Effect of Microscopic Noise on Front Propagation

We study the effect of the noise due to microscopic fluctuations on the position of a one dimensional front propagating from a stable to an unstable region in the “linearly marginal stability case.” By simulating a very simple system for which the effective number N of particles can be as large as N=10150, we measure the N dependence of the diffusion constant DN of the front and the shift of its velocity vN. Our results indicate that DN∼(log N)−3. They also confirm our recent claim that the shift of velocity scales like vmin−vN≃K(log N)−2 and indicate that the numerical value of K is very close to the analytical expression Kapprox obtained in our previous work using a simple cut-off approximation.

[1]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[2]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[3]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[4]  James S. Langer,et al.  Propagating pattern selection , 1983 .

[5]  M. Bramson.  Convergence of solutions of the Kolmogorov equation to travelling waves , 1983 .

[6]  Eshel Ben-Jacob,et al.  Pattern propagation in nonlinear dissipative systems , 1985 .

[7]  A. Kerstein.  Computational study of propagating fronts in a Lattice-gas model , 1986 .

[8]  Maury Bramson,et al.  Microscopic selection principle for a diffusion-reaction equation , 1986 .

[9]  Pierre Collet,et al.  Instabilities and fronts in extended systems , 1990 .

[10]  B. Derrida,et al.  Directed polymers in a random medium : l / d expansion and the n-tree approximation , 2022 .

[11]  Francesco Petruccione,et al.  Fluctuation effects on wave propagation in a reaction-diffusion process , 1994 .

[12]  Francesco Petruccione,et al.  The Macroscopic Limit in a Stochastic Reaction-Diffusion Process , 1995, Europhysics Letters (EPL).

[13]  Mareschal,et al.  Langevin approach to a chemical wave front: Selection of the propagation velocity in the presence of internal noise. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Riordan,et al.  Fluctuations and stability of fisher waves. , 1995, Physical review letters.

[15]  Mareschal,et al.  Fluctuation effects on chemical wave fronts. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Sokolov,et al.  Front Propagation and Local Ordering in One-Dimensional Irreversible Autocatalytic Reactions. , 1996, Physical review letters.

[17]  Casademunt,et al.  External fluctuations in front propagation. , 1996, Physical review letters.

[18]  Velocity Selection for Propagating Fronts in Superconductors. , 1996, Physical review letters.

[19]  Bernard Derrida,et al.  Shift in the velocity of a front due to a cutoff , 1997 .

[20]  D. ben-Avraham.  Fisher waves in the diffusion-limited coalescence process A + A ⇄ A , 1998, cond-mat/9805180.

[21]  L. Sander,et al.  Front propagation: Precursors, cutoffs, and structural stability , 1998, patt-sol/9802001.

[22]  J. M. Sancho,et al.  Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise , 1998 .

[23]  Herbert Levine,et al.  Fluctuation-induced diffusive instabilities , 1998, Nature.

[24]  W. Saarloos.  Three basic issues concerning interface dynamics in nonequilibrium pattern formation , 1998, patt-sol/9801002.

[25]  C. Dellago,et al.  Largest Lyapunov Exponent for Many Particle Systems at Low Densities , 1997, chao-dyn/9710020.

[26]  A. Lemarchand,et al.  Perturbation of local equilibrium by a chemical wave front , 1998 .

[27]  José M. Sancho,et al.  NOISE-INDUCED FRONTS , 1999 .

[28]  Herbert Levine,et al.  Interfacial velocity corrections due to multiplicative noise , 1999 .

[29]  A. Lemarchand,et al.  Different description levels of chemical wave front and propagation speed selection , 1999 .

[30]  Raymond Kapral,et al.  Fluctuation effects on quadratic autocatalysis fronts , 1999 .

[31]  Ebert,et al.  Subdiffusive fluctuations of "pulled" fronts with multiplicative noise , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Topological transitions and freezing in XY models and Coulomb gases with quenched disorder: renormalization via traveling waves , 1999, cond-mat/9908335.

[33]  Sokolov,et al.  Front propagation in one-dimensional autocatalytic reactions: the breakdown of the classical picture at small particle concentrations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[34]  A. Lemarchand.  Selection of an Attractor in a Continuum of Stable Solutions: Descriptions of a Wave Front at Different Scales , 2000 .

[35]  William H. Press,et al.  Numerical recipes in C , 2002 .