Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions.

Efficient and accurate integration of stochastic (partial) differential equations with multiplicative noise can be obtained through a split-step scheme, which separates the integration of the deterministic part from that of the stochastic part, the latter being performed by sampling exactly the solution of the associated Fokker-Planck equation. We demonstrate the computational power of this method by applying it to the most absorbing phase transitions for which Langevin equations have been proposed. This provides precise estimates of the associated scaling exponents, clarifying the classification of these nonequilibrium problems, and confirms or refutes some existing theories.

[1]  M. A. Muñoz,et al.  Numerical study of the Langevin theory for fixed-energy sandpiles. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  L. Canet,et al.  Nonperturbative renormalization-group study of reaction-diffusion processes. , 2003, Physical review letters.

[3]  F. Wijland.  Infinitely-many absorbing-state nonequilibrium phase transitions , 2003, cond-mat/0307232.

[4]  H. Hinrichsen,et al.  Epidemic processes with immunization. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  F. Wijland.  Universality class of nonequilibrium phase transitions with infinitely many-absorbing-states , 2002, cond-mat/0209202.

[6]  G. Ódor.  Universality classes in nonequilibrium lattice systems , 2002, cond-mat/0205644.

[7]  H. Hinrichsen,et al.  Critical coarsening without surface tension: the universality class of the voter model. , 2001, Physical review letters.

[8]  H. Hilhorst,et al.  Reply to ``Comment on `Critical behavior of a two-species reaction-diffusion problem' '' , 2000, cond-mat/0008126.

[9]  H. Janssen.  Comment on "Critical behavior of a two-species reaction-diffusion problem". , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Alessandro Vespignani,et al.  Field theory of absorbing phase transitions with a nondiffusive conserved field , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  de Freitas Je,et al.  Critical behavior of a two-species reaction-diffusion problem , 2000 .

[12]  Rossi,et al.  Universality class of absorbing phase transitions with a conserved field , 2000, Physical review letters.

[13]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

[14]  M. A. Muñoz,et al.  Paths to self-organized criticality , 1999, cond-mat/9910454.

[15]  H. Hilhorst,et al.  Fluctuation-Induced First-Order Transition in a Nonequilibrium Steady State , 1999, cond-mat/9910351.

[16]  H. Levine,et al.  Interfacial velocity corrections due to multiplicative noise , 1998, cond-mat/9811020.

[17]  M. A. Muñoz,et al.  DRIVING, CONSERVATION, AND ABSORBING STATES IN SANDPILES , 1998, cond-mat/9806249.

[18]  M. A. Muñoz,et al.  Phase Structure of Systems with Infinite Numbers of Absorbing States , 1998 .

[19]  M. A. Muñoz,et al.  Numerical analysis of a Langevin equation for systems with infinite absorbing states , 1997 .

[20]  Germany,et al.  Wilson renormalization of a reaction–diffusion process , 1997, cond-mat/9706197.

[21]  Peter Grassberger,et al.  Spreading in media with long-time memory , 1997 .

[22]  G. Grinstein,et al.  Critical behavior of systems with many absorbing states. , 1996, Physical review letters.

[23]  Dickman,et al.  Hyperscaling in the Domany-Kinzel cellular automaton. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Dickman.  Numerical study of a field theory for directed percolation. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Iwan Jensen.  Critical behavior of the pair contact process. , 1993, Physical review letters.

[26]  Schaub,et al.  Effects of pollution on critical population dynamics. , 1989, Physical review. A, General physics.

[27]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[28]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[29]  Dietrich Stoyan,et al.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences.@@@Elements of Applied Stochastic Processes, 2nd Edition.@@@Comparison Methods for Queues and Other Stochastic Models.@@@Parameter Estimation for Stochastic Processes. , 1986 .

[30]  L. Peliti.  Path integral approach to birth-death processes on a lattice , 1985 .

[31]  M. Doi.  Stochastic theory of diffusion-controlled reaction , 1976 .

[32]  W. Feller.  TWO SINGULAR DIFFUSION PROBLEMS , 1951 .

[33]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[34]  P. Grassberger,et al.  Fock Space Methods for Identical Classical Objects , 1980 .