Yves Elskens

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Papers by Yves Elskens

Controlling chaos in area-preserving maps

by Yves Elskens and marco pettini

Physica D: Nonlinear Phenomena, 2005

We describe a method of control of chaos that occurs in area-preserving maps. This method is base... more

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Transport and large scale stochasticity for a nonperiodic generalisation of the standard map

Lecture Notes in Physics, 1995

In the standard map, accelerator modes generate ballistic transport, which may dominate over anom... more

Kinetic Theory for Plasmas and Wave-Particle Hamiltonian Dynamics

Lecture Notes in Physics, 2002

The plasma limit is characterized by the fact that a particle interacts with many partners. The d... more The plasma limit is characterized by the fact that a particle interacts with many partners. The dynamics leads to the Vlasov equation in the mean-field limit. Collective behaviour in the N-body system is naturally described by M M N M{\mathbf{ }} \ll {\mathbf{ }}\mathcal{N} N wavelike degrees of freedom, which behave as harmonic oscillator interacting with a population of N

Ergodic theory of the mixmaster model in higher space-time dimensions

by Marc Henneaux and Yves Elskens

Nuclear Physics B, 1987

The dynamical theory of the mixmaster model in spacetime dimensions greater than four is undertak... more The dynamical theory of the mixmaster model in spacetime dimensions greater than four is undertaken. A new parametrization of the Kasner exponents is displayed, which reduces the mixmaster map to a translation or a dilating inversion (followed by an appropriate permutation), depending on the original value of the Kasner exponents. The concept of the ``Kasner era&amp;amp;#39;&amp;amp;#39; is generalized. Particular attention

From instability to irreversibility

Proceedings of the National Academy of Sciences, 1986

A canonical procedure transforming the unitary evolution group U. in a contracting semigroup W. f... more A canonical procedure transforming the unitary evolution group U. in a contracting semigroup W. for phase-space ensembles has been developed for Kolmogorov dynamical systems in a series of recent papers. This paper investigates the physical meaning of this transformation. We stress that, for sufficiently unstable dynamical systems in which phase-space points are identified with an arbitrary but finite precision, one must take into account the undiscernibility of trajectories having the same asymptotic behavior in the future. The fundamental objects of our description are thus bundles of converging trajectories. We show that such an ensemble, corresponding to initial conditions whose support has finite measure, is then represented by a distribution function (called a Boltzmann ensemble) that evolves to equilibrium under the action of a markovian semigroup. The usual Gibbs-Koopman ensembles satisfying the Liouville equation are recovered as a singular limit. This work validates Boltzmann's intuition for a class of unstable dynamical systems and appears as a step toward the derivation of equations exhibiting irreversibility at a microscopic level.

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Nonlinear dynamics of a charged dust grain in a plasma

Physics of Plasmas, 1997

For an arbitrary monotonic charging function, the dynamics of a dust grain is dissipative and ene... more For an arbitrary monotonic charging function, the dynamics of a dust grain is dissipative and energy is a Liapunov function. In an arbitrary external potential two types of equilibria exist. The first type, with uncharged grain, is always unstable. The second type of equilibrium, admitting states of both positive and negative charge, can be marginally stable; stability depends on the local potential. Under spatially uniform (constant or time-dependent) potentials, motion is free while the charge adapts to the potential. For a spatially oscillating potential, the phase space is that of the simple pendulum with one additional degree of freedom, the charge. Dissipation in the charging process forbids periodic behavior and ensures the existence of attractors: A grain is at stable equilibrium only when charged positively and trapped in a potential well, or when charged negatively on top of a hill. The small oscillations near a stable equilibrium decay weakly, and the grain charge oscillates at twice the oscillation frequency. {copyright} {ital 1997 American Institute of Physics.}

Invariance of stretching and expansion spectra, and Oseledec' multiplicative ergodic theorem

Physica D: Nonlinear Phenomena, 1997

The local stretching rate a expresses the logarithmic divergence of trajectories along the local ... more The local stretching rate a expresses the logarithmic divergence of trajectories along the local unstable manifold in phase space. We show that its distribution S(a) in an ergodic component is invariant with respect to the initial condition (almost surely) but depends on the coordinate system in which this dynamics is studied; the same applies to its moments. Only the average f aS(a) da, which is the largest Liapunov exponent, is invariant with respect to changes of coordinates. The same argument applies to higher-dimensional expansion spectra.

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Reversible dynamics and the macroscopic rate law for a solvable Kolmogorov system: The three bakers' reaction

Journal of Statistical Physics, 1985

We investigate a piecewise linear (area-preserving) map T describing two coupled baker transforma... more We investigate a piecewise linear (area-preserving) map T describing two coupled baker transformations on two squares, with coupling parameter 0 ~< c ~< 1. The resulting dynamical system is Kolmogorov for any c ~ 0. For rational values of c, we construct a generating partition on which T induces a Markov chain. This Markov structure is used to discuss the decay of correlation functions: exponential decay is found for a class of functions related to the partition. Explicit results are given for c = 2 -~. The macroscopic analog of our model is a leaking process between two (badly) stirred containers: according to the Markov analysis, the corresponding progress variable decays exponentially, but the rate coefficients characterizing this decay are not those determined from the one-way flux across the cell boundary. The validity of the macroscopic rate law is discussed.

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Microscopic derivation of a Markovian Master equation in a deterministic model of chemical reaction

Journal of Statistical Physics, 1984

We consider a (deterministic, conservative) one-dimensional system of colored hard points, changi... more We consider a (deterministic, conservative) one-dimensional system of colored hard points, changing color each time they hit one another with a relative velocity above a threshold. In the limit of rare reactions, the N-particle color distribution follows a Markovian birth-and-death process. Using the reaction rate as an intrinsic time scale, we also obtain the reaction-diffusion equation for a test particle in this hydrodynamic limit. Explicit results are given for a discrete and a Maxwellian velocity distribution.

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Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes

Communications in Nonlinear Science and Numerical Simulation, 2010

The one-dimensional motion of N particles in the field of many incoherent waves is revisited nume... more The one-dimensional motion of N particles in the field of many incoherent waves is revisited numerically. When the wave complex amplitudes are independent, with a gaussian distribution, the quasilinear approximation is found to always overestimate transport and to become accurate in the limit of infinite resonance overlap.

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Kinetic limit of N-body description of wave-particle self-consistent interaction

by M.-c. Firpo and Yves Elskens

The system of N particles xi^N=(x_1, v_1, ... x_N,v_N) interacting with M waves Zn = An exp( i ph... more The system of N particles xi^N=(x_1, v_1, ... x_N,v_N) interacting with M waves Zn = An exp( i phi_n) has been used to study the dynamics of Landau damping [1] and warm beam-plasma instability [2]. Given initial data (Z(0), xi^N(0)), it evolves according to hamiltonian dynamics to (Z^N(t), xi^N(t)). In the limit N arrow ∞, the hamiltonian dynamics generates a

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Kinetic Limit of N-Body Description of Wave–Particle Self-Consistent Interaction

by M.-c. Firpo and Yves Elskens

Journal of Statistical Physics, 2000

CITATIONS 24 READS 30 2 authors, including: Some of the authors of this publication are also work... more CITATIONS 24 READS 30 2 authors, including: Some of the authors of this publication are also working on these related projects: Magnetic reconnection View project Marie-Christine Firpo École Polytechnique 68 PUBLICATIONS 809 CITATIONS SEE PROFILE All content following this page was uploaded by Marie-Christine Firpo on 15 December 2016. The user has requested enhancement of the downloaded file.

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Weak turbulence and structure evolution in N -body Hamiltonian systems with long-range force

Physical Review E, 1998

The dynamics of a family of one-dimensional spatially periodic systems of N classical particles i... more The dynamics of a family of one-dimensional spatially periodic systems of N classical particles interacting by a repulsive pair force is investigated. This force is the long-range part of the one-dimensional Coulomb interaction; the family includes the mean-field Hamiltonian rotator model. Initial conditions generating turbulent structures are considered. These structures are density holes in (x,v) space that produce a non-Gaussian probability distribution of fluctuations of the particle distribution function f(x,v,t). These density holes appear in a velocity domain where f(x,v) has large derivative ∂vf as predicted by the kinetic theory of clumps in plasmas. Their evolution is shown to be controlled by the motion of the particles in the (x,v) space domain swept by the separatrix associated with the longest-range coupling field components, which implies that their lifetime is proportional to the number N of particles. The relaxation time of the velocity distribution function of tagged particles in the system (for various initial conditions) is also shown to be quite insensitive to the presence of turbulent structures and to spatial scales smaller than the Debye length.

Annihilation kinetics in the one-dimensional ideal gas

Physical review. A, 1985

We consider the annihilation reaction A+A-->0 on the real line with an initial equilibrium dis... more We consider the annihilation reaction A+A-->0 on the real line with an initial equilibrium distribution of particles A (points or rods); the particles move freely before annihilating, with independent initial velocities. For the dichotomic distribution of velocities P(v=c)=1-P(v=-c)=p, we determine the fraction S(t) of particles surviving at time t: (1) If p=(1/2), S(t)~t-1/2 by a central-limit effect; (2) if p!=(1/2), S(t)~||2p-1||+At-3/2exp(-Bt) with known constants A and B. We also determine the asymptotic spatial distribution of surviving particles. From these results we derive some bounds on the decay of S(t) for other velocity distributions, and we compare them to the decay of S(t) for diffusive (Brownian) motion.

Microscopic Dynamics of Plasmas and Chaos

Plasma Physics and Controlled Fusion, 2003

Some of the key intellectual foundations of plasma physics are in danger of becoming a lost art. ... more Some of the key intellectual foundations of plasma physics are in danger of becoming a lost art. Fortunately, however, this threat recedes with the publication of this valuable book. It renders accessible those aspects of theoretical plasma physics that are best approached from the perspectives of classical mechanics, in both its early nineteenth century and late twentieth century manifestations. Half

Intuitive and rigorous derivation of spontaneous emission and Landau damping of Langmuir waves through classical mechanics

Physics of Plasmas, 1996

Classical mechanics provides the intuitive and unified description of spontaneous emission, Landa... more Classical mechanics provides the intuitive and unified description of spontaneous emission, Landau growth and damping of Langmuir waves, the cold beam–plasma instability, and van Kampen modes. This is done by studying the interaction between M weak modes of a plasma without resonant particles and N quasiresonant particles, which leads to an exactly solvable high-dimensional Floquet problem. Growth corresponds to an

Explicit reduction of N-body dynamics to self-consistent particle–wave interaction

Physics of Plasmas, 1998

The one-dimensional (1-D) spatially periodic system of N classical particles, interacting via a C... more The one-dimensional (1-D) spatially periodic system of N classical particles, interacting via a Coulomb-like repulsive long-range force, is studied using classical mechanics. The usual Bohm-Gross dispersion relation for the collective modes is obtained in the absence of quasiresonant particles. In the presence of R quasiresonant particles, the evolution equations for M long-wavelength modes are coupled to the particles&amp;amp;#39; motion through

Proof of quasilinear equations in the chaotic regime of the weak warm beam instability

Physics Letters A, 2002

The diffusion coefficient is proved rigorously to take on the quasilinear value for the chaotic m... more The diffusion coefficient is proved rigorously to take on the quasilinear value for the chaotic motion of an electron in a prescribed set of strongly overlapping Langmuir waves with random phases. Natural approximations show this result to extend to the self-consistent chaotic motion of many particles in a set of many Langmuir waves corresponding to the weak warm beam instability.

Phase Transition in the Collisionless Damping Regime for Wave-Particle Interaction

by Yves Elskens and Marie-christine Firpo

Physical Review Letters, 2000

Gibbs statistical mechanics is derived for the Hamiltonian system coupling self-consistently a wa... more Gibbs statistical mechanics is derived for the Hamiltonian system coupling self-consistently a wave to N particles. This identifies Landau damping with a regime where a second order phase transition occurs. For nonequilibrium initial data with warm particles, a critical initial wave intensity is found : above it, thermodynamics predicts a finite wave amplitude in the limit N → ∞ ; below it, the equilibrium amplitude vanishes. Simulations support these predictions providing new insight on the long-time nonlinear fate of the wave due to Landau damping in plasmas.

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Kinetic Theory and Large-N Limit for Wave-Particle Self-Consistent Interaction

by Yves Elskens and Marie-christine Firpo

Physica Scripta, 1998

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