Statistical Mechanics of Assemblies of Coupled Oscillators

Ford, G. W.; Kac, Mark; Mazur, Péter

doi:10.1063/1.1704304

Cited by 884 publications

(610 citation statements)

References 11 publications

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“…However, in order to avoid confusion with the usual Fokker-Planck equation approach to their problem, in which the irrelevant degrees of freedom are assumed to be in a specific macroscopic state (usually a canonical one, see e.g. [6,7]), the word booster was adopted instead. Therefore, the general goal of BMWG was to derive the thermostatistical properties of a system coupled to a (finite or infinite) bath whose thermal properties arise naturally from dynamics, bridging the gap between mechanics and thermodynamics much like the ideas of Boltzmann.…”

Section: The Dynamical Approach Of Bmwgmentioning

confidence: 99%

Does the Boltzmann Principle Need a Dynamical Correction?

Adib

2004

Journal of Statistical Physics

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In an attempt to derive thermodynamics from classical mechanics, an approximate expression for the equilibrium temperature of a finite system has been derived [M. Bianucci, R. Mannella, B. J. West, and P. Grigolini, Phys. Rev. E 51, 3002 (1995)] which differs from the one that follows from the Boltzmann principle S = k ln Ω(E) via the thermodynamic relation 1/T = ∂S/∂E by additional terms of "dynamical" character, which are argued to correct and generalize the Boltzmann principle for small systems (here Ω(E) is the area of the constant-energy surface). In the present work, the underlying definition of temperature in the Fokker-Planck formalism of Bianucci et al. is investigated and shown to coincide with an approximate form of the equipartition temperature. Its exact form, however, is strictly related to the "volume" entropy S = k ln Φ(E) via the thermodynamic relation above for systems of any number of degrees of freedom (Φ(E) is the phase space volume enclosed by the constant-energy surface). This observation explains and clarifies the numerical results of Bianucci et al. and shows that a dynamical correction for either the temperature or the entropy is unnecessary, at least within the class of systems considered by those authors. Explicit analytical and numerical results for a particle coupled to a small chain (N ∼ 10) of quartic oscillators are also provided to further illustrate these facts.

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Section: The Dynamical Approach Of Bmwgmentioning

confidence: 99%

Does the Boltzmann Principle Need a Dynamical Correction?

Adib

2004

Journal of Statistical Physics

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“…Mechanical models of a particle immersed in a heat bath were introduced by Ford, Kac, and Mazur (15,14) and Zwanzig (54) as simple models to study kinetics and irreversible statistical mechanics. The ''heat bath'' is a collection of n particles which interact with a ''distinguished'' particle through springs; the heat bath particles are assumed to have random initial data distributed according to the laws of statistical mechanics.…”

Section: Introductionmentioning

confidence: 99%

Fractional Kinetics in Kac–Zwanzig Heat Bath Models

Kupferman

2004

Journal of Statistical Physics

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We study a variant of the Kac-Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As n Q . the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro-differential equation-a generalized Langevin equation (GLE) with power-law memory kernel and driven by 1/f a -noise. The limiting process exhibits fractional sub-diffusive behaviour. We further consider the approximation of nonMarkovian processes by higher-dimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a so-called fractional Fokker-Planck equation in the present context. All results are supported by direct numerical experiments.

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“…This property is illustrated by exact calculations for a spin-boson model. Dynamics of open quantum systems has long been a subject of study in diverse fields [1,2]. Recent interest in quantum computing has focused attention [3,4,5] on quantifying environmental effects that cause small deviations from the isolated-system quantum dynamics.…”

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confidence: 99%

Additivity of decoherence measures for multiqubit quantum systems

2004

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We introduce new measures of decoherence appropriate for evaluation of quantum computing designs. Environment-induced deviation of a quantum system's evolution from controlled dynamics is quantified by a single numerical measure. This measure is defined as a maximal norm of the density matrix deviation. We establish the property of additivity: in the regime of the onset of decoherence, the sum of the individual qubit error measures provides an estimate of the error for a several-qubit system. This property is illustrated by exact calculations for a spin-boson model. Dynamics of open quantum systems has long been a subject of study in diverse fields [1,2]. Recent interest in quantum computing has focused attention [3,4,5] on quantifying environmental effects that cause small deviations from the isolated-system quantum dynamics. During short time intervals of "quantum-gate" functions, environment-induced relaxation/decoherence effects must be kept below a certain threshold in order to allow fault-tolerant quantum error correction [6]. The reduced density matrix of the quantum system, with the environment traced out, is usually evaluated within some approximation scheme, e.g., [7]. In this work, we introduce a new, additive measure of the deviation from the "ideal" density matrix of an isolated system. For a single two-state system (a qubit) this measure is calculated explicitly for the environment modeled as a bath of harmonic modes [8], e.g., phonons. Furthermore, we establish that for a several-qubit system, the introduced measure of decoherence is approximately additive for short times. It can be estimated by summing up the deviation measures of the constituent qubits, without the need to carry out a many-body calculation, similar to the approximate additivity expected for relaxation rates of exponential approach to equilibrium at large times.In most quantum computing proposals, quantum algorithms are implemented in the following way. Evolution of the qubits is governed by a Hamiltonian consisting of single-qubit operators and of two-qubit interaction terms [9,10]. Some parameters of the Hamiltonian can be controlled externally to implement the desired algorithm. During each cycle of the computation the Hamiltonian remains constant. This ideal model does not include the influence of the environment on the computation, which necessitates quantum error correction. The latter involves non-unitary operations [6] and cannot be described as Hamiltonian-governed dynamics.The accepted approach to evaluate environmentally induced decoherence involves a model in which each qubit is coupled to a bath of environmental modes [8]. The reduced density matrix of the system, with the bath modes traced out, then describes the time-dependence of the system dynamics [3,7,11]. Because of the interaction with environment, after each cycle the state of the qubits will be slightly different from the ideal. The resulting error accumulates at each cycle, so that largescale quantum computation is not possible without implementing fault-t...

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Statistical Mechanics of Assemblies of Coupled Oscillators

Cited by 884 publications

References 11 publications

Does the Boltzmann Principle Need a Dynamical Correction?

Does the Boltzmann Principle Need a Dynamical Correction?

Fractional Kinetics in Kac–Zwanzig Heat Bath Models

Additivity of decoherence measures for multiqubit quantum systems

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