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Reaction-diffusion processes described by three-state quantum chains and integrability

Silvio R. Dahmen, Bethe Ansatz

Published Jan 1, 1995• 12 citations
OpenAlex

Abstract

The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrödinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain. Within this approach we search for all 3-state integrable quantum chains whose spectra are known and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the Uq ̂ SU(2)-invariant model with external fields and the 3-state UqSU(P/M)-invariant Perk-Schultz models with external fields. A nonlocal similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction. Key words: Reaction-diffusion models, integrable quantum chains,

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Reaction-diffusion processes described by three-state quantum chains and integrability

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The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrodinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain with known properties. Within this approach we search for three-state integrable quantum chains with known spectra and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the U q SU(2)-invariant model with external fields and the three-state U q SU(P/M)-invariant Perk-Schultz models with external fields. A non-local similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction for N-state Hamiltonians (N>2).

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Institute of Physics eBooks•Nov 26, 2004•0 citations

Reaction-diffusion processes and their connection with integrable quantum spin chains

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This chapter focuses on the quantum Hamiltonian formulation of nonequilibrium processes which, in turn, is based on the master equation. It discusses a few basic facts about Hecke algebras. The chapter shows that the integrability of certain singlespecies reaction–diffusion processes, through their relation to integrable vertex models. It reviews some further methods such as spectral and partial integrability, the free-fermion technique, similarity transformations or diffusion algebras. The chapter shows how the introduced concept of local scale-invariance might become useful in the description of nonequilibrium ageing phenomena. A class of non-equilibrium models which are particularly simple to formulate are the so-called reaction–diffusion processes. The description of reaction–diffusion processes with pair reactions in low dimensions requires a truly microscopic approach beyond kinetic reaction–diffusion equations while these equations may well turn out to be adequate for multi-particle reactions. In trying to extract explicit information on certain reaction–diffusion systems, the integrability of the quantum Hamiltonian H plays a central role.

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The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrödinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain. Within this approach we search for all 3-state integrable quantum chains whose spectra are known and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the Uq ̂ SU(2)-invariant model with external fields and the 3-state UqSU(P/M)-invariant Perk-Schultz models with external fields. A nonlocal similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction. Key words: Reaction-diffusion models, integrable quantum chains,
The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrodinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain with known properties. Within this approach we search for three-state integrable quantum chains with known spectra and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the U q SU(2)-invariant model with external fields and the three-state U q SU(P/M)-invariant Perk-Schultz models with external fields. A non-local similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction for N-state Hamiltonians (N>2).
This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schrodinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.
This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains.Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems.Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field kinetic equations inapplicable.Starting from the master equation rewritten as a Schrödinger equation with imaginary time, the associated quantum Hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains.The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz.Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly.An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of nonequilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.
This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schrödinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.
This chapter focuses on the quantum Hamiltonian formulation of nonequilibrium processes which, in turn, is based on the master equation. It discusses a few basic facts about Hecke algebras. The chapter shows that the integrability of certain singlespecies reaction–diffusion processes, through their relation to integrable vertex models. It reviews some further methods such as spectral and partial integrability, the free-fermion technique, similarity transformations or diffusion algebras. The chapter shows how the introduced concept of local scale-invariance might become useful in the description of nonequilibrium ageing phenomena. A class of non-equilibrium models which are particularly simple to formulate are the so-called reaction–diffusion processes. The description of reaction–diffusion processes with pair reactions in low dimensions requires a truly microscopic approach beyond kinetic reaction–diffusion equations while these equations may well turn out to be adequate for multi-particle reactions. In trying to extract explicit information on certain reaction–diffusion systems, the integrability of the quantum Hamiltonian H plays a central role.
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.