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arXiv:cond-mat/0009157 · Brazilian Journal of Physics

Exact solution of asymmetric diffusion with N classes of particles of arbitrary size and hierarchical order

Francisco Castilho Alcaraz, R. Z. Bariev

Published Dec 1, 2000• 32 citations
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Abstract

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 the lattice spacing. The solution is derived via a Bethe ansatz of nested type.

DOI
10.1590/s0103-97332000000400004

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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 the lattice spacing. The solution is derived via a Bethe ansatz of nested type.
The exact solution of the asymmetric exclusion problem with first- and scond-class particles is presented. In this model the particles (size 1) of both classes are located at lattice points, and diffuse with equal asymmetric rates, but particles in the first class do not distinguish those in the second class from holes (empty sites). We generalize and solve exactly this model by considering molecules in the first and second class with sizes s1 and s2 (s1, s2 = 0, 1, 2, ...), in units of lattice spacing, respectively. The solution is derived by a Bethe ansatz of nested type. We give a simple pedagogical presentation of the Bethe ansatz solution of the problem which can easily be followed by a reader not specialized in exactly integrable models.
The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schrodinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0; 1; 2; ...) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have the diffusion of particles belonging to N distinct classes of particles (c = 1; ... ; N), with hierarchical order and arbitrary sizes, is also presented. Our matrix product ansatz asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the ansatz depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model.
A generalization of the simple exclusion asymmetric model is introduced. In this model an arbitrary mixture of molecules with distinct sizes s=0,1,2, ..., in units of lattice space, diffuses asymmetrically on the lattice. A related surface growth model is also presented. Variations of the distribution of the molecules sizes may change the excluded volume almost continuously. We solve the model exactly through the Bethe ansatz and the dynamical critical exponent z is calculated from the finite-size corrections of the mass gap of the related quantum chain. Our results show that for an arbitrary distribution of molecules, the dynamical critical behavior is on the Kardar-Parizi-Zhang universality.
A one-dimensional asymmetric diffusion model on a periodic chain is proposed. The model is defined in terms of the master equation. In contrast with the asymmetric simple exclusion process, particles are not subject to the exclusion interaction: each lattice site can accommodate more than one particle. The model is solved by the Bethe ansatz method and the resulting Bethe equation is analyzed in the thermodynamic limit. The finite size correction of the energy gap is calculated to be ${O(L}^{\ensuremath{-}3/2}),$ where L denotes the length of the chain.
We define a family of asymmetric processes for particles on a one-dimensional lattice, depending on a continuous parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$, interpolating between the completely asymmetric processes (for $\ensuremath{\lambda}=1)$ and the n=1n=1n=1 drop-push models (for $\ensuremath{\lambda}=0)$. For arbitrary $\ensuremath{\lambda}$, the model describes an exclusion process, in which a particle pushes its right neighboring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation.