Quantum Dynamics with Bohmian Trajectories

We describe the advantages and disadvantages of numerical methods when Bohmian trajectory-grids are used for numerical simulations of quantum dynamics. We focus on the crucial non crossing property of Bohmian trajectories, which numerically must be paid careful attention to. Failure to do so causes instabilities or leads to false simulations.Comment: 17 pages, 18 figures; some typos corrected, 4 figures added, some paragraphs extended, source code extende

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density $\to\infty$). Our estimates allow us to interchange the thermodynamic with the mean-field limit.Comment: 27 pages, typos corrected, a few more explanations adde

Dynamics of Sound Waves in an Interacting Bose Gas

We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a box of volume $\Lambda$ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, $\rho=\frac{N}{\Lambda}$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume $\Lambda$ and small ratio $\frac{\Lambda}{\rho}$. The initial state of the gas is assumed to be close to a \textit{product state} of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of $\Lambda$. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio $\frac{\Lambda}{\rho}$. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.Comment: 42 page

Remarks on the derivation of Gross-Pitaevskii equation with magnetic Laplacian

The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the "projection counting" scheme

Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction

Using a new method it is possible to derive mean field equations from the microscopic $N$ body Schr\"odinger evolution of interacting particles without using BBGKY hierarchies. In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii equation which is usually derived assuming positivity of the interaction. The new method for dealing with mean field limits presented in [6] allows us to relax this condition. The price we have to pay for this relaxation is however that we have to restrict the scaling behavior to $\beta<1/3$ and that we have to assume fast convergence of the reduced one particle marginal density matrix of the initial wave function $\mu^{\Psi_0}$ to a pure state $|\phi_0><\phi_0|$

Effective non-linear dynamics of binary condensates and open problems

We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Sch\"odinger equations. After reviewing and commenting our proof in the mean field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.Comment: Corrected typos, updated reference

Linear programming approach for solving stochastic control problem on networks with discounted transition costs

Секция 10. Теоретическая информатикаThe infinite horizon stochastic control problem on network with expected total discounted cost optimization criterion is studied. A linear programming approach for solving this problem on networks is developed. Moreover, a polynomial tim

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

We consider the time evolution of a system of $N$ identical bosons whose interaction potential is rescaled by $N^{-1}$. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit $N \to \infty$ the quantum $N$-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum $N$-body dynamics to the Hartree dynamics.Comment: Typos correcte

In vitro cell compatibility and antibacterial activity of microencapsulated doxycycline designed for improved localized therapy of septic arthritis

OBJECTIVES: For the treatment of septic arthritis in large animals, the local application of antibiotics as a slow release system may be an appropriate means to reach high local bioactivity and low systemic side effects and drug residues. In this study, doxycycline microspheres were developed and tested in vitro for their drug-release properties, suitability for intra-articular application and antimicrobial activity. METHODS: The development of a slow release system was achieved by microencapsulation of the drug into poly(lactide-co-glycolide) microspheres by a novel ultrasonic atomization method. Drug elution was evaluated from microspheres dispersed in elution medium at pre-defined time points by HPLC. Joint-tissue compatibility was tested on cultured bovine synoviocytes by evaluating the expression of pro-inflammatory cytokine mRNA and the production of nitric oxide (NO). Finally, the antimicrobial activity of the released antibiotic was assessed with gram-negative and gram-positive bacteria exposed to release medium sampled at days 1, 7 and 12 after microsphere suspension. RESULTS: An adequate size of the microspheres, sufficient stabilization of doxycycline in aqueous environment and drug release (25 mg microspheres in 4 mL medium) above MIC for bacteria usually isolated in bovine and equine joints were obtained over 15 days. Although the cytokine mRNA expression reflected the excellent tissue compatibility, the results with NO yielded contradictory results. Antimicrobial tests of the release medium proved to match perfectly the activity of non-encapsulated, free doxycycline as reported in the literature. CONCLUSIONS: The newly developed doxycycline delivery system achieved the target specifications and is ready for in vivo testin