Showing papers by "Jürg Fröhlich published in 2016"

Gibbs measures of nonlinear Schr\"odinger equations as limits of many-body quantum states in dimensions $d \leq 3$

Abstract: We prove that Gibbs measures of nonlinear Schrodinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body quantum thermal states that we consider are the grand canonical ensemble for $d = 1$ and an appropriate modification of the grand canonical ensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise

Abstract: We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrodinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.

Indirect Acquisition of Information in Quantum Mechanics

Abstract: Long sequences of successive direct (projective) measurements or observations of just a few “uninteresting” physical quantities pertaining to a quantum system, such as clicks of some detectors, may reveal indirect, but precise and unambiguous information on the values of some very “interesting” observables of the system. In this paper, the mathematics underlying this claim is developed; i.e., we attempt to contribute to a mathematical theory of indirect and, in particular, non-demolition observations and measurements in quantum mechanics. Our attempt leads us to make some novel uses of classical notions and results of probability theory, such as the “algebra of functions measurable at infinity”, the Central Limit Theorem, results concerning relative entropy and its role in the theory of large deviations, etc.

The preparation of states in quantum mechanics

Abstract: The important problem of how to prepare a quantum mechanical system, S, in a specific initial state of interest—e.g., for the purposes of some experiment—is addressed. Three distinct methods of state preparation are described. One of these methods has the attractive feature that it enables one to prepare S in a preassigned initial state with certainty, i.e., the probability of success in preparing S in a given state is unity. This method relies on coupling S to an open quantum-mechanical environment, E, in such a way that the dynamics of S∨E pulls the state of S towards an “attractor,” which is the desired initial state of S. This method is analyzed in detail.

The quest for laws and structure

Abstract: The purpose of this paper is to illustrate, on some concrete examples, the quest of theoretical physicists for new laws of Nature and for appropriate mathematical structures that enables them to formulate and analyze new laws in as simple terms as possible and to derive consequences therefrom. The examples are taken from thermodynamics, atomism and quantum theory.

Indirect acquisition of information in quantum mechanics: states associated with tail events

Abstract: The problem of reconstructing information on a physical system from data acquired in long sequences of direct (projective) measurements of some simple physical quantities - histories - is analyzed within quantum mechanics; that is, the quantum theory of indirect measurements, and, in particular, of non-demolition measurements is studied. It is shown that indirect measurements of time-independent features of physical systems can be described in terms of quantum-mechanical operators belonging to an algebra of asymptotic observables. Our proof involves associating a natural measure space with certain sets of histories of a system and showing that quantum-mechanical states of the system determine probability measures on this space. Our main result then says that functions on that space of histories measurable at infinity (i.e., functions that only depend on the tails of histories) correspond to operators in the algebra of asymptotic observables.

System and method for applying pulsed electromagnetic fields

Abstract: Systems and methods for applying a pulsed electromagnetic fields. A system for applying a pulsed electromagnetic field (PEMF) to a cell, comprising: a sensor for obtaining a characteristic of the cell; a memory module having stored therein a plurality of characteristics and PEMF efficacy window data, each characteristic having its corresponding PEMF efficacy window data; a pulse generator coupled to a set of PEMF coils and configured to generate an output of electrical pulses to drive the set of PEMF coils; and a controller in communication with the sensor, the memory module and the pulse generator, wherein the controller is configured to: retrieve, from the memory module, the PEMF efficacy window data that corresponds to the characteristic of the cell obtained by the sensor; and control the output of the pulse generator based on the retrieved PEMF efficacy window data such that the set of PEMF coils apply a PEMF in accordance with the PEMF efficacy window data.