KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

In this paper, one-dimensional (1D) nonlinear wave equations

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Structures of Equatorial Envelope Rossby Wave Under a Generalized External Forcing

The cubic nonlinear Schroedinger (NLS for short) equation with a generalized external heating source is derived for large amplitude equatorial envelope Rossby wave in a shear flow. And then various periodic structures for these equatorial cnvelope Rossby waves are obtained with the help of a new transformation, Jacobi elliptic functions,and elliptic equation. It is shown that different types of resonant phase-locked diabatic heating play different roles in structures of equatorial envelope Rossby wave.     

Transient and Stationary Simulations for a Quantum Hydrodynamic Model

The transient and stationary characteristics of a one-dimensional quantum hydrodynamic model are comparatively studied for semiconductor charge transport in a resonant tunnelling diode. When the bias is not small, our numerical results show a deviation of the asymptotic transient solutions from the stationary ones. A dynamic instability accounts for such deviation. The stationary quantum hydrodynamic model is therefore unsuitable in general for simulating quantum devices.     

Evolution of Quantum State for Mesoscopic Circuits with Dissipation

Based on the maximum entropy principle, we present a density matrix of mesoscopic RLC circuit to make it possible to analyze the connection of the initial condition with temperature. Our results show that the quantum state evolution is closely related to the initial condition, and that the system evolves to generalized coherent state if it is in ground state initially, and evolves to squeezed state if it is in excited state initially.     

A Geometric Approach to Time Evolution Operators of Lie Quantum Systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrödinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Heat Flux Boundary Conditions for a Lattice Boltzmann Equation Model

A method simplifying the external force term for the Boltzmann equation is proposed,using the Chapman-Enskog asymptotic expansion technique.The relationship between the perturbation strength parameter and the Grashof number is presented.A heat flux boundary condition for the temperature field simulation is developed according to the transforming ralations between the cold and hot particles on the boundary wall.The effects of buoyant force and the heat flux boundary condition are implemented with the two-particle systems of D2Q9 and D3Q15,and the numerical results agree well with the experimental result of silicone oil.     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Non-Triviality of a Discrete Bak–Sneppen Evolution Model

Consider the following evolution model, proposed in ref. 1 by Bak and Sneppen. Put N vertices on a circle, spaced evenly. Each vertex represents a certain species. We associate with each vertex a random variable, representing the state or fitness of the species, with values in [0,1]. The dynamics proceeds as follows. Every discrete time step, we choose the vertex with minimal fitness, and assign to this vertex, and to its two neighbours, three new independent fitnesses with a uniform distribution on [0,1]. A conjecture of physicists, based on simulations, is that in the stationary regime, the one-dimensional marginal distributions of the fitnesses converges, when N, to a uniform distribution on (f,1), for some threshold f<1. In this paper we consider a discrete version of this model, proposed in ref. 2. In this discrete version, the fitness of a vertex can be either 0 or 1. The system evolves according to the following rules. Each discrete time step, we choose an arbitrary vertex with fitness 0. If all the vertices have fitness 1, then we choose an arbitrary vertex with fitness 1. Then we update the fitnesses of this vertex and of its two neighbours by three new independent fitnesses, taking value 0 with probability 0<q<1, and 1 with probability p=1–q. We show that if q is close enough to one, then the mean average fitness in the stationary regime is bounded away from 1, uniformly in the number of vertices. This is a small step in the direction of the conjecture mentioned above, and also settles a conjecture mentioned in ref. 2. Our proof is based on a reduction to a continuous time particle system.     

On the Distribution of Free Path Lengths¶for the Periodic Lorentz Gas

Consider the domain

Butterfly Effect in a System of Quantum Dots in the Sachdev–Ye–Kitaev Model

JETP Letters - A method has been proposed to calculate the out-of-time time ordered correlator in the generalization of the Sachdev–Ye–Kitaev model with a nonzero spatial dimension. The...     

Crystallization of a “Boltzmann” System of Quantum Hard Spheres

The crystallization of a quantum system of hard spheres has been considered for the case where the effects of Bose and Fermi statistics can be neglected. It has been found that the quantum crystal–liquid equilibrium line almost completely coincides with the classical line except for the range of very low temperatures. Because of this nontrivial conclusion, the efficiency of quantum fluctuations at the melting of matter is questionable.     

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. Received: 7 February 2000 / Accepted: 7 July 2000     

Classical and Non-relativistic Limits of a Lorentz-Invariant Bohmian Model for a System of Spinless Particles

A completely Lorentz-invariant Bohmian model has been proposed recently for the case of a system of non-interacting spinless particles, obeying Klein-Gordon equations. It is based on a multi-temporal formalism and on the idea of treating the squared norm of the wave function as a space-time probability density. The particle’s configurations evolve in space-time in terms of a parameter σ with dimensions of time. In this work this model is further analyzed and extended to the case of an interaction with an external electromagnetic field. The physical meaning of σ is explored. Two special situations are studied in depth: (1) the classical limit, where the Einsteinian Mechanics of Special Relativity is recovered and the parameter σ is shown to tend to the particle’s proper time; and (2) the non-relativistic limit, where it is obtained a model very similar to the usual non-relativistic Bohmian Mechanics but with the time of the frame of reference replaced by σ as the dynamical temporal parameter.     

Conservation Quantities of the Explicit Symplectic Scheme for Time-evolution of Quantum System

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Solutions for a Mode Ⅲ Growing Crack in a Piezoelectric Plane Under Two Kinds of Electric Boundary Conditions ＊

The solutions for a mode Ⅲ crack growing along an arbitrary propagation path in a piezoelectric plane are studied under the impermeable surface condition and the electrical contact surface condition respectively. According to the two kinds of electric boundary conditions, the Hilbert and Riemann boundary value problems in a half-plane including opening smooth arc are obtained from the theoretical analysis. Moreover, the equipollence of the solution formed under these two electric boundaries is proved, and unified solutions for the stress and electric displacement distribution in the crack-tipfield of the piezoelectric plane are achieved.     

Quantum Statistics of a Forced Oscillator with a Time－Dependent Driving Force

Quantum statistics of a forced harmonic oscillator acted upon by a time-dependent external force are derived usingthe Wilcox trick and the time-dependent inhomogeneous Bogoliubov transformation formalism. The internal energy, fluctuation of the particle-number average and entropy of this nonequilibrium system are presentedexplicitly.