8]. In fact, the multi-species asymmetric simple exclusion process can be considered in a more general context—that is, the coloured six vertex model [
9].
Another direction of generalizing the ASEP and other models studied in the integrable probability is to make the jump rates inhomogeneous. It is known that the Bethe ansatz method is still applicable to some single-species model with inhomogeneous jump rates. The basic idea of using the Bethe ansatz in the ASEP is that the generator of the ASEP is a similarity transformation of the XXZ quantum spin system. Considering that the Bethe ansatz is a method to find eigenvalues and eigenvectors of a certain class of quantum spin systems, we use the Bethe ansatz to find the solution of the forward equation of a certain class of Markov processes, that is, the transition probabilities of the processes.
Of course, for some particle models, the Bethe ansatz method cannot be used. For the background of Bethe ansatz, see [
10,
11]. It is known that the Bethe ansatz is applicable to some generalization of the ASEP. For example, the transition probability and the current distribution of the totally asymmetric simple exclusion process (TASEP) with particle-dependent rates were studied in [
12], and the transition probabilities and some asymptotic results for the
q-deformed totally asymmetric zero range process with site-dependent rates were studied in [
13,
14].
In this paper, we consider the multi-species totally asymmetric simple exclusion processes with
N particles in which particles move to the right and each species
l is allowed to have its own rate
. Following the notations used in [
4], let
with
represent the positions of particles, and let
be a permutation of a multi-set
with elements taken from
and
representing the species of the
leftmost particle. Then, the state of an
N-particle system is denoted by
Let us write
for the transition probability from
at
to
at a later time
t. For fixed
X and
Y,
is regarded as a matrix element of an
matrix denoted by
whose columns and rows are labelled by
, respectively. Throughout this paper, given an
matrix, we assume that its rows (
) and columns (
) are labelled by
and that these labels are listed lexicographically, unless stated otherwise. The main result of this paper is that the multi-species TASEP with species-dependent rates is an integrable model, and we provide a formula analogous to (2.12) in [
4] using the Bethe ansatz method.
Statement of the Results
We first introduce a few objects to state the main theorem. Define an
matrix
with
where
Remark 1. The form of the matrix (1) was obtained by induction via similar arguments to Sections 2.1 and 2.2 in [3], which treats a special case, and the motivation of (2) is given in Section 2.1. Finding the form of (1) with (2) is the key idea of this paper. Let
be the simple transposition that interchanges the number at the
th slot and the number at the
st slot. If
maps a permutation
to
, we write
when necessary. Corresponding to a simple transposition
, we define
matrix
by the tensor product of matrices,
where
is the
identity matrix. For a permutation
in the symmetric group
written
for some
, we define
Here, is well defined, that is, is unique regardless of the representation of by simple transpositions. This well-definedness is due to the following lemma.
Lemma 1. The following consistency relations are satisfied.
- (a)
- (b)
- (c)
The relations in Lemma 1 with for all l are already known for the multi-species ASEP.
Remark 2. The definitions of and are motivated by the arguments for in Sections 2.1 and 2.2 in [3], which treats a special case. Let
be the
diagonal matrix whose
-element is given by
where
let
be the
diagonal matrix whose
-element is given by
, and let
be the
diagonal matrix whose
-element is given by
where
s are the initial positions. In the next theorem, the integral of a matrix implies that the integral is taken element-wise, and ⨍ implies
.
Theorem 1. Let be given as in (5) and c be a positively oriented circle centred at the origin with a radius less than bl for all l in the complex plane . Then, the matrix of the transition probabilities of the multi-species TASEP with species-dependent rates is Remark 3. If , in other words, all N particles belong to different species, and the species are initially arranged in ascending order, then the system is the same as the TASEP with particle-dependent rates studied in [12]. Hence, the transition probability can be expressed as a determinant (see Theorem 1 in [12]). Remark 4. Theorem 1 partially extends (2.12) in [4]. In other words, (6) with is equal to (2.12) in [4] with . The proofs of Lemma 1 and Theorem 1 are given in the next section.
2. Proof of Theorem 1
In order to prove that the -element of the right-hand side of (6) is , we should show that the -element satisfies its forward equation and the initial condition .
2.1. Forward Equations
We first study the two-particle systems, which will be building-blocks for the formulas for
N-particle systems. When
, the forward equations of
are straightforward because two particles act as
free particles. Hence, the forward equations of
are expressed as
where the derivative of the matrix
on the left-hand side implies the matrix of the derivatives of elements of
. The matrices
and
account for the
probability current in the states
by a particle’s jump to the right next site, which is empty. On the other hand, when
, if two particles belong to different species, two particles may swap their positions. For example, if the initial state is
, the system cannot be at
at any later time
t. Hence, the forward equation of
is
On the other hand,
for all
t, because the model is totally asymmetric. If the initial state is
, the forward equation of
is
and the forward equation of
is
Hence, the forward equations of
are expressed as
Here, the matrix
accounts for
probability current going in the states
by the species-2 particle’s jump from the state
. Similarly, the matrix
accounts for
probability current going out of the states
by species-2 particle’s jump to the state
. Equations (7) and (8) imply that, if
is a
matrix whose elements are functions on
, then the forward equation of
for any
is in the form of the
-element of
subject to the
-element of
Now, we extend the argument for two-particle systems to
N-particle systems. The matrices
and
in (7) for two-particle systems are generalized to
where
is the diagonal matrix,
The matrix
in (8) is generalized to an
matrix
with
and let
The matrix
in (8) is generalized to
matrix
with
and let
All forward equations of
may be expressed as a matrix equation. For example, if
for all
i, then the forward equation of
is the
-element of
and if
and
for all
,
For other configurations of
, the form of the matrix of the forward equations may be different from (9) and (10). However, as in other Bethe ansatz applicable models, if
is an
matrix whose elements
are functions on
, then the forward equation of
for any
is in the form of the
-element of
subject to the
-element of
for all
.
2.2. Solutions of the Forward Equations via Bethe Ansatz
The
-element of (11) is
Assume the separation of variables to write
. Then, the equation of the spatial variables is
for some constant
with respect to
. Then, we observe that, for any
,
solves (13) if and only if
Based on the observation in the above, assume that the matrix
is invertible and that it is decomposed as
where
is an
diagonal matrix where
are functions of time only. Hence, from (11), we obtain
Both sides of (15) must be a diagonal matrix
whose elements are some constants with respect to
. Thus, we obtain the matrix equation for spatial variables
and the matrix equation for the time variable
Lemma 2. Let be an diagonal matrix with Then, for any ,where is an arbitrary invertible matrix whose elements are constants with respect to is a solution of (16) if and only if is given by Proof. First, we observe that
because
is a diagonal matrix whose
-element is
. We observe that
and
Now, we prove the statement. Suppose that (17) is a solution of (16). Substituting (17) into (16) and then dividing both sides by
, then
Multiplying by
on both sides, we obtain
and thus, the
-element of
is given by (18). The second part of the proof can be done via the reverse way of the first part of the proof. □
The previous lemma implies that the general solution of (16) is given by
2.3. Boundary Conditions
Now, (19) should satisfy the spatial part of the
boundary condition (12), that is,
for
. Extending the technique used in [
4], we will find the formulas of
in (19), which satisfy (20). Define an
diagonal matrix,
and recall the definition of
in (1). Then, we observe that
Lemma 3. Iffor all even permutations σ and , then (20) is satisfied. Proof. Substituting (19) into (20), we obtain
If we express (22) as a sum over the alternating group
However, (23) is satisfied if
for each even permutation
, which is equivalent to (21). □
In fact, (3) and (5) implies that the assumption of Lemma 3 is satisfied, hence (19) with in (5) satisfies (20).
2.4. Consistency Relations—Proof of Lemma 1
Lemma 1 confirms that the multi-species TASEP is integrable even when the rates are species-dependent.
2.4.1. Proof of Lemma 1 (a)
This equality clearly holds because both sides are equal to . □
2.4.2. Proof of Lemma 1 (b)—Yang-Baxter Equation
If we re-arrange the columns and the rows of the
matrices in (24) so that all their labels from the same multi-set
are grouped together, then the matrices in (24) become block-diagonal (See
Figure 1).
Let
be the sub-matrix of
whose rows and columns are labelled by the permutations of the multi-set
, and similarly, we define
. Then, in order to show (24), it suffices to show
for each multi-set
whose elements are from
because all matrices are block-diagonal matrices in the same form. If
, (25) is equivalent to
which is trivially true. If
, then (25) is equivalent to
which can be easily verified by direct computation. Similarly, the other two cases of (25) for
and for the case that all
are distinct can be verified by direct computation. □
2.4.3. Proof of Lemma 1 (c)
It suffices to show that
. Let us re-arrange the rows and the columns in the same way as in the proof of Lemma 1 (b) to make
and
block-diagonal. Then, each block on the diagonal of
is either a
matrix or a
matrix. The
sub-matrix of
consisting of the row
and the column
is
. The
sub-matrix of
consisting of the rows
and the columns
with
is
Similarly, the
sub-matrix of
consisting of the row
and the column
is
, and the
sub-matrix of
consisting of the rows
and the columns
with
is
It is trivial that
, and this can be verified,
by the direct computation. □
2.5. Initial Condition
The contour integral of (19) multiplied by
from the left and by
from the right, that is, the right-hand side of (6) still satisfies (11) and (12). (The contour is the one introduced in Theorem 1). Hence, it remains to show that all transition probabilities
satisfy the initial condition
where
is the
-element of
. We will show that the integral with the identity permutation in the sum satisfies (26), and other integral terms with non-identity permutations are zero.
Proof. If is the identity permutation, then is the identity matrix. Hence, if , then the integral is zero. It is easy to see that if and for all i, then the integral is 1. If and for some i (recall that our model is totally asymmetric), then the integral becomes zero when integrating with respect to . Now, suppose that is not the identity permutation. Note that the factors in are from (1), all poles arising from , if any, are outside the contours. There exists an i such that because each and is not the identity permutation. Hence, integrating with respect to i, the integral is 0. □