A mathematical aspect of a tunnel-junction for spintronic qubit

We consider the Dirac particle that lives in the 1-dimensional configuration space consisting of two quantum wires and a junction between the two. We regard the spin of a Dirac particle as spintronic qubit. We give concrete formulae explicitly expressing the one-to-one correspondence between every self-adjoint extension of the minimal Dirac operator and its corresponding boundary condition of the wave functions of the Dirac particle. We then show that all the boundary conditions can be classified into just two types. The two types are characterized by whether the electron passes through the junction or not. We also show how the tunneling produces its own phase factor and what is the relation between the phase factor and the spintronic qubit in the tunneling boundary condition.     

对稳态NUT- Kerr-Newman黑洞的量子隧穿特征的研究

运用Parikh的量子隧穿模型, 研究了NUT- Kerr- Newman黑洞的量子隧穿辐射特征. 研究结果表明, 当考虑能量守恒与角动量守恒时, 稳态NUT- Kerr- Newman黑洞的真实谱不再是纯热谱, 视界处粒子的隧穿率与Bekenstein-Hawking熵有关, 且满足量子力学中的幺正性原理.     

A Particle Migrating Randomly on a Sphere

Consider a particle moving on the surface of the unit sphere in R ³ and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model.     

Resonances and tunneling in a quantum wire

We consider a problem in mathematical scattering theory related to the ballistic conductance model. The model under investigation describes the charge propagation in a quantum wire. We assume that the charge carrier has a spin and take the Rashba spin-orbital interaction into account. We study the conductance resonances generated by the weak quantum-wire interaction with the quasistationary state of a parallel-connected quantum dot or with the tunneling through a series-connected quantum dot. Such a quantum dot is usually the control element. We present sufficient conditions for the spatial symmetry of the system to ensure that the quasistationary state of the quantum dot generates a conductance resonance. We assume that the conductance is related to the scattering matrix by the Landauer formula. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 92–102, April, 2006.     

Dynamical Phase Transition for a Quantum Particle Source

We analyze the time evolution describing a quantum source for non-interacting particles, either bosons or fermions. The growth behavior of the particle number (trace of the density matrix) is investigated, leading to spectral criteria for sublinear or linear growth in the fermionic case, but also establishing the possibility of exponential growth for bosons. We further study the local convergence of the density matrix in the long time limit and prove the semi-classical limit.     

Tunneling through a quantum dot in a quantum waveguide

The problem is considered of scattering in a system consisting of a quantum waveguide and a quantum dot weakly coupled to the waveguide. It is assumed that the quantum waveguide is described by the Pauli equations, and the Rashba spin-orbit interaction is taken into account. The possibility of tunneling through the quantum dot is proved.     

Tunneling cosmological state and the origin of Higgs inflation in the standard model

For the tunneling cosmological state, we propose a path integral formulation admitting a consistent renormalization and renormalization-group improvement in particle physics applications of quantum cosmology with heavy massive quantum fields. We apply this formulation to the inflationary cosmology driven by the standard-model Higgs boson playing the role of an inflaton with a strong nonminimal coupling to gravity. A complete cosmological scenario is thus obtained, embracing the formation of initial conditions for the inflationary background in the form of a sharp probability peak in the distribution of the inflaton field and the ongoing generation of the cosmic microwave background spectrum on this background. We also discuss the status of the no-boundary and tunneling states in a cosmology driven by massless fields conformally coupled to gravity.     

Cosmological quantum tunneling and the holographic principle

The probable trajectory of the ground-state wave function of the universe arises through quantum tunneling by gravitational instantons. We calculate the quantum tunneling rate for an (n>2)-dimensional closed Friedmann-Robertson-Walker universe with a positive cosmological constant. In four dimensions, we use the holographic principle to relate the tunneling rate to the maximal entropy of the early universe after quantum tunneling.     

Kinetic limits for a class of interacting particle systems

Summary We study one dimensional particle systems in which particles travel as independent random walks and collide stochastically. The collision rates are chosen so that each particle experiences finitely many collisions per unit time. We establish the kinetic limit and derive the discrete Boltzmann equation for the macroscopic particle density.     

Numerical study of the quantum euler-poisson model

A numerical study of the isothermal quantum Euler-Poisson model for potential flow is presented. The stationary model consists of nonlinear elliptic equations of degenerate type with quadratic growth in the gradient. The equations are decoupled in a Gummel-type manner and convergence of this scheme is proven for small applied voltages. Numerical simulations of a resonant tunneling structure are presented and the zero relaxation time limit is performed numerically.     

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model that I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle’s position distribution converges to a Gaussian under diffusive rescaling.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Center-of-mass tomography and probability representation of quantum states for tunneling

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 371–387, February, 2005.     

Quantum object as a system with memory

A quantum theory model is suggested in which non-Markovian processes play an important part. It is assumed that any quantum object is formed of local nuclei, or carriers of particle properties, and a nonlocal wave field—the memory carrier. Within the framework of this model, the solution of the problem of quantum measurements is presented. The Einstein-Podolsky-Rosen paradox and the Bell inequality are also considered.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 264–272, February, 1996.Translated by V. M. Volosov.     

The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. in [7]. The model is obtained as the diffusive limit of the quantum Liouville–BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.     

Optimal hierarchical system of a grid road network

This paper develops a simple analytical model for determining the hierarchical system of road networks. The model is based on a grid road network where roads are classified into three types according to road widths and travel speeds. We derive the optimal ratios of road areas that minimize the average and maximum travel time. Minimizing the average travel time provides an efficient solution, whereas minimizing the maximum travel time provides an equitable solution. Both of the solutions are expressed in terms of road widths and travel speeds. As an application of the grid network model, we evaluate the hierarchical system of the road network of Tokyo.     

Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section. © 2000 John Wiley & Sons, Inc.     

Embedding quasi laminar 1D flame profiles to model turbulent premixed combustion with a joint PDF method

A new model for turbulent premixed combustion is presented which is based on a joint velocity composition probability density function (JPDF) method. The key idea is a scale separation approach. The method combines the model by Bray, Moss and Libby [1] (BML) for premixed combustion with the flamelet approach for nonpremixed combustion. Here, a Lagrangian formulation of the BML model is considered. The progress variable used by the BML model becomes a computational particle property and its value is triggered by the arrival of the flame front at the particle's position. Similar as in the flamelet approach we assume that the smallest eddies are not small enough to disturb the reactive diffusive flame structure. To resolve the (embedded) quasi laminar flame structure, a flame residence time is introduced. With that residence time, the evolution of the particle composition, including enthalpy, can be determined from precomputed laminar 1D flames. The main challenge with this approach is to model the probability that an embedded flamefront arrives at the particle location, which is necessary to close the chemical source term. Numerical experiments of a turbulent premixed flame show good agreement with experimental data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstructing freeway travel times with a simplified network flow model alternating the adopted fundamental diagram

The present study summarises the travel time reconstruction performance of a network flow model by explicitly analysing the adopted fundamental diagram relation under congested and un-congested traffic patterns. The incorporated network flow model uses a discrete meso-simulation approach in which the anisotropic property of traffic flow and the uniform acceleration of vehicle packets are explicitly considered. The flow performances on link-route dynamics have been derived by reasonably alternating the adopted two-phase, i.e., congested and un-congested, fundamental relation of traffic flow. The linear speed–density relation with the creeping speed assumption is substituted with the triangular flow–density relation in order to investigate the performance of the network flow model in varying flow patterns. Applying the anisotropic mesoscopic model, the measure of travel time is obtained as a link performance from a simplified dynamic network loading process. Travel time reconstruction performance of the network flow model is sought considering the actual measures that are obtained by a probe vehicle, in addition to reconstructions by a macroscopic network flow model. The main improvements on travel time reconstruction process are encountered in terms of the computation load within the explicit analyses by the alternation of adopted two-phase fundamental diagram. Although the accuracies of the flow model with the adoption of two different fundamental diagrams are hard to differentiate, the computational burden of the simulation process by the triangular fundamental diagram is found to be considerably different.     

Tunneling of the hard‐core model on finite triangular lattices

We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.