Time Evolution of Two Dimensional Systems with Infinitely Many Particles Mutually Interacting via Very Singular Forces

Buttà, Paolo; Cavallaro, Guido; Marchioro, Carlo

doi:10.1007/s10955-012-0461-6

Cited by 4 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, this case can be treated as in [8,Appendix] (where the specific choice σ 1 = 2 and σ 2 = 1 is detailed), via an a priori estimate on the growth in time of the energy density for these finite systems, which allows to perform the limit. It is worthwhile to mention that a priori energy estimates are the key ingredient also in several results concerning the dynamics of infinitely many particles moving in a continuum, see, e.g., [1,5,6,9,[12][13][14][15][16]. Moreover, in the case of quasi-one-dimensional systems, such estimates turn out to be good enough to obtain not trivial results on the long time behavior, see, e.g., [3,4,10,11].…”

Section: Notation and Statement Of The Resultsmentioning

confidence: 99%

Dynamics of Infinite Classical Anharmonic Crystals

Buttà

Marchioro

2016

J Stat Phys

Self Cite

View full text Add to dashboard Cite

Abstract. We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential V and is subjected to a restoring force of potential U . We assume that U and V are even nonnegative polynomials of degree 2σ 1 and 2σ 2 . We study the time evolution of this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case σ 1 < 2σ 2 − 1 of some already known results obtained for σ 1 ≥ 2σ 2 − 1.

show abstract

Section: Notation and Statement Of The Resultsmentioning

confidence: 99%

Dynamics of Infinite Classical Anharmonic Crystals

Buttà

Marchioro

2016

J Stat Phys

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is not obvious that this system cannot have a blow-up, that is a collapse of infinite mass in a finite region and/or an unbounded growth of the velocity. Actually for point particles the problem has been solved many years ago, when the problem of the dynamics of infinitely many particles has been faced (see [1,2,5,6,7,11,12,18,20,21,22,27,28,29,32,33,39,45,46,47] and for a short review [9]). The solution depends on the dimensions and the shape of the region in which the motion happens (the unbounded cylinder case is treated in [6]).…”

Section: Introduction Statement Of the Problem And Main Resultsmentioning

confidence: 99%

Time Evolution of an Infinitely Extended Vlasov System with Singular Mutual Interaction

2015

View full text Add to dashboard Cite

We study the time evolution of an infinitely extended system in the mean field approximation, governed by the Vlasov equation. This system is confined in an unbounded cylinder by an external force singular on the border. The mutual interaction is assumed singular at short distance as 1/r α with α < 2/3 (or α < 1 in case of an external Lorentz force) and with a short range. The initial density is assumed bounded. Differently from studies which assume initial data compact in space and/or in velocities, here we consider a system having infinite mass and an exponential bound on the velocities, according to the Maxwell-Boltzmann law.

show abstract

“…The existence of the time evolution for systems composed by infinitely many particles moving according to Newton's laws of motion is a classical issue in non-equilibrium statistical mechanics, and several studies have been devoted to this subject, see, e.g., [3,7,8,11,12,14,20,21,24,25,27]. For a summary, see for instance Appendix 1 of [9].…”

Section: Introductionmentioning

confidence: 99%