The problem of ballistically controlled annihilation is revisited for general initial velocity di... more The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t)~t-ξ and the root-mean-square velocity v¯~t-γ in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents ξ and γ are obtained in arbitrary dimension as a function of the parameter μ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.
Physical Review E Statistical Nonlinear and Soft Matter Physics, Mar 12, 2010
The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relati... more The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature.
Physica a Statistical Mechanics and Its Applications, Feb 1, 1997
The dynamics of reaction-diffusion systems in low dimensions is often driven by fluctuations and ... more The dynamics of reaction-diffusion systems in low dimensions is often driven by fluctuations and a simple-minded description in terms of rate equations is not sufficient. Moreover, the emergence of complex patterns in such systems can involve simultaneously several new mechanisms as, for example, aggregation and precipitation. We shall show that the properties of these complex systems can be well understood in the framework of a mesoscopic description based on cellular automata models.
Recently some two-dimensional models with double symmetric absorbing states were shown to share t... more Recently some two-dimensional models with double symmetric absorbing states were shown to share the same critical behaviour that was called the voter universality class. We show, that for an absorbing-states Potts model with finite but further than nearest neighbour range of interactions the critical point is splitted into two critical points: one of the Ising type, and the other of the directed percolation universality class. Similar splitting takes place in the three-dimensional nearest-neighbour model.
The problem of ballistically controlled annihilation is revisited for general initial velocity di... more The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t)~t-ξ and the root-mean-square velocity v¯~t-γ in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents ξ and γ are obtained in arbitrary dimension as a function of the parameter μ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.
Physical Review E Statistical Nonlinear and Soft Matter Physics, Mar 12, 2010
The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relati... more The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature.
Physica a Statistical Mechanics and Its Applications, Feb 1, 1997
The dynamics of reaction-diffusion systems in low dimensions is often driven by fluctuations and ... more The dynamics of reaction-diffusion systems in low dimensions is often driven by fluctuations and a simple-minded description in terms of rate equations is not sufficient. Moreover, the emergence of complex patterns in such systems can involve simultaneously several new mechanisms as, for example, aggregation and precipitation. We shall show that the properties of these complex systems can be well understood in the framework of a mesoscopic description based on cellular automata models.
Recently some two-dimensional models with double symmetric absorbing states were shown to share t... more Recently some two-dimensional models with double symmetric absorbing states were shown to share the same critical behaviour that was called the voter universality class. We show, that for an absorbing-states Potts model with finite but further than nearest neighbour range of interactions the critical point is splitted into two critical points: one of the Ising type, and the other of the directed percolation universality class. Similar splitting takes place in the three-dimensional nearest-neighbour model.
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Papers by Michel Droz