Giada Basile - Academia.edu (original) (raw)

Papers by Giada Basile

Physical Review Letters, May 22, 2006

Anomalous large thermal conductivity has been observed numerically and experimentally in oneand t... more Anomalous large thermal conductivity has been observed numerically and experimentally in oneand two-dimensional systems. There is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimension 1 and 2 if momentum is conserved, while it remains finite in dimension d ≥ 3. We consider a system of harmonic oscillators perturbed by a non-linear stochastic dynamics conserving momentum and energy. We compute explicitly the time correlation function of the energy current CJ (t), and we find that it behaves, for large time, like t −d/2 in the unpinned cases, and like t −d/2−1 when an on site harmonic potential is present. This result clarifies the role of conservation of momentum in the anomalous thermal conductivity in low dimensions.

HAL (Le Centre pour la Communication Scientifique Directe), Jul 1, 2011

We prove diffusive behaviour of the energy fluctuations in a system of harmonic oscillators with ... more We prove diffusive behaviour of the energy fluctuations in a system of harmonic oscillators with a stochastic perturbation of the dynamics that conserves energy and momentum. The results concern pinned systems or lattice dimension d ≥ 3, where the thermal diffusivity is finite.

Springer eBooks, 2016

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains... more We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.

arXiv (Cornell University), Nov 24, 2021

We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a ... more We analyze the large deviations for a discrete energy Kac-like walk. In particular, we exhibit a path, with probability exponentially small in the number of particles, that looses energy.

I present some results obtained together with D. Benedetto and L. Bertini on a gradient flow form... more I present some results obtained together with D. Benedetto and L. Bertini on a gradient flow formulation of linear kinetic equations, in terms of an entropy dissipation inequality. The setting includes the current as a dynamical variable. As an application I discuss the diffusive limit of linear Boltzmann equations and show that the rescaled entropy inequality asymptotically provides the corresponding inequality for heat equation.

Http Www Theses Fr, 2007

PARIS-DAUPHINE-BU (751162101) / SudocSudocFranceItalyFRI

International Journal of Modern Physics B, 2004

We show that the equilibrium macroscopic entropy of a generic non-reversible Kawasaki–Glauber dyn... more We show that the equilibrium macroscopic entropy of a generic non-reversible Kawasaki–Glauber dynamics is a non-local functional of the density. This implies that equilibrium correlations extend to macroscopic distances.

We consider lattice dynamics with a small stochastic perturbation of order ε and prove that for a... more We consider lattice dynamics with a small stochastic perturbation of order ε and prove that for a space-time scale of order ε −1 the Wigner function evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/ √ t, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method.

$Research paper thumbnail of Convergence of a kinetic equation to a fractional diffusion equation$

A linear Boltzmann equation is interpreted as the forward equation for the probability density of... more A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y (t)) on (T × R), where T is the one-dimensional torus. K(t) is a autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y (t) is an additive functional of K, defined as t 0 v(K(s))ds, where |v| ∼ 1 for small k. We prove that the rescaled process N −2/3 Y (N t) converge in distribution to a symmetric Lévy process, stable with index α = 3/2.

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed... more Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collisions (Kac’s model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov’s theorem to the microcanonical ensemble and large deviations for the Kac’s model in the microcanonical setting.

Abstract. We consider a point particle moving in a random distribution of obstacles described by ... more Abstract. We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation. 1.

Mathematics in Engineering, 2022

Journal of Statistical Physics, 2021

We introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but n... more We introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.

Abstract: We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it... more Abstract: We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t−d/2 in the unpinned case and like t−d/2−1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases. 1.

Kinetic & Related Models, 2019

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface... more We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature T in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution ρ(t, y) of a heat equation with the boundary condition ρ(t, 0) ≡ T .

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2020

We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling... more We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.

Communications in Mathematical Physics, 2015

We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that,... more We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick's law, with the diffusion coefficient determined by the Green-Kubo formula.

Lecture Notes in Physics, 2016

We consider a class of continuous time Markov chains on a compact metric space that admit an inva... more We consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with absorbing states. We prove the joint large deviation principle for the empirical measure and flow. Due to the lack of uniform ergodicity, the zero level set of the rate function is not a singleton. As corollaries, we obtain the Donsker-Varadhan rate function for the empirical measure and a variational expression of the rate function for the empirical flow.