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Exact Solution of Asymmetric Diffusion With N Classes of Particles of Arbitrary Size and Hierarchical Order

The exact solution of the asymmetric exclusion problem with N distinct classes of particles (c = 1,2,...,N), with hierarchical order is presented. In this model the particles (size 1) are located at lattice points, and diffuse with equal asymmetric rates, but particles in a class c do not distinguish those in the classes c'>c from holes (empty sites). We generalize and solve exactly this model by considering the molecules in each distinct class c =1,2,...,N with sizes s_c (s_c = 0,1,2,...), in units of lattice spacing. The solution is derived by a Bethe ansatz of nested type.


I. INTRODUCTION
The similarity between the master equation describing time fluctuations in nonequilibrium problems and the Schrödinger equation describing the quantum fluctuations of quantum spin chains turns out to be fruitful for both areas of research [1]- [15]. Since many quantum chains are known to be exactly integrable through the Bethe ansatz, this provides exact information on the related stochastic model. At the same time the classical physical intuition and probabilistic methods successfully applied to nonequilibrium systems give new insights into the understanding of the physical and algebraic properties of quantum chains.
An example of this fruitful interchange is the problem of asymmetric diffusion of hardcore particles on the one dimensional lattice ( see [16][17][18] for reviews). This model is related to the exactly integrable anisotropic Heisenberg chain in its ferromagnetic regime [19] (XXZ model). However if we demand this quantum chain to be invariant under a quantum group symmetry U q (SU(2)), we have to introduce, for the equilibrium statistical system, unusual surface terms, which on the other hand have a nice and simple interpretation for the related stochastic system [3,4].
In the area of exactly integrable models it is well known that one of the possible extensions of the spin- 1 2 XXZ chain to higher spins, is the anisotropic spin-S Sutherland model (grading ǫ 1 = ǫ 2 = . . . = ǫ 2s+1 = 1) [20]. On the other hand in the area of diffusion limited reactions a simple extension of the asymmetric diffusion problem is the problem of diffusion with particles belonging to N distinct classes (c = 1, 2, . . . , N) with hierarchical order [22]- [24] .
In this problem a mixture of hard-core particles diffuses on the lattice. Particles belonging to a class c (c = 1, . . . , N) ignore the presence of those in classes c ′ > c, i. e., they see them in the same way as they see the holes (empty sites). In [3] it was shown that for open boundary conditions the anisotropic spin-1 Sutherland model and this last stochastic model, in the case N = 2, are exactly related. The Hamiltonian governing the quantum or time fluctuations of both models being given in terms of generators of a Hecke algebra, invariant under the quantum group U q SU (3). In fact this relation can be extended to arbitrary values of N, and the quantum chain associated to the stochastic model is invariant under the quantum U q (SU(N + 1)) group. In this paper we are going to derive through the Bethe ansatz the exact solution of the associated quantum chain, in a closed lattice. Recently [15] (see also [14]) we have shown that without losing its exact integrability, we can consider the problem of asymmetric diffusion with an arbitrary mixture of molecules with different sizes (even zero), as long they do not interchange positions, that is, there is no reactions. In this paper we are going to extend the asymmetric diffusion problem with N type of particle with hierarchical order, to the case where the particles in each class have an arbitrary size, in units of the lattice spacing. Unlike the case of asymmetric diffusion problem, we have in this case a nested Bethe ansatz [25]. A pedagogical presentation for the simplest case N = 2 was presented in [26].
The paper is organized as follows. In the next section we introduce the generalized asymmetric model with N type of particle with hierarchical order and derive the associated quantum chain. In section 3 the Bethe ansatz solution of the model is presented. Finally in section 4 we present our conclusions, with some possible generalizations of the stochastic problem considered in this paper, and some perspectives on future work.

CLASSES OF PARTICLES WITH HIERARCHICAL ORDER
A simple extension of the asymmetric exclusion model, where hard-core particles diffuse on the lattice, is the problem where a mixture of particles belonging to different classes (c = 1, 2, . . . , N) diffuses on the lattice. This problem in the case where we have only N = 2 classes was used to describe shocks [22]- [24] in nonequilibrium and also has a stationary probability distribution that can be expressed via the matrix-product ansatz [27]. In [28] it was also shown that the stationary state of the case N = 3 can also be expressed by the matrix-product ansatz. In this model we have n 1 , n 2 , . . . n N molecules belonging to the classes c = 1, 2, . . . , N, respectively. All classes of molecules diffuse asymmetrically, but with the same asymmetrical rates, whenever they encounter empty sites (holes) at nearestneighbor sites. However, when molecules of different classes, c and c ′ (c < c ′ ), are at their minimum separation, the molecules of class c exchange position with the same rate as they diffuse, and consequently the molecules in the class c see no difference between molecules belonging to the classes c ′ > c and holes.
We now introduce a generalization of the above model, where instead of having unit size, The time evolution of the probability distribution P ({β}, t), of a given configuration {β} is given by the master equation diffusion to the left and interchange of particles As we see from (4), particles belonging to a given class c interchange positions with those of class c ′ > c with the same rate as they interchange positions with the empty sites (diffusion).
We should remark however that unless the particles in class c ′ have unit size (s c = 1), the net effect of these particles in those of class c is distinct from the effect produced by the holes, since as the result of the exchange the particles in class c will move by s c ′ lattice size units, accelerating its diffusion.
The master equation (1) can be written as a Schrödinger equation in Euclidean time (see Ref. [3] for general application for two body processes) if we interpret |P >≡ P ({β}, t) as the associated wave function. If we represent β i as |β > i the vector |β > 1 ⊗|β > 2 ⊗ · · · ⊗ |β > L will give us the associated Hilbert space. The process (2)-(4) gives us the Hamiltonian (see Ref. [3] for general applications) with and periodic boundary conditions. The matrices E α,β are (N + 1) × (N + 1) matrices with a single nonzero element (E α,β ) i,j = δ α,i δ β,j (α, β, i, j = 0, . . . , N). The projector P in (6), projects out from the associated Hilbert space the vectors |{β} > which represent forbidden positions of the molecules due to their finite size, which mathematically means that for all

III. THE BETHE ANSATZ EQUATIONS
We present in this section the exact solution of the general quantum chain (6). A pedagogical presentation for the particular case where N = 2 was presented in [26].
Due to the conservation of particles in the diffusion and interchange processes the total number of particles n 1 , n 2 , . . . , n N in each class are good quantum numbers and consequently we can split the associated Hilbert space into block disjoint sectors labeled by the numbers n 1 , n 2 . . . n N (n i = 0, 1, . . . ; i = 1, . . . , N). We therefore consider the eigenvalue equation where |n 1 , n 2 , . . . , n N >= and n = N i=1 n i is the total number of particles. In (10) |x 1 , Q 1 ; . . . ; x n , Q n > means the configuration where a particle of class Q i (Q i = 1, 2, . . . , N) is at position x i (x i = 1, . . . , L).
Before getting the results for general values of n let us consider initially the cases where we have 1 or 2 particles. n = 1. For one particle on the chain, in any class c = 1, 2, . . . , N, as a consequence of the translational invariance of (6) it is simple to verify directly that the eigenfunctions are the momentum-k eigenfunctions with and energy given by n =2. For two particles of classes Q 1 and Q 2 (Q 1 , Q 2 = 1, 2, . . . , N) on the lattice, the eigenvalue equation (9) gives us two distinct relations depending on the relative location of the particles. The first relation applies to the case in which a particle of class Q 1 (size s Q 1 ) is at position x 1 and a particle Q 2 (size s Q 2 ) is at position x 2 , where x 2 > x 1 + s Q 1 . We obtain in this case the relation where we have used the relation ǫ + + ǫ − = 1. This last equation can be solved promptly by the ansatz with energy where are free parameters to be fixed. In (16) the summation is over the permutations P = P 1 , P 2 of (1,2). The second relation applies when In this case instead of (15) we have If we now substitute the ansatz (16) with the energy (17), the constants A Q 1 ,Q 2

12
and A Q 1 ,Q 2

21
, initially arbitrary, should now satisfy where D l,m = −(ǫ + + ǫ − e i(k l +km) ). (20) At this point it is convenient to consider separately the case where Q 1 = Q 2 from those and the cases Q 1 = Q 2 give us the equations Performing the above summation we obtain, after lengthy but straightforward algebra, the following relation among the amplitudes Equations (21) and (22) can be written in a compact form with where we have introduced the S matrix. From (21) and (22) this S matrix has only N(2N −1) non zero elements, namely Equations (23) do not fix the "wave numbers" k 1 and k 2 . In general, these numbers are complex, and are fixed due to the cyclic boundary condition which from (16) give the relations This last equation, when solved by exploiting (23) where the sum extends over all permutations P of the integers 1, 2, . . . , n, and n = N i=1 n i is the total number of particles.
The application of the translation operator in the above wave functions implies that (10) are also eigenfunctions of the momentum operator with eigenvalues p = n j=1 k j = 2πl L , (l = 0, 1, . . . , L − 1).
We can verify by a long and straightforward calculation that for arbitrary number of classes N and values of the sizes s 1 , s 2 , . . . , s N , the S matrix (25), satisfies the Yang-Baxter equation (34), and consequently we may use relations (31) and (33) to obtain the eigenenergies of the Hamiltonian (6). Applying relation (31) n times on the right of equation (33) we obtain a relation between the amplitudes with the same ordering in the lower indices: where we have introduced the harmless extra sum (see [26] for ilustrations of the above equations). In order to fix the values of {k j } we should solve (35), i.e., we should find the eigenvalues Λ(k) of the matrix with periodic boundary condition The Bethe-ansatz equations which fix the set {k l } will be given from (35) by e −ik j N = (−1) n−1 n l=1 Ξ l,j Λ(k j ), j = 1, . . . , n.
The matrix T (k) has dimension N n × N n and can be interpreted as the transfer matrix of a inhomogeneous N(2N −1)-vertex model in a two dimensional lattice with periodic boundary condition in the horizontal direction (n sites). Due to the special form of the S matrix (25) the eigenvalues of (37) are invariant under a local gauge transformation where for each factor S(k P l , k) in (37): where φ (l) α (l = 1, . . . , L; α = 1, . . . , N) are arbitrary functions. If we perform the transformation (40) with the special choice the equivalent transfer matrix to be diagonalized is given bỹ where with the twisted boundary conditioñ The matrixS in (43) and (44) is obtained from those in (25) by taking the size of all particles equal to the unity. In this way the problem is transformed into the evaluation of the eigenvalues of a regular (all particles with size 1) inhomogeneous transfer matrix T 0 with n(2N − 1) non-zero vertex and twisted boundary condition.

Diagonalization of T 0 (k)
The simplest way to diagonalize T 0 is through the introduction of the monodromy matrix M(k) [25], which is a transfer matrix of the inhomogeneous vertex model under consideration, where the first and last link in the horizontal direction are fixed to the values µ 1 and The monodromy matrix M for µ 1 , ν 1 , µ n+1 , ν n+1 = 1, 2, . . . , N.This relation follows directly from successive applications of the Yang-Baxter equations (34) (see [26], for a graphical representation of these equations).
In order to proceed we need now to diagonalize the new transfer matrix T 1 (k), that is and then (58) give us where, using the fact thatS N,N N,N (k l , k) = 1, In order to proof that Λ (0) and |K (1) α , F > are the eigenvalues and eigenvectors of T 0 (k), we 1 , . . . , k (1) m 1 } by requiring that the "unwanted terms" in (61) vanish. Although for N = 2 this calculation is not complicated [26] for arbitrary N it is not simple. Since the expression (62) for the eigenvalues should be valid for arbitrary values of k we can obtain j ) in an alternative way from the following trick [31]. At k = k l ′ ) , j = 1, . . . , m 1 .
Notice that our result in (63) does not depend on the particular ordering of the additional variables k (1) j (j = 1, . . . , m 1 ). This means that if instead of the ordering chosen in (54), we chose the reverse order, namely, j . This value can be obtained from (65) which gives a generalization of (67) The condition (63) is then replaced by where now we need to find the relations that fix {k (2) j }. Iterating this process we find the generalization of (69) , (j = 1, 2, . . . , m l ; l = 0, 1, . . . , N − 2).
Equations (67)  (ǫ − e ik (0) and and m l = N −l j=1 n j , l = 0, . . . , N (m 0 = N, m N = 0). It is interesting to observe that in the particular case where n 2 = n 3 = . . . = n N = 0 we obtain the Bethe-ansatz equations, recently derived [15] (see also [14]), for the asymmetric diffusion problem with particles of size s 1 . Also the case s 1 = s 2 = . . . = s N = 1 give us the corresponding Bethe ansatz equations for the standard problem of N type of particles in hierarchical order. The Bethe-ansatz solution in the particular case of N=2 with a a single particle of class 2 (n 1 = n − 1, n 2 = 1) was derived recently [32]. The Bethe-ansatz equations for the fully asymmetric problem is obtained by setting in (72)-(73) ǫ + = 1 and ǫ − = 0.

IV. CONCLUSIONS AND GENERALIZATIONS
We obtained through the Bethe ansatz the exact solution of the problem where particles belonging to N distinct classes with hierarchical order diffuse as well interchange positions with rates depending on their relative hierarchy. We show that the exact solution can also be derived in the general case where the particles have arbitrary sizes. s c ′ = 0, contrary to the case s c ′ > 1, where they "accelerate" the diffusion of the particles in classes c < c ′ , now they "retard" the diffusive motion of these particles. The quantum Hamiltonian in the cases where the particles have size zero is obviously not given by (6) but can be written in terms of spin S = ∞ quantum chains. Another further extension of our model is obtained by considering an arbitrary mixture of molecules, where molecules in the same hierarchy may have distinct sizes. The results presented in [15] correspond to the particular case of this generalization where N = 1 (simple diffusion). For general N the S matrix we obtain in (25) is also a solution of the Yang-Baxter equation (34), but the diagonalization of the transfer matrix of the associated inhomogeneous vertex model is more complicated. The Bethe-ansatz equations in the case of the asymmetric diffusion, with particles of unit size [10,11], or with arbitrary size [15], were used to obtain the finite-size corrections of the mass gap G N of the associated quantum chain. The real part of these finite-size corrections are governed by the dynamical critical exponent z, i. e., The calculation of the exponent z for the model presented in this paper, with particles of