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Influence des effets de taille finie sur la propagation d'un front & Distribution de l'énergie libre d'un polymère dirigé en milieu aléatoire

Abstract : In the first part, we have studied the effect of noise on the speed of fronts described by equations similar to Fisher-Kolmogorov's equation. These equations usually appear as the limit of a stochastic model involving N particles when N goes to infinity. They have many solutions, but the marginaly stable velocity v^* is selected for a localized initial condition. We have shown that if we take into account the discreteness of the microscopic system by adding a cut-off of order 1/N in the tail of the front, then, whatever the initial conditions are, the propagating velocity v_N is close to v^* and the difference v^* - v_N is of order (log N)^(-2). These results can be applied to the stochastic model: by doing simulations involving up to 10^14 particles, we have observed a correction to the speed which is compatible with the one obtained in the model with a cut-off. With the method we used, we can also recover Bramson's results on the influence of initial conditions on the velocity of a front. In the second part, we have studied directed polymers in a random medium when the width of this medium is finite. The replica method reduces the calculation of the free energy of a directed polymer to a quantum problem with n particles in interaction. This model can be solved using the Bethe Ansatz, but the solutions must be extended to non-integer n in order to relate to the free energy of a polymer. We have presented a method that allowed us to calculate the exact first cumulants of this free energy. Moreover, for a periodic transverse dimension, one can calculate all those cumulants when the width of the system is large and thus determine the distribution of the free energy. This distribution is the same as the one found in the ASEP model and seems to be an universal property of the KPZ equation.
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https://tel.archives-ouvertes.fr/tel-00000922
Contributor : Éric Brunet <>
Submitted on : Friday, December 7, 2001 - 12:12:21 PM
Last modification on : Thursday, October 29, 2020 - 3:01:32 PM
Long-term archiving on: : Tuesday, June 2, 2015 - 3:05:16 PM

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  • HAL Id : tel-00000922, version 1

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Éric Brunet. Influence des effets de taille finie sur la propagation d'un front & Distribution de l'énergie libre d'un polymère dirigé en milieu aléatoire. Analyse de données, Statistiques et Probabilités [physics.data-an]. Université Paris-Diderot - Paris VII, 2000. Français. ⟨tel-00000922⟩

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