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Propagating fronts are seen in varieties of non-equilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and mean-field dynamics of the fronts were mostly understood in late 1980s and 1990s. On the other hand, although the earliest work on the effect of fluctuations on propagating fronts dates back to early 1980s, the subject of fluctuating fronts did not reach its adolescence until the mid 1990s. From there onwards the last few years witnessed a surge in activities in the effect of fluctuations on propagating fronts. Scores of papers have been written on this subject since then, contributing to a significant maturity of our understanding, and only recently a full picture of fluctuating fronts has started to emerge. This review is an attempt to collect all the works on fluctuating (propagating) fronts in a coherent and cogent manner in proper perspective. It is based on the idea of making our knowledge in this field available to a broader audience, and it is also expected to help to collect bits and pieces of loose thread-ends together for possible further investigation.
Propagating fronts are seen in varieties of non-equilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and mean-field dynamics of the fronts were mostly understood in late 1980s and 1990s. On the other hand, although the earliest work on the effect of fluctuations on propagating fronts dates back to early 1980s, the subject of fluctuating fronts did not reach its adolescence until the mid 1990s. From there onwards the last few years witnessed a surge in activities in the effect of fluctuations on propagating fronts. Scores of papers have been written on this subject since then, contributing to a significant maturity of our understanding, and only recently a full picture of fluctuating fronts has started to emerge. This review is an attempt to collect all the works on fluctuating (propagating) fronts in a coherent and cogent manner in proper perspective. It is based on the idea of making our knowledge in this field available to a broader audience, and it is also expected to help to collect bits and pieces of loose thread-ends together for possible further investigation.
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=10 150 , we measure the N dependence of the diffusion constant D N of the front and the shift of its velocity v N . Our results indicate that D N t(log N ) &3 . They also confirm our recent claim that the shift of velocity scales like v min &v N & K(log N ) &2 and indicate that the numerical value of K is very close to the analytical expression K approx obtained in our previous work using a simple cut-off approximation.
Montroll’s approach to diffusion-controlled annihilation reactions recently generalized by the present authors to account for the simultaneous displacement of two walkers, is extended by including more complex kinetic schemes and many-body effects. The mean walklength to reaction and the spatial organization of the reactants in a finite planar lattice is evaluated analytically and by Monte Carlo simulations in two representative schemes involving, respectively, a single autocatalytic reaction and an autocatalytic reaction coupled to isomerization. While in the first scheme the results are in qualitative (though not quantitative) accord with mean-field predictions, marked qualitative differences with mean-field behavior are found in the second scheme.
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