On functional approach to classical mechanics

Piskovskiy, Evgeny

doi:10.1134/s2070046611030095

Cited by 3 publications

(8 citation statements)

References 8 publications

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“…If the considered solutions belong to the class of continuously differentiable functions, then the Boltzmann-Enskog equation is irreversible. The qualitative differences in solutions of evolution equations depending on the considered functional space is analysed in [21][22][23], it is crucial in so called functional mechanics [24][25][26][27][28][29][30][31].…”

Section: Discussionmentioning

confidence: 99%

“…Let us consider the simplest nontrivial case N = 2. Then equation (29) for f ε is reduced to We see, that the corrections to the Boltzmann-Enskog equation can be expressed by additional factors ζ (depending on the initial distribution f 0 ). As we see in Section 4, if ε → 0, then these factors matter only on infinitesimal time intervals.…”

Section: Generalized Enskog Equationmentioning

confidence: 99%

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Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

Трушечкин

2014

Kinetic &Amp; Related Models

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N. N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if the special regularization to the delta-functions is applied. This allows to reduce the nonlinear Boltzmann-Enskog equation to the BBGKY hierarchy of linear equations in this particular case.Also we show that similar functions are exact smooth solutions for the recently proposed generalized Enskog equation. They can be referred to as "particle-like" or "soliton-like" solutions and are analogues of multisoliton solutions of the Korteweg-de Vries equation. arXiv:1307.4741v2 [math-ph]Boltzmann-Enskog kinetic equation. It is a generalization of the Boltzmann equation for the case of moderately dense (rather than dilute) gases. As the Boltzmann equation, it describes the time-irreversible behaviour and entropy production [5,6]. However, N. N. Bogolyubov [7] (see also [8]) discovered that this equation has timereversible microscopic solutions as well. They have form of sums of delta-functions and correspond to trajectories of individual hard spheres. Another interest feature of these solutions (along with their reversibility) is that the Boltzmann-Enskog kinetic equation is believed to give only an approximate aggregate description of the gas dynamics in terms of a single-particle distribution function. The Bogolyubov's result shows that the Boltzmann-Enskog equation can describe the gas dynamics in terms of trajectories of individual hard spheres as well.Also the Vlasov kinetic equation is known to have microscopic solutions. They were discovered by A. A. Vlasov himself [9].However, the justification of these solutions for the Boltzmann-Enskog equation is performed by Bogolyubov on the "physical" level of rigour. In particular, Bogolyubov did not discuss the products of generalized functions in the collision integral.A way to give a rigorous sense to the microscopic solutions is the use of a regularization for delta-functions. A rather general type of regularization and a direct proof of the existence of the limit for the collision integral is proposed in [10]. But the proof is rather involved.Here we present a special kind of regularization. The crucial observation is that, under this regularization, the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres. This allows to reduce the nonlinear Boltzmann-Enskog equation to the BBGKY hierarchy of linear equations in this particular case. This gives a rigorous sense to the microscopic solutions in a simp...

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Section: Discussionmentioning

confidence: 99%

Section: Generalized Enskog Equationmentioning

confidence: 99%

Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

Трушечкин

2014

Kinetic &Amp; Related Models

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“…Functional mechanics [11,12,13,14,15,16] also claims that the Newton equation (or Hamilton equations) is not fundamental even in the classical nonrelativistic world.…”

Section: Reconciliation Of the Kinetic Equations With The Microscopic...mentioning

confidence: 99%

“…Recently Igor Volovich proposed functional mechanics [11,12] (see also [13,14,15,16]) for solving the irreversibility problem. The functional mechanics states that the fundamental notion of mechanics is not a material point, but an (integrable) probability density function.…”

Section: Introductionmentioning

confidence: 99%

Derivation of the particle dynamics from kinetic equations

Трушечкин

2012

P-Adic Num Ultrametr Anal Appl

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We consider the microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, we construct the so called approximate microscopic solutions. These solutions are continuously differentiable and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the time-reversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.Comment: 18 pages; some misprints have been corrected, some references have been adde

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“…We then apply the obtained results to a problem of classical mechanics: deciding whether one should prefer recently suggested functional classical mechanics [17,18] (see also [19,20,21,22,23,24]) to traditional Newtonian one. The basic concept of functional mechanics is not a material point or a trajectory but a probability density function in a phase space.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical description of collapses and revivals of quantum wave packets in bounded domains

Trushechkin,

Volovich

2013

Preprint

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We study a special kind of semiclassical limit of quantum dynamics on a circle and in a box (infinite potential well with hard walls) as the Planck constant tends to zero and time tends to infinity. The results give detailed information about all stages of evolution of quantum wave packets: semiclassical motion, collapses, revivals, as well as intermediate stages. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most of the time close to uniform. This fact was previously known only from numerical calculations.We apply the obtained results to a problem of classical mechanics: deciding whether recently suggested functional classical mechanics is preferable to traditional Newtonian one from the quantum-mechanical point of view. To do this, we study the semiclassical limit of the Husimi functions of quantum states. We show that functional mechanics remains valid at larger time scales than Newtonian one and, therefore, is preferable.Finally, we analyse the quantum dynamics in a box in case when the size of the box is known with a random error. We show that, in this case, the probability distribution of the position of a quantum particle is not almost periodic, but tends to a limit distribution as time indefinitely increases.

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On functional approach to classical mechanics

Cited by 3 publications

References 8 publications

Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres

Derivation of the particle dynamics from kinetic equations

Semiclassical description of collapses and revivals of quantum wave packets in bounded domains

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