arXiv subjects

Renato Feres

Publications and source records attributed to Renato Feres.

At least 19 recordsLinked to original sources

arXiv2026

Entropy production in Knudsen thermodynamics of compartmented systems

We investigate entropy production and nonequilibrium transport in a class of random dynamical systems modeling a Knudsen gas confined to a compartmented container. The system consists of a single particle undergoing random billiard motion, with collisions leading to random reflection or transmission through semi-reflecting, compartment-separating walls. The stationary entropy production rate is first expressed as the relative entropy between forward and time-reversed path measures. Under a reciprocity assumption, used to introduce temperature into the general random billiard system, it is shown how this information-theoretic definition reduces to the classical thermodynamic formula: the mean energy transferred to the walls divided by their local temperatures, a stochastic Clausius relation. We then develop a modular analysis of compartmented systems. Each open compartment is characterized by a compartment scattering operator and its sojourn statistics. The sequence of compartment entrance states defines a Markov chain whose stationary distribution is used in a renewal-reward theorem to assemble the compartment contributions into the global entropy production rate. The framework is illustrated with a series of examples of increasing complexity governed by a generalized Maxwell-Smoluchowski scattering operator and amenable to detailed and explicit analysis. Such operators are defined by a few parameters: temperature, a partial thermal accommodation, the height of potential barriers, and a porosity coefficient. The central example is a cyclic three-compartment system consisting of two thermal walls at different temperatures and a potential barrier. For full thermal accommodation we obtain closed-form expressions for the entropy production rate and net probability circulation around the cycle, revealing a thermal ratchet effect analogous to thermal transpiration.

math-ph↗

arXiv2026

Pulse-response analysis of a simple reaction-advection-diffusion equation

We analyze a reaction-advection-diffusion (RAD) equation arising in pulse-response studies of transport and reaction in a narrow reactor tube. A short pulse of gas is injected near one end of the reactor, while unreacted gas and reaction products are detected at the outlet. Particular attention is given to the effect of a constant axial advection velocity as a minimal extension of the standard diffusion model. For a single pulse, we obtain analytical expressions for the exit-flow response and for experimentally accessible characteristics, including its moments and peak properties, as functions of the P\'eclet number and the reaction rate. The corresponding diffusion-advection model without reaction defines a standard transport curve that may be used as a baseline for identifying chemical activity. For a first-order irreversible reaction, the reactive and nonreactive exit-flow curves satisfy a simple exponential factorization, allowing the reaction rate to be extracted directly from their ratio. We also extend the single-pulse analysis to a uniform periodic train of pulses. Using Poisson summation, we derive explicit expressions for the Fourier coefficients of the asymptotic periodic exit flow and relate them to the Laplace transform and moments of the single-pulse response. Finally, a stochastic interpretation in terms of reflected diffusion, first-passage times, and exponential killing provides probabilistic meaning for several of the analytical results and unifies the single-pulse, periodic-response, and reaction aspects of the model.

math.AP↗

arXiv2026

Nonholonomic billiards and bounded motion in cylinders

A widely used mathematical model for the bouncing motion of an ideally elastic ball -- referred to in previous work by the first two authors and collaborators as a {\em no-slip billiard} system -- exhibits some notable dynamical behavior that is not well-understood. For example, under certain initial conditions, the axial component of the position of the center of the ball moving inside a vertical solid cylinder under constant gravitational force does not accelerate downward as might be expected but remains bounded. There is not as yet, as far as we know, any analytical study of the bouncing ball dynamics, under gravity, in general cylinders (not necessarily having a circular cross-section) in $\mathbb{R}^3$. In this paper, we propose an approach by comparing the no-slip system with a smooth approximation of it that we call {\em nonholonomic billiards}. It consists of a $4$-dimensional ball rolling on the solid $3$-dimensional cylinder. We first review earlier work on no-slip billiards and their connection with nonholonomic (rolling) systems, explain how nonholonomic billiards approximate the no-slip kind (after work by the first two authors and B. Zhao), and illustrate the relationship with a few numerical case studies that demonstrate the utility of the soft (nonholonomic) system as a helpful tool for exploring the dynamics of no-slip billiard systems.

math.DS↗

arXiv2024

Determination of output composition in reaction-advection-diffusion systems on network reactors

We consider reaction-transport processes in open reactors in which systems of first order reactions involving a number of gas species and solid catalysts can occur at localized active regions. Reaction products flow out of the reactor into vacuum conditions and are collected at an exit boundary. The output composition problem (OCP) is to determine the composition (molar fractions) of the collected gas after the reactor is fully emptied. We provide a solution to this problem in the form of a boundary-value problem for a system of time-independent partial differential equations. We then consider network-like reactors, which can be approximated by a network consisting of a collection of nodes and 1-dimensional branches, with reactions taking place at nodes. For these, it is possible to solve the OCP in a simple and effective way, giving explicit formulas for the output composition as a function of the reaction coefficients and parameters associated with the geometric configuration of the system. Several examples are given to illustrate the method.

physics.chem-ph↗

arXiv2023

Chaotic lensed billiards

Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form $C1_{\mathcal{A}}$, where $C$ is a real valued constant and $1_{\mathcal{A}}$ is the indicator function of an open subset $\mathcal{A}$ of the billiard table whose boundaries (of $\mathcal{A}$ and the table) are piecewise smooth. Trajectories are polygonal lines that undergo either reflection or refraction at the boundary of $\mathcal{A}$ depending on the angle of incidence. After laying out some basic concepts and general facts, in particular reviewing the optical/mechanical analogy that motivates these billiard models, we explore how their dynamical properties depend on the potential parameter $C$ using a number of families of examples. In particular, we explore numerically the Lyapunov exponents for these parametric families and highlight the more salient common properties that distinguish them from standard billiard systems. We further justify some of these properties by characterizing lensed billiards in terms of switching dynamics between two open (standard) billiard subsystems and obtaining mean values associated to orbit sojourn in each subsystem.

nlin.CD↗

arXiv2023

Revisiting Maxwell-Smoluchowski theory: low surface roughness in straight channels

The Maxwell-Smoluchowski (MS) theory of gas diffusion is revisited here in the context of gas transport in straight channels in the Knudsen regime of large mean free path. This classical theory is based on a phenomenological model of gas-surface interaction that posits that a fraction $\vartheta$ of molecular collisions with the channel surface consists of diffuse collisions, i.e., the direction of post-collision velocities is distributed according to the Knudsen Cosine Law, and a fraction $1-\vartheta$ undergoes specular reflection. From this assumption one obtains the value $\mathcal{D}=\frac{2-\vartheta}{\vartheta}\mathcal{D}_K$ for the self-diffusivity constant, where $\mathcal{D}_K$ is a reference value corresponding to $\vartheta=1$. In this paper we show that $\vartheta$ can be expressed in terms of micro- and macro-geometric parameters for a model consisting of hard spheres colliding elastically against a rigid surface with prescribed microgeometry. Our refinement of the MS theory is based on the observation that the classical surface scattering operator associated to the microgeometry has a canonical velocity space diffusion approximation by a generalized Legendre differential operator whose spectral theory is known explicitly. More specifically, starting from an explicit description of the effective channel surface microgeometry -- a concept which incorporates both the actual surface microgeometry and the molecular radius -- and using this operator approximation, we show that $\vartheta$ can be resolved into easily obtained geometric parameters.

math.DS↗

arXiv2021

Knudsen diffusivity in random billiards: spectrum, geometry, and computation

We develop an analytical framework and numerical approach to obtain the coefficient of self-diffusivity for the transport of a rarefied gas in channels in the limit of large Knudsen number. This framework provides a method for determining the influence of channel surface microstructure on the value of diffusivity that is particularly effective when the microstructure exhibits relatively low roughness. This method is based on the observation that the Markov transition (scattering) operator determined by the microstructure, under the condition of weak surface scattering, has a universal form given, up to a multiplicative constant, by the classical Legendre differential operator. We also show how characteristic numbers of the system -- namely geometric parameters of the microstructure, the spectral gap of a Markov operator, and the tangential momentum accommodation coefficient of a commonly used model of surface scattering -- are all related. Examples of microstructures are investigated to illustrate the relation of these quantities numerically and analytically.

math.DS↗

arXiv2020

Exact discretization of harmonic tensors

Lyons and Sullivan have shown how to discretize harmonic functions on a Riemannian manifold $M$ whose Brownian motion satisfies a certain recurrence property called $\ast$-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, under certain restrictions on the holonomy of the connection, the lifted diffusion on the orthonormal frame bundle has the same $\ast$-recurrence property as the original Brownian motion. This observation permits us to reduce to the discretization of ordinary harmonic functions by a device called scalarization.

math.DG↗

arXiv2020

Entropy Production in Random Billiards

We introduce a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smolukowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters.

math-ph↗

arXiv2018

Rolling and no-slip bouncing in cylinders

The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics.

math.DS↗

arXiv2016

Stability of periodic orbits in no-slip billiards

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards--planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutkin) and linear stability of periodic orbits for a Sinai-type billiard (Wojtkowski) for no-slip billiards. We show among other facts that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than $π$, no-slip billiard dynamics will always contain elliptic period-$2$ orbits; (ii) polygonal no-slip billiards always admit small invariant open sets and thus cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of a Sinai billiard must contain linearly stable periodic orbits of period $2$ and, more generally, we provide a curvature threshold at which a commonly occurring period-$2$ orbit shifts from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in a class of polygonal billiards.

math.DS↗

arXiv2016

No-slip billiards in dimension two

We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot readily be constructed using the common techniques for generating chaos in standard billiards.

math.DS↗

arXiv2015

Explicit formulas for reaction probability in reaction-diffusion experiments

A computational procedure is developed for determining the conversion probability for reaction-diffusion systems in which a first-order catalytic reaction is performed over active particles. We apply this general method to systems on metric graphs, which may be viewed as 1-dimensional approximations of 3-dimensional systems, and obtain explicit formulas for conversion. We then study numerically a class of 3-dimensional systems and test how accurately they are described by model formulas obtained for metric graphs. The optimal arrangement of active particles in a 1-dimensional multiparticle system is found, which is shown to depend on the level of catalytic activity: conversion is maximized for low catalytic activity when all particles are bunched together close to the point of gas injection, and for high catalytic activity when the particles are evenly spaced.

cond-mat.soft↗

arXiv2015

Reaction-diffusion on metric graphs and conversion probability

Motivated by a problem in heterogeneous catalysis, we study a model for irreversible first-order reactions in which gas transport occurs only by diffusion, and reaction occurs only at a small number of well-localized sites. The main problem is to determine fractional conversion in terms of the diffusion coefficient and geometric properties of the reactor. We formulate an appropriate stochastic model for this problem, and then show that, when the domain is composed of a network of thin tubes, reasonably explicit formulas can be obtained.

math.PR↗

arXiv2015

Differential Geometry of Rigid Bodies Collisions and Non-standard Billiards

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A boundary condition is the assignment of a collision map at each tangent space on the boundary of $M$ that gives the post-collision state of the system as a function of the pre-collision state. Our main result is a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions can be stated in geometric language by making explicit a family of vector subbundles of the tangent bundle to the boundary of $M$: the diagonal, non-slipping, and impulse subbundles. Collision maps at a boundary configuration are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. The space of such maps is naturally identified with the union of Grassmannians of $k$-dimensional subspaces of $\mathbb{R}^{n-1}$, $0\leq k\leq n-1$, each subspace specifying the directions of contact roughness. We then consider non-standard billiard systems, defined by fixing the position of one of the bodies and allowing boundary conditions different from specular reflection. We also make a few observations of a dynamical nature for simple examples of non-standard billiards and provide a sufficient condition for the billiard map on the space of boundary states to preserve the canonical (Liouville) measure on constant energy hypersurfaces.

math.DS↗

arXiv2013

Diffusivity in multiple scattering systems

We consider random flights of point particles inside $n$-dimensional channels of the form $\mathbb{R}^{k} \times \mathbb{B}^{n-k}$, where $\mathbb{B}^{n-k}$ is a ball of radius $r$ in dimension $n-k$. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator $P$ that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in $P$ and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are {\em natural} (definition below), which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of $P$. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of $P$ and compute, in the case of 2-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).

math.PR↗

arXiv2012

Multiple scattering in random mechanical systems and diffusion approximation

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,η) of square-integrable functions over the (lower) half-space H in R^m, where η is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.

math-ph↗

arXiv2012

From billiards to thermodynamics

We explore some beginning steps in stochastic thermodynamics of billiard-like mechanical systems by introducing extremely simple and explicit random mechanical processes capable of exhibiting steady-state irreversible thermodynamical behavior. In particular, we describe a Markov chain model of a minimalistic heat engine and numerically study its operation and efficiency.

math-ph↗