We consider a conservative system of stochastic PDE’s, namely a one dimensional phase field model... more We consider a conservative system of stochastic PDE’s, namely a one dimensional phase field model perturbed by an additive space–time white noise. We prove a global existence and uniqueness result in a space of continuous functions on + × . This result is obtained by extending previous results of Doering [3] on the stochastic Allen–Cahn equation.
Interface dynamics describes the evolution of systems after phase segregation. In a quenching exp... more Interface dynamics describes the evolution of systems after phase segregation. In a quenching experiment a system is initially in thermodynamic equilibrium with a reservoir which is then cooled down below the critical temperature of the system. This process is usually so fast that the macroscopic state of the system does not change significantly. After the cooling it is no longer in equilibrium with the reservoir and one usually models the successive evolution by supposing its state still stationary but unstable. It is therefore very sensitive to external perturbations as those of stochastic nature coming from the reservoir. As soon as the state changes, because of its instability, the deterministic driving forces internal to the system take over, driving it toward the stable phases and the phase separation phenomena take place. Equilibrium is not reached yet at this stage. When the system is spatially extended it reaches locally a thermodynamic stable phase, but since there are several equally accessible phases, there is no reason for the equilibria of faraway regions to coincide. The typical picture at the end of this stage is then a collection of clusters of different phases with interfaces in between. The successive stage of the evolution is the interface dynamics which describes the competition between phases. A rigorous derivation of the whole picture from microscopic models has not yet been carried out in a systematic and rigorous theory as in equilibrium statistical mechanics. In the last years however some results have been obtained in the context of particular and simpler models. This thesis focuses on interface dynamics referring to two specific models with non conserved order parameter and with two symmetric stable phases. The evolution of the interfaces is ruled by the motion by mean curvature
Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli st... more Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli studenti della Laurea Triennale in Matematica dell’Università Sapienza di Roma
. In this paper we prove a weak large deviation principle for the empiricaldistribution of Ising ... more . In this paper we prove a weak large deviation principle for the empiricaldistribution of Ising spins in d 2 dimensions when the interaction is determined by a Kacpotential and the temperature is below the critical value. We prove that the rate functionis proportional to the area of the interface by a factor which identifies the surface tension.Its value is
We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dir... more We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.
We consider a stochastic \begin{document}$ N $\end{document}-particle system on a torus in which ... more We consider a stochastic \begin{document}$ N $\end{document}-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration.
Mathematical Models and Methods in Applied Sciences
We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Pa... more We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes,...
We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl w... more We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl when the initial vorticity is very concentrated in N disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in N disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.
We study the time evolution of a system of infinitely many charged particles confined by an exter... more We study the time evolution of a system of infinitely many charged particles confined by an external magnetic field in an unbounded cylindrical conductor and mutually interacting via the Coulomb force. We prove the existence, uniqueness and quasi-locality of the motion. Moreover, we give some nontrivial bounds on its long time behavior.
We consider a conservative system of stochastic PDE’s, namely a one dimensional phase field model... more We consider a conservative system of stochastic PDE’s, namely a one dimensional phase field model perturbed by an additive space–time white noise. We prove a global existence and uniqueness result in a space of continuous functions on + × . This result is obtained by extending previous results of Doering [3] on the stochastic Allen–Cahn equation.
Interface dynamics describes the evolution of systems after phase segregation. In a quenching exp... more Interface dynamics describes the evolution of systems after phase segregation. In a quenching experiment a system is initially in thermodynamic equilibrium with a reservoir which is then cooled down below the critical temperature of the system. This process is usually so fast that the macroscopic state of the system does not change significantly. After the cooling it is no longer in equilibrium with the reservoir and one usually models the successive evolution by supposing its state still stationary but unstable. It is therefore very sensitive to external perturbations as those of stochastic nature coming from the reservoir. As soon as the state changes, because of its instability, the deterministic driving forces internal to the system take over, driving it toward the stable phases and the phase separation phenomena take place. Equilibrium is not reached yet at this stage. When the system is spatially extended it reaches locally a thermodynamic stable phase, but since there are several equally accessible phases, there is no reason for the equilibria of faraway regions to coincide. The typical picture at the end of this stage is then a collection of clusters of different phases with interfaces in between. The successive stage of the evolution is the interface dynamics which describes the competition between phases. A rigorous derivation of the whole picture from microscopic models has not yet been carried out in a systematic and rigorous theory as in equilibrium statistical mechanics. In the last years however some results have been obtained in the context of particular and simpler models. This thesis focuses on interface dynamics referring to two specific models with non conserved order parameter and with two symmetric stable phases. The evolution of the interfaces is ruled by the motion by mean curvature
Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli st... more Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli studenti della Laurea Triennale in Matematica dell’Università Sapienza di Roma
. In this paper we prove a weak large deviation principle for the empiricaldistribution of Ising ... more . In this paper we prove a weak large deviation principle for the empiricaldistribution of Ising spins in d 2 dimensions when the interaction is determined by a Kacpotential and the temperature is below the critical value. We prove that the rate functionis proportional to the area of the interface by a factor which identifies the surface tension.Its value is
We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dir... more We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.
We consider a stochastic \begin{document}$ N $\end{document}-particle system on a torus in which ... more We consider a stochastic \begin{document}$ N $\end{document}-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration.
Mathematical Models and Methods in Applied Sciences
We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Pa... more We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes,...
We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl w... more We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl when the initial vorticity is very concentrated in N disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in N disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.
We study the time evolution of a system of infinitely many charged particles confined by an exter... more We study the time evolution of a system of infinitely many charged particles confined by an external magnetic field in an unbounded cylindrical conductor and mutually interacting via the Coulomb force. We prove the existence, uniqueness and quasi-locality of the motion. Moreover, we give some nontrivial bounds on its long time behavior.
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Papers by Paolo Buttà