General solutions for multispin two-time correlation and response functions in the Glauber-Ising chain

The kinetic Glauber–Ising spin chain is one of the very few exactly solvable models of non-equilibrium statistical mechanics. Nevertheless, existing solutions do not yield tractable expressions for two-time correlation and response functions of observables involving products of more than one or two spins. We use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions. From this general solution follow closed expressions for arbitrary multispin two-time correlation and response functions, for the case where the system is quenched from equilibrium at Ti > 0 to some arbitrary T ≥ 0. By way of application, we give the results for two- and four-spin two-time correlation and response functions. From the standard mapping, these also imply new exact results for two-time particle correlation and response functions in one-dimensional diffusion limited annihilation.

[1]  ENERGY FLOW, PARTIAL EQUILIBRATION, AND EFFECTIVE TEMPERATURES IN SYSTEMS WITH SLOW DYNAMICS , 1997, cond-mat/9611044.

[2]  J. Brey,et al.  Stretched exponential decay at intermediate times in the one-dimentional Ising model at low temperatures , 1993 .

[3]  E. Lippiello,et al.  Fluctuation dissipation ratio in the one-dimensional kinetic ising model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  J. Brey,et al.  Aging in the one-dimensional Ising model with Glauber dynamics , 1997 .

[5]  Cugliandolo,et al.  Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. , 1993, Physical review letters.

[6]  Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Quantum corrections to the Weizsäcker-Williams gluon distribution function at small x. , 1995, Physical review. D, Particles and fields.

[8]  R. Glauber.  Time‐Dependent Statistics of the Ising Model , 1963 .

[9]  Response of non-equilibrium systems at criticality: exact results for the Glauber-Ising chain , 1999, cond-mat/9911348.

[10]  J. Spouge,et al.  Exact solutions for a diffusion-reaction process in one dimension. , 1988, Physical review letters.

[11]  Effective temperatures in simple non-mean-field systems , 2001, cond-mat/0111241.

[12]  Exact solution for the generating function of correlators of the kinetic Glauber-Ising model , 1998 .

[13]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[14]  The duality relation between Glauber dynamics and the diffusion - annihilation model as a similarity transformation , 1996, cond-mat/9609292.

[15]  Irwin Oppenheim,et al.  Decay of correlations. III. Relaxation of spin correlations and distribution functions in the one-dimensional ising lattice , 1970 .

[16]  H. Hinrichsen,et al.  Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results , 1995 .

[17]  Fluctuation-dissipation relations in the activated regime of simple strong-glass models. , 2001, Physical review letters.

[18]  Glenn H. Fredrickson,et al.  Kinetic Ising model of the glass transition , 1984 .

[19]  B. U. Felderhof.  Spin relaxation of the Ising chain , 1971 .

[20]  C. Newman,et al.  Aging in 1D discrete spin models and equivalent systems. , 2001, Physical review letters.

[21]  Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Brey,et al.  Low-temperature relaxation in the one-dimensional Ising model. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Amar,et al.  Diffusion annihilation in one dimension and kinetics of the Ising model at zero temperature. , 1990, Physical review. A, Atomic, molecular, and optical physics.