Poincaré resonances and the extension of classical dynamics

Petrosky, Tomio

doi:10.1016/0960-0779(95)00042-9

Cited by 76 publications

(73 citation statements)

References 24 publications

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“…In this paper we do call this equivalence into question. However, we do not elect to formulate a grand theory to establish the difference between the two points of view, this has been done by Petrosky and Prigogine [1,2], but rather we take the more modest approach of establishing an inconsistency based on a simple physical problem.…”

Section: Introductionmentioning

confidence: 99%

“…A numerical treatment of the phase space density requires, by necessity, a transformation of the Liouville density function σ (x, t) into a probability, through the relation p (x i , t) = σ (x i , t) δx i (1) where the x-axis is divided into N equal parts indexed by j = 1, 2, ..., N, δx i is a small phase space interval (x i , x i + δx i ) and the probability density is normalized to unity at all times…”

Section: Introductionmentioning

confidence: 99%

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Strange kinetics: conflict between density and trajectory description

2002

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We study a process of anomalous diffusion, based on intermittent velocity fluctuations, and we show that its scaling depends on whether we observe the motion of many independent trajectories or that of a Liouville-like equation driven density. The reason for this discrepancy seems to be that the Liouville-like equation is unable to reproduce the multi-scaling properties emerging from trajectory dynamics. We argue that this conflict between density and trajectory might help us to define the uncertain border between dynamics and thermodynamics, and that between quantum and classical physics as well.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strange kinetics: conflict between density and trajectory description

2002

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“…It is well known that the collision operator plays a central role in nonequilibrium statistical mechanics, as seen in a Boltzmann equation or a Fokker-Planck equation [22,27]. A striking finding of the theory is that the spectrum of the collision operator is identical to that of the Liouvillian, which signifies the direct link between the microscopic dynamics governed by the Liouvillian and the phenomenological kinetic theory [24,26,28]. Since the collision operator is a non-Hermitian operator, the eigenvalues can take complex values which reflect the dissipation of the system.…”

Section: Introductionmentioning

confidence: 79%

Nonequilibrium transport on a quantum molecular chain in terms of the complex Liouvillian spectrum

Tanaka

Kanki²,

Petrosky³

2011

Phys. Rev. E

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The transport process in a molecular chain in a nonequilibrium stationary state is theoretically investigated. The molecule is interacting at both ends with thermal baths of different temperatures, while no dissipation mechanism is contained inside the molecular chain. We have first obtained the nonequilibrium stationary state outside the Hilbert space in terms of the complex spectral representation of Liouvillian. The nonequilibrium stationary state is obtained as an eigenstate of the Liouvillian, which is constructed through the collision invariant of the kinetic equation. The eigenstate of the Liouvillian contains information on the spatial correlation between the molecular chain and the thermal baths. While energy flow in the nonequilibrium state which is due to the first-order correlation can be described by the Landauer formula, the particle current due to the second-order correlation cannot be described by the Landauer formula. The present method provides a simple way to evaluate the energy transport in a molecular chain in a nonequilibrium situation.

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“…To describe dissipative processes we need an extension of L H outside the Hilbert space. 12,13) Therefore the formulation of dynamics for non-integrable systems involves an extension of the statistical description outside the Hilbert space.…”

Section: Resonances and Poincaré's Theoremmentioning

confidence: 99%

“…12,13,15) The dynamics is decomposed into a set of independent "subdynamics" by Π (ν) . Instead a set of equations with invariant or oscillating solutions, we obtain now a set of Markovian kinetic equations…”

Section: λ Transformationmentioning

confidence: 99%

Irreversibility, Probabilities and Dressed Unstable States in Quantum Mechanics

Petrosky

Ordóñez

2003

J. Phys. Soc. Jpn.

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The problem of the meaning of quantum unstable states including their dressing is considered. The formulation given in terms of density matrices outside the Hilbert space is used. A dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states is obtained. Due to resonance singularities that appear in the frequency denominators, quantum unstable systems are categorized as non-integrable systems in the sense of Poincaré. The excited unstable state is derived from the stable states through a suitable analytic continuation of the denominators. It is given by an irreducible density matrix with broken time-symmetry. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short time process and the long time process. The long time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.

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Poincaré resonances and the extension of classical dynamics

Cited by 76 publications

References 24 publications

Strange kinetics: conflict between density and trajectory description

Strange kinetics: conflict between density and trajectory description

Nonequilibrium transport on a quantum molecular chain in terms of the complex Liouvillian spectrum

Irreversibility, Probabilities and Dressed Unstable States in Quantum Mechanics

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