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Register a new userA Survey on the Classical Limit of Quantum Dynamical Entropies
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Added by
Valerio Cappellini Dr.
Publication date
2007
fields
Physics
and research's language is
English
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We analyze the behavior of quantum dynamical entropies production from sequences of quantum approximants approaching their (chaotic) classical limit. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail and a semi-classical analysis is performed on it using coherent states, fulfilling an appropriate dynamical localization property. Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai invariant is found only over time scales that are logarithmic in the quantization parameter.
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We study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection. We assume that the three constants of motion are the Hamiltonian, $H$, with the standard form of a kinetic part plus a potential dependent on the position only, the $z$-component of the angular momentum, $L$, and a Hamiltonian-like constant, $widetilde H$, for which the kinetic part is quadratic in the momenta. We find the explicit form of these potentials compatible with complete integrability. The classical equations of motion, written in terms of two arbitrary potential functions, is separated in oblate spheroidal coordinates. The quantization of such systems leads to a set of two differential equations that can be presented in the form of spheroidal wave equations.
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, that we define similarly to the Holevo capacity, but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
Dynamical systems with $epsilon$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in $mathbb{R}^n$. Based on the theory of random perturbations of dynamical systems and the WKB approximation respectively, results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the large deviation principle of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $ttoinfty$ and $epsilonto 0$. Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the $epsilon$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.
This work is concerned with the minimization of quantum entropies under local constraints of density, current, and energy. The problem arises in the work of Degond and Ringhofer about the derivation of quantum hydrodynamical models from first principles, and is an adaptation to the quantum setting of the moment closure strategy by entropy minimization encountered in kinetic equations. The main mathematical difficulty is the lack of compactness needed to recover the energy constraint. We circumvent this issue by a monotonicity argument involving energy, temperature and entropy, that is inspired by some thermodynamical considerations.
In this paper, we prove a power-law version dynamical localization for a random operator $mathrm{H}_{omega}$ on $mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schrodinger equation $$mathrm{i}partial_{t}u=mathrm{H}_{omega}u, quad u in ell^2(mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $tgeq 0$.
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