arXiv subjects

Federico Bonetto

Publications and source records attributed to Federico Bonetto.

At least 19 recordsLinked to original sources

arXiv2026

A Kac system interacting with two heat reservoirs: the shearing case

We study a system formed by $M$ particles moving in 3 dimensions and interacting with two heat reservoirs, each with $N\gg M$ particles. The system and the reservoirs interact via random collisions and thus evolve via a Kac-type master equation. The initial state of the reservoirs is given by two non-centered Maxwellian distributions; they have temperature $T_+$ and $T_-$ and have average velocity $\vec p_+$ and $\vec p_-$, respectively. We prove that, for times shorter than $\sqrt{N}/M$, the interaction with the two reservoirs is well-approximated by the interaction with two shearing {\it dynamic} Maxwellian thermostats (i.e. heat reservoirs with $N=\infty$). As a byproduct of our analysis, we obtain a uniform in time approximation when $T_+=T_-$ and $\vec p_+=\vec p_-$.

math-ph↗

arXiv2025

Improved Approximation of Infinite Thermostat by Finite Reservoir Using the 3D Kac Model

In this paper, we study a system of $M$ particles interacting with a reservoir of $N$ particles, where $N >> M$, and compare this setup to one where the $M$-particle system interacts with a thermostat of infinite particles. Our goal is to prove a suitable upper bound, uniform in time, on the distance between the states of these two setups, given an initial Maxwellian state for both the reservoir and thermostat. Previous work has analyzed this problem using the one-dimensional Kac Model of gas collisions and an $L^2$ norm to define distance; the result was a bound which scaled with $M/\sqrt{N}$. In this paper, we use the $L^2$ norm and the three-dimensional generalization of the Kac Model to prove a bound whose long-term behavior scales with $M/N$.

math-ph↗

arXiv2025

Approach to equilibrium for a particle interacting with a harmonic thermal bath

We study the long time evolution of the position-position correlation function $C_{\alpha,N}(s,t)$ for a harmonic oscillator (the {\it probe}) interacting via a coupling $\alpha$ with a large chain of $N$ coupled oscillators (the {\it heat bath}). At $t=0$ the probe and the bath are in equilibrium at temperature $T_P$ and $T_B$, respectively. We show that for times $t$ and $s$ of the order of $N$, $C_{\alpha,N}(s,t)$ is very well approximated by its limit $C_{\alpha}(s,t)$ as $N\to\infty$. We find that, if the frequency $\Omega$ of the probe is in the spectrum of the bath, the system appears to thermalize, at least at higher order in $\alpha$. This means that, at order 0 in $\alpha$, $C_\alpha(s,t)$ equals the correlation of a probe in contact with an ideal stochastic {\it thermostat}, that is forced by a white noise and subject to dissipation. In particular we find that $\lim_{t\to\infty} C_\alpha(t,t)=T_B/\Omega^2$ while that $\lim_{\tau\to\infty} C_\alpha(\tau,\tau+t)$ exists and decays exponentially in $t$. Notwithstanding this, at higher order in $\alpha$, $C_{\alpha}(s,t)$ contains terms that oscillate or vanish as a power law in $|t-s|$. That is, even when the bath is very large, it cannot be thought of as a stochastic thermostat. When the frequency of the bath is far from the spectrum of the bath, no thermalization is observed.

cond-mat.stat-mech↗

arXiv2025

A Kac system interacting with two heat reservoirs

We study a system formed by $M$ particles moving in 3 dimension and interacting with 2 heat reservoirs with $N>>M$ particles each. The system and the reservoirs evolve and interact via random collision described by a Kac-type master equation. The initial state of the reservoirs is given by 2 Maxwellian distributions at temperature $T_+$ and $T_-$. We show that, for times much shorter than $\sqrt{N}$ the interaction with the reservoirs is well approximated by the interaction with 2 Maxwellian thermostats, that is, heat reservoirs with $N=\infty$. As a byproduct, if $T_+=T_-$ we extend the results in \cite{BLTV} to particles in 3 dimension.

math-ph↗

arXiv2025

Synchronization and averaging in partially hyperbolic systems with fast and slow variables

We study a family of dynamical systems obtained by coupling an Anosov map on the two-dimensional torus -- the chaotic system -- with the identity map on the one-dimensional torus -- the neutral system -- through a dissipative interaction. We show that the two systems synchronize: the trajectories evolve toward an attracting invariant manifold, and the full dynamics is conjugated to its linearization around the invariant manifold. As a byproduct, we obtain that there exists a unique exponentially mixing physical measure. When the interaction is small, the evolution of the variable which describes the neutral system is very close to the identity; hence, it appears as a slow variable with respect to the variable which describes the chaotic system, and which is wherefore named the fast variable. We demonstrate that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable. More precisely, we prove that the invariant manifold is in probability close to the fixed point of the averaged dynamics and that the difference between the exact evolution of the slow variable, seen from the invariant manifold, and its averaged evolution is in probability exponentially decreasing for arbitrarily large times.

nlin.CD↗

arXiv2024

Analiticity of the Lyapunov exponents of perturbed toral automorphisms

We consider a dynamical system generated by an analytic perturbation $A_\varepsilon$ of an analytic Anosov diffeomorphism $A_0$ of $\TTT^d$. We show that, if $A_0$ admit a splitting of $\mathrm T\mathds T^d$ in $k$ invariant subspaces, there exists a {\it partial conjugation} $\mathcal H_\e$ of $dA_\e$ and $dA_0$ that preserves the splitting and is analytic in $\e$. This show that the splitting can be extended to $A_\e$. As an application of this results, we obtain that the Lyapunov exponents, if non degenerate, are analytic functions of the perturbation.

math.DS↗

arXiv2022

Grand Canonical Evolution for the Kac Model

We study a model of random colliding particles interacting with an infinite reservoir at fixed temperature and chemical potential. Interaction between the particles is modeled via a Kac master equation \cite{kac}. Moreover, particles can leave the system toward the reservoir or enter the system from the reservoir. The system admits a unique steady state given by the Grand Canonical Ensemble at temperature $T=β^{-1}$ and chemical potential $χ$. We show that any initial state converges exponentially fast to equilibrium by computing the spectral gap of the generator in a suitable $L^2$ space and by showing exponential decrease of the relative entropy with respect to the steady state. We also show propagation of chaos and thus the validity of a Boltzmann-Kac type equation for the particle density in the infinite system limit.

math-ph↗

arXiv2020

Decay of Information for the Kac Evolution

We consider a system of $M$ particles in contact with a heat reservoir of $N\gg M$ particles. The evolution in the system and the reservoir, together with their interaction, are modeled via the Kac's Master Equation. We chose the initial distribution with total energy $N+M$ and show that if the reservoir is initially in equilibrium, that is if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decay exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in [2].

math-ph↗

arXiv2019

Autonomous evolution of electron speeds in a thermostatted system: exact results

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere. This verifies previously conjectured behavior. Our results are based on splitting the system's evolution into a "slow" process and an independent "noise". We show that the noise, suitably rescaled, converges a Brownian motion, enhanced in the sense of rough paths. Then we employ the Itô-Lyons continuity theorem to identify the limit of the slow process.

math-ph↗

arXiv2018

Non-integrable fermionic chains near criticality

We compute the Drude weight and the critical exponents as functions of the density in non-integrable generalizations of XXZ or Hubbard chains, in the critical zero temperature regime where Luttinger liquid description breaks down and Bethe ansatz cannot be used. Even in the regions where irrelevant terms dominate, no difference between integrable and non integrable models appear in exponents and conductivity. Our results are based on a fully rigorous two-regime multiscale analysis and a recently introduced partially solvable model.

cond-mat.stat-mech↗

arXiv2018

A Dynamic Analysis of Nash Equilibria in Search Models with Fiat Money

We study the rise in the acceptability fiat money in a Kiyotaki-Wright economy by developing a method that can determine dynamic Nash equilibria for a class of search models with genuine heterogenous agents. We also address open issues regarding the stability properties of pure strategies equilibria and the presence of multiple equilibria. Experiments illustrate the liquidity conditions that favor the transition from partial to full acceptance of fiat money, and the effects of inflationary shocks on production, liquidity, and trade.

econ.EM↗

arXiv2018

A study of deformation localization in nonlinear elastic lattices

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where the onset of non-uniform, quasi-static deformation patterns was associated with the loss of convexity of the interaction potential, and where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These observations are here conducted on other lattice structures, with the goal of identifying models of reduced complexity that are able to provide insight into the key parameters that govern the onset of instability-induced localization. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs. Results illustrate that depending on the choice of spring constants and their relative values, the lattice exhibits in-plane or out-of plane instabilities leading to folding and unfolding. This model is further simplified by considering the one-dimensional case of a spring-mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and illustrates its hysteretic and loading path-dependent behaviors. Finally, the lattice is further reduced to a minimal four mass model which undergoes a folding/unfolding process qualitatively similar to the same process in the central part of a longer chain, helping our understanding of localization in more complex systems. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode-shapes at the bifurcation points.

cond-mat.soft↗

arXiv2017

Entropy decay for the Kac evolution

We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $M\le N$ particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelson's hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.

math-ph↗

arXiv2016

Uniform Approximation of a Maxwellian Thermostat by Finite Reservoirs

We study the evolution of a system of M particles in contact with a large reservoir of N>>M particles. The reservoir is initially in equilibrium at temperature T=1/β. The evolution of the system and reservoir is described via a suitable Kac-style collision process. We show that for large N, this evolution can be effectively described by replacing the reservoir with a Maxwellian thermostat at temperature T. This description provides an approximation that is uniform in time both in a suitable L^2 norm and in the Gabetta-Toscani-Wennberg (GTW) distance.

math-ph↗

arXiv2014

The Kac Model Coupled to a Thermostat

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $β$. The system admits the canonical distribution at inverse temperature $β$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.

math-ph↗

arXiv2013

Entropic Chaoticity for the Steady State of a Current Carrying System

The steady state for a system of N particle under the influence of an external field and a Gaussian thermostat and colliding with random "virtual" scatterers can be obtained explicitly in the limit of small field. We show the sequence of steady state distribution, as N varies, forms a chaotic sequence in the sense that the k particle marginal, in the limit of large N, is the k-fold tensor product of the 1 particle marginal. We also show that the chaoticity properties holds in the stronger form of entropic chaoticity.

nlin.CD↗

arXiv2012

Speed Distribution of N Particles in the Thermostated Periodic Lorentz Gas with a Field

We study the long time evolution and stationary speed distribution of N point particles in 2D moving under the action of an external field E, and undergoing elastic collisions with either a fixed periodic array of convex scatterers, or with virtual random scatterers. The total kinetic energy of the N particles is kept fixed by a Gaussian thermostat which induces an interaction between the particles. We show analytically and numerically that for weak fields this distribution is universal, i.e. independent of the position or shape of the obstacles or the nature of the stochastic scattering. Our analysis is based on the existence of two time scales; the velocity directions become uniformized in times of order unity while the speeds change only on a time scale of O(|E|^-2).

nlin.CD↗

arXiv2011

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.

nlin.CD↗