Quantum-Classical Correspondence for Two Interacting Particles in a One-Dimensional Box

Meza-Montes, L.; Izrailev, F. M.; Ulloa, Sergio E.

doi:10.1002/1521-3951(200007)220:1<721::aid-pssb721>3.3.co;2-s

Cited by 4 publications

(7 citation statements)

References 5 publications

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“…whereC =D kγi gives the transformation at collisions with the walls withD k from Eq. (7). δτ f is the delay in the collision time of the (infinitesimal) nearby trajectory with respect to the collision time of the reference trajectory.…”

Section: A Dynamics In Tangent Space Lyapunov Exponents (Le) and Tmentioning

confidence: 99%

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Origin of chaos in soft interactions and signatures of nonergodicity

2007

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The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios gamma=m(2)/m(1) of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with gamma , the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincaré sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.

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Section: A Dynamics In Tangent Space Lyapunov Exponents (Le) and Tmentioning

confidence: 99%

“…A Yukawa interaction between particles is assumed. Such a system has been considered classically [6] and quantum mechanically [7] for the case of equal masses. In order to calculate the spectrum of Lyapunov Exponents (LEs), the dynamics in tangent space is determined explicitly.…”

Section: Introductionmentioning

confidence: 99%

Origin of chaos in soft interactions and signatures of nonergodicity

2007

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“…The results presented in Refs. [64,65,66,67,68,70,69,72,71,73] give evidence for a quantum-classical correspondence of the strength function. In the classical limit, the meaning of the strength function is just a projection of the energy surface of H 0 onto that of H, the fact that simplifies the analysis of quantum systems having a well defined classical limit.…”

Section: 24)mentioning

confidence: 88%

“…The quantum-classical correspondence for the strength functions, as well as for the envelopes of eigenstates, has been thoroughly studied in Refs. [64,65,66,67,68,69,70,71,72,73,73] for various models of interacting particles.…”

Section: Definitionsmentioning

confidence: 99%

Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles

Borgonovi,

Izrailev,

Santos

et al. 2016

Preprint

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This review is devoted to the problem of thermalization in a small isolated conglomerate of interacting constituents. A variety of physically important systems of intensive current interest belong to this category: complex atoms, molecules (including biological molecules), nuclei, small devices of condensed matter and quantum optics on nano-and micro-scale, cold atoms in optical lattices, ion traps. Physical implementations of quantum computers, where there are many interacting qubits, also fall into this group. Statistical regularities come into play through inter-particle interactions, which have two fundamental components: mean field, that along with external conditions, forms the regular component of the dynamics, and residual interactions responsible for the complex structure of the actual stationary states. At sufficiently high level density, the stationary states become exceedingly complicated superpositions of simple quasiparticle excitations. At this stage, regularities typical of quantum chaos emerge and bring in signatures of thermalization. We describe all the stages and the results of the processes leading to thermalization, using analytical and massive numerical examples for realistic atomic, nuclear, and spin systems, as well as for models with random parameters. The structure of stationary states, strength functions of simple configurations, and concepts of entropy and temperature in application to isolated mesoscopic systems are discussed in detail. We conclude with a schematic discussion of the time evolution of such systems to equilibrium.

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“…The Hamilton equations, Eqs. (11), are numerically solved to obtain, as a function of time t, a sequence x i t; p i t (i 1; 2; t 0; 1; 2; . .…”

mentioning

confidence: 99%

Quantum-Classical Correspondence for Two Interacting Particles in a One-Dimensional Box

Meza-Montes

Izrailev

Ulloa

2000

phys. stat. sol. (b)

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We study the model of two interacting particles moving in a 1D box, paying main attention to the quantum‐classical correspondence for the average shape of quantum eigenstates and for the local density of states (LDOS). We show that if the classical motion is chaotic, in a deep semi‐classical region of a quantum system, both the shape of eigenstates and of the LDOS coincide with their classical analogs, on average. However, individual eigenstates exhibit quite large fluctuations which may not be treated as statistical ones. Thus, comparison of quantum quantities to the classical ones allows one to detect quantum effects of localization which for conservative systems emerge in the energy space.

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Quantum-Classical Correspondence for Two Interacting Particles in a One-Dimensional Box

Cited by 4 publications

References 5 publications

Origin of chaos in soft interactions and signatures of nonergodicity

Origin of chaos in soft interactions and signatures of nonergodicity

Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles

Quantum-Classical Correspondence for Two Interacting Particles in a One-Dimensional Box

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