Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

Dittrich, Thomas

doi:10.3390/e21030286

Cited by 6 publications

(7 citation statements)

References 96 publications

(268 reference statements)

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“…Looking only at the central bistable system, the random sequence thus generated amounts to a production of one bit of entropy per run of the experiment. Here, another aspect of the Hamiltonian dynamics of closed systems comes in handy, the conservation of entropy under canonical transformations [ 16 ]. It implies that the entropy in the random sequence cannot be produced by the central system but must originate somewhere else in the total system.…”

Section: Introductionmentioning

confidence: 99%

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Dittrich

Martı́nez

2020

Entropy

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We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

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Section: Introductionmentioning

confidence: 99%

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Dittrich

Martı́nez

2020

Entropy

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“…. Contour plots of the potential (16) with parameters a = b = 0.15, m = 1, ω = 0.4, without coupling (a), with coupling g = 0.06, but not including the counter term g 2 X 2 /(2mω 2 ) (b), and with this term (c). Colour code ranges from red (negative) through white (zero) through blue (positive).…”

Section: Double Well Coupled To a Single Harmonic Oscillator Or A Few...mentioning

confidence: 99%

“…Looking only at the central bistable system, the random sequence thus generated amounts to a productioncof one bit of entropy per run of the experiment. Here, another aspect of the Hamiltonian dynamics of closed systems comes in handy, the conservation of entropy under canonical transformations [16]. It implies that the entropy in the random sequence cannot be produced by the central system but must originate somewhere else in the total system.…”

Section: Introductionmentioning

confidence: 99%

Toppling pencils -- Macroscopic Randomness from Microscopic Fluctuations

Dittrich,

Martínez

2020

Preprint

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We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N = 1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the KAM scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

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“…Chaos seems to be chaotic but has a delicate internal structure, which is a unique and ubiquitous phenomenon in nonlinear systems [21]. Randomness, ergodicity, and regularity are the most typical characteristics of chaos, enabling it to traverse all states within a given range without repeating according to one of its own "laws."…”

Section: Chaos Initializationmentioning

confidence: 99%

An Improved ABC Algorithm and Its Application in Bearing Fault Diagnosis with EEMD

Chen

Xiao

2019

Algorithms

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The Ensemble Empirical Mode Decomposition (EEMD) algorithm has been used in bearing fault diagnosis. In order to overcome the blindness in the selection of white noise amplitude coefficient e in EEMD, an improved artificial bee colony algorithm (IABC) is proposed to obtain it adaptively, which providing a new idea for the selection of EEMD parameters. In the improved algorithm, chaos initialization is introduced in the artificial bee colony (ABC) algorithm to insure the diversity of the population and the ergodicity of the population search process. On the other hand, the collecting bees are divided into two parts in the improved algorithm, one part collects the optimal information of the region according to the original algorithm, the other does Levy flight around the current global best solution to improve its global search capabilities. Four standard test functions are used to show the superiority of the proposed method. The application of the IABC and EEMD algorithm in bearing fault diagnosis proves its effectiveness.

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Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

Cited by 6 publications

References 96 publications

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Toppling pencils -- Macroscopic Randomness from Microscopic Fluctuations

An Improved ABC Algorithm and Its Application in Bearing Fault Diagnosis with EEMD

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