Exact diffusion constant for the one-dimensional partially asymmetric exclusion model

We calculate exactly the diffusion constant associated with the fluctuations of the current for the partial asymmetric exclusion model on a ring with an arbitrary number of particles and holes. We also give the diffusion constant of a tagged particle on that ring. Our approach extends, using the deformed harmonic oscillator algebra, a result already known for the fully asymmetric case. In the limit of weak asymmetry, we extract from our exact expression the crossover between the Edwards - Wilkinson and the Kardar - Parisi - Zhang equations in (1 + 1) dimensions.

[1]  Kim.  Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  T. Liggett.  Interacting Particle Systems , 1985 .

[3]  B. Derrida,et al.  Exact solution of the totally asymmetric simple exclusion process: Shock profiles , 1993 .

[4]  P. M. Richards.  Theory of one-dimensional hopping conductivity and diffusion , 1977 .

[5]  Sven Sandow,et al.  On Uq[SU(2)]-Symmetric Driven Diffusion , 1994 .

[6]  Haye Hinrichsen,et al.  ON MATRIX PRODUCT GROUND STATES FOR REACTION-DIFFUSION MODELS , 1996 .

[7]  Bernard Derrida,et al.  Exact diffusion constant for one-dimensional asymmetric exclusion models , 1993 .

[8]  Universality in surface growth: Scaling functions and amplitude ratios. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[9]  Microscopic derivation of Kardar-Parisi-Zhang Equation , 1996, cond-mat/9601067.

[10]  Yicheng Zhang,et al.  Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics , 1995 .

[11]  P. Ferrari,et al.  MICROSCOPIC STRUCTURE OF TRAVELLING WAVES IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS , 1991 .

[12]  Liu,et al.  Time-reversal invariance and universality of two-dimensional growth models. , 1987, Physical review. B, Condensed matter.

[13]  P. Meakin,et al.  Universal finite-size effects in the rate of growth processes , 1990 .

[14]  Peter Fulde,et al.  Theoretical models for superionic conductors , 1980 .

[15]  D. Huse,et al.  Huse, Henley, and Fisher respond. , 1985, Physical review letters.

[16]  J. Krug.  Statistical physics of growth processes , 1995 .

[17]  S. Sandow,et al.  Partially asymmetric exclusion process with open boundaries. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  André Robert Dabrowski,et al.  Stable Limits for Associated Random Variables , 1994 .

[19]  B. Derrida,et al.  Exact diffusion constant of a one-dimensional asymmetric exclusion model with open boundaries , 1995 .

[20]  H. van Beijeren.  Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems , 1991 .

[21]  Spohn,et al.  Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[22]  B. Derrida,et al.  Exact solution of a 1d asymmetric exclusion model using a matrix formulation , 1993 .

[23]  Stinchcombe Rb,et al.  Application of operator algebras to stochastic dynamics and the Heisenberg chain. , 1995 .

[24]  G. Schütz,et al.  Operator Algebra for Stochastic Dynamics and the Heisenberg Chain , 1995 .

[25]  F. Spitzer.  Interaction of Markov processes , 1970 .

[26]  Schütz,et al.  Application of operator algebras to stochastic dynamics and the Heisenberg chain. , 1995, Physical review letters.

[27]  Beate Schmittmann,et al.  Statistical mechanics of driven diffusive systems , 1995 .

[28]  Current fluctuations for the asymmetric simple exclusion process , 1994 .

[29]  A. J. Macfarlane,et al.  On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q , 1989 .

[30]  G. Mil’shtein,et al.  Interaction of Markov Processes , 1972 .

[31]  Eytan Domany,et al.  An exact solution of a one-dimensional asymmetric exclusion model with open boundaries , 1992 .

[32]  Takashi Nagatani,et al.  Shock formation and traffic jam induced by a crossing in the 1D asymmetric exclusion model , 1993 .

[33]  Bernard Derrida,et al.  Exact Steady State Properties of the One Dimensional Asymmetric Exclusion Model , 1994 .

[34]  Lebowitz,et al.  Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[35]  L. C. Biedenharn,et al.  The quantum group SUq(2) and a q-analogue of the boson operators , 1989 .

[36]  Fabian H.L. Essler,et al.  Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries , 1995 .

[37]  C. Kipnis.  Central Limit Theorems for Infinite Series of Queues and Applications to Simple Exclusion , 1986 .

[38]  Krug,et al.  Boundary-induced phase transitions in driven diffusive systems. , 1991, Physical review letters.

[39]  R. Kutner,et al.  Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas , 1985 .

[40]  Pablo A. Ferrari.  The Simple Exclusion Process as Seen from a Tagged Particle , 1986 .

[41]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[42]  Herbert Spohn,et al.  Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors , 1984 .

[43]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[44]  S. Edwards,et al.  The surface statistics of a granular aggregate , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[45]  Maury Bramson,et al.  Shocks in the asymmetric exclusion process , 1988 .

[46]  Pablo A. Ferrari,et al.  Self-diffusion in one-dimensional lattice gases in the presence of an external field , 1985 .

[47]  Krug,et al.  Amplitude universality for driven interfaces and directed polymers in random media. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[48]  Sander,et al.  Ballistic deposition on surfaces. , 1986, Physical review. A, General physics.

[49]  S. Majumdar,et al.  Tag diffusion in driven systems, growing interfaces, and anomalous fluctuations. , 1991, Physical review. B, Condensed matter.