First-principles quantum dynamics in interacting Bose gases: I. The positive P representation

The performance of the positive P phase-space representation for exact many- body quantum dynamics is investigated. Gases of interacting bosons are considered, where the full quantum equations to simulate are of a Gross-Pitaevskii form with added Gaussian noise. This method gives tractable simulations of many-body systems because the number of variables scales linearly with the spatial lattice size. An expression for the useful simulation time is obtained, and checked in numerical simulations. The dynamics of first-, second- and third-order spatial correlations are calculated for a uniform interacting 1D Bose gas subjected to a change in scattering length. Propagation of correlations is seen. A comparison is made with other recent methods. The positive P method is particularly well suited to open systems as no conservation laws are hard-wired into the calculation. It also differs from most other recent approaches in that there is no truncation of any kind.

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus allows first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness of the basis and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anticommuting Grassmann variables. Furthermore, because of the overcompleteness of the basis, the phase-space distribution can always be chosen positive. This has important consequences for the sign problem in fermion physics.

Many-body quantum dynamics of polarization squeezing in optical fibers

We report new experiments that test quantum dynamical predictions of polarization squeezing for ultrashort photonic pulses in a birefringent fiber, including all relevant dissipative effects. This exponentially complex many-body problem is solved by means of a stochastic phase-space method. The squeezing is calculated and compared to experimental data, resulting in excellent quantitative agreement. From the simulations, we identify the physical limits to quantum noise reduction in optical fibers. The research represents a significant experimental test of first-principles time-domain quantum dynamics in a one-dimensional interacting Bose gas coupled to dissipative reservoirs.

Computational mechanics of molecular systems:quantifying high-dimensional dynamics by distribution of Poincaré recurrence times

A framework that connects computational mechanics and molecular dynamics has been developed and described. As the key parts of the framework, the problem of symbolising molecular trajectory and the associated interrelation between microscopic phase space variables and macroscopic observables of the molecular system are considered. Following Shalizi and Moore, it is shown that causal states, the constituent parts of the main construct of computational mechanics, the e-machine, define areas of the phase space that are optimal in the sense of transferring information from the micro-variables to the macro-observables. We have demonstrated that, based on the decay of their Poincare´ return times, these areas can be divided into two classes that characterise the separation of the phase space into resonant and chaotic areas. The first class is characterised by predominantly short time returns, typical to quasi-periodic or periodic trajectories. This class includes a countable number of areas corresponding to resonances. The second class includes trajectories with chaotic behaviour characterised by the exponential decay of return times in accordance with the Poincare´ theorem.

Transition from time-continuous systems to discrete mappings

Detailed transport studies in plasmas require the solution of the time evolution of many different initial positions of test particles in the phase space of the systems to be investigated. To reduce this amount of numerical work, one would like to replace the integration of the time-continues system with a mapping.

Quantifying the complexity of chaos in multibasin multidimensional dynamics of molecular systems

The simulated classical dynamics of a small molecule exhibiting self-organizing behavior via a fast transition between two states is analyzed by calculation of the statistical complexity of the system. It is shown that the complexity of molecular descriptors such as atom coordinates and dihedral angles have different values before and after the transition. This provides a new tool to identify metastable states during molecular self-organization. The highly concerted collective motion of the molecule is revealed. Low-dimensional subspaces dynamics is found sensitive to the processes in the whole, high-dimensional phase space of the system. © 2004 Wiley Periodicals, Inc.

Dynamic complexity of chaotic transitions in high-dimensional classical dynamics:Leu-Enkephalin folding

Leu-Enkephalin in explicit water is simulated using classical molecular dynamics. A ß-turn transition is investigated by calculating the topological complexity (in the "computational mechanics" framework [J. P. Crutchfield and K. Young, Phys. Rev. Lett., 63, 105 (1989)]) of the dynamics of both the peptide and the neighbouring water molecules. The complexity of the atomic trajectories of the (relatively short) simulations used in this study reflect the degree of phase space mixing in the system. It is demonstrated that the dynamic complexity of the hydrogen atoms of the peptide and almost all of the hydrogens of the neighbouring waters exhibit a minimum precisely at the moment of the ß-turn transition. This indicates the appearance of simplified periodic patterns in the atomic motion, which could correspond to high-dimensional tori in the phase space. It is hypothesized that this behaviour is the manifestation of the effect described in the approach to molecular transitions by Komatsuzaki and Berry [T. Komatsuzaki and R.S. Berry, Adv. Chem. Phys., 123, 79 (2002)], where a "quasi-regular" dynamics at the transition is suggested. Therefore, for the first time, the less chaotic character of the folding transition in a realistic molecular system is demonstrated. © Springer-Verlag Berlin Heidelberg 2006.

On the Structure of Spatial Branching Processes

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.

Lie-szimmetriák egy közgazdasági alkalmazása (An economic application of the Lie symmetries)

Ebben a tanulmányban ismertetjük a Nöther-tétel lényegi vonatkozásait, és kitérünk a Lie-szimmetriák értelmezésére abból a célból, hogy közgazdasági folyamatokra is alkalmazzuk a Lagrange-formalizmuson nyugvó elméletet. A Lie-szimmetriák dinamikai rendszerekre történő feltárása és viselkedésük jellemzése a legújabb kutatások eredményei e területen. Például Sen és Tabor (1990), Edward Lorenz (1963), a komplex kaotikus dinamika vizsgálatában jelent®s szerepet betöltő 3D modelljét, Baumann és Freyberger (1992) a két-dimenziós Lotka-Volterra dinamikai rendszert, és végül Almeida és Moreira (1992) a három-hullám interakciós problémáját vizsgálták a megfelelő Lie-szimmetriák segítségével. Mi most empirikus elemzésre egy közgazdasági dinamikai rendszert választottunk, nevezetesen Goodwin (1967) ciklusmodelljét. Ennek vizsgálatát tűztük ki célul a leírandó rendszer Lie-szimmetriáinak meghatározásán keresztül. / === / The dynamic behavior of a physical system can be frequently described very concisely by the least action principle. In the centre of its mathematical presentation is a specic function of coordinates and velocities, i.e., the Lagrangian. If the integral of the Lagrangian is stationary, then the system is moving along an extremal path through the phase space, and vice versa. It can be seen, that each Lie symmetry of a Lagrangian in general corresponds to a conserved quantity, and the conservation principle is explained by a variational symmetry related to a dynamic or geometrical symmetry. Briey, that is the meaning of Noether's theorem. This paper scrutinizes the substantial characteristics of Noether's theorem, interprets the Lie symmetries by PDE system and calculates the generators (symmetry vectors) on R. H. Goodwin's cyclical economic growth model. At first it will be shown that the Goodwin model also has a Lagrangian structure, therefore Noether's theorem can also be applied here. Then it is proved that the cyclical moving in his model derives from its Lie symmetries, i.e., its dynamic symmetry. All these proofs are based on the investigations of the less complicated Lotka Volterra model and those are extended to Goodwin model, since both models are one-to-one maps of each other. The main achievement of this paper is the following: Noether's theorem is also playing a crucial role in the mechanics of Goodwin model. It also means, that its cyclical moving is optimal. Generalizing this result, we can assert, that all dynamic systems' solutions described by first order nonlinear ODE system are optimal by the least action principle, if they have a Lagrangian.

Improving the predictability of time series by chaotic parameter estimation

Limited literature regarding parameter estimation of dynamic systems has been identified as the central-most reason for not having parametric bounds in chaotic time series. However, literature suggests that a chaotic system displays a sensitive dependence on initial conditions, and our study reveals that the behavior of chaotic system: is also sensitive to changes in parameter values. Therefore, parameter estimation technique could make it possible to establish parametric bounds on a nonlinear dynamic system underlying a given time series, which in turn can improve predictability. By extracting the relationship between parametric bounds and predictability, we implemented chaos-based models for improving prediction in time series. ^ This study describes work done to establish bounds on a set of unknown parameters. Our research results reveal that by establishing parametric bounds, it is possible to improve the predictability of any time series, although the dynamics or the mathematical model of that series is not known apriori. In our attempt to improve the predictability of various time series, we have established the bounds for a set of unknown parameters. These are: (i) the embedding dimension to unfold a set of observation in the phase space, (ii) the time delay to use for a series, (iii) the number of neighborhood points to use for avoiding detection of false neighborhood and, (iv) the local polynomial to build numerical interpolation functions from one region to another. Using these bounds, we are able to get better predictability in chaotic time series than previously reported. In addition, the developments of this dissertation can establish a theoretical framework to investigate predictability in time series from the system-dynamics point of view. ^ In closing, our procedure significantly reduces the computer resource usage, as the search method is refined and efficient. Finally, the uniqueness of our method lies in its ability to extract chaotic dynamics inherent in non-linear time series by observing its values. ^

High energy three -body breakup of three -nucleon systems

The main goal of this dissertation was to study two- and three-nucleon Short Range Correlations (SRCs) in high energy three-body breakup of 3He nucleus in 3He(e, e'NN) N reaction. SRCs are characterized by quantum fluctuations in nuclei during which constituent nucleons partially overlap with each other. ^ A theoretical framework is developed within the Generalized Eikonal Approximation (GEA) which upgrades existing medium-energy methods that are inapplicable for high momentum and energy transfer reactions. High momentum and energy transfer is required to provide sufficient resolution for probing SRCs. GEA is a covariant theory which is formulated through the effective Feynman diagrammatic rules. It allows self-consistent calculation of single and double re-scatterings amplitudes which are present in three-body breakup processes. The calculations were carried out in detail and the analytical result for the differential cross section of 3He(e, e'NN)N reaction was derived in a form applicable for programming and numerical calculations. The corresponding computer code has been developed and the results of computation were compared to the published experimental data, showing satisfactory agreement for a wide range of values of missing momenta. ^ In addition to the high energy approximation this study exploited the exclusive nature of the process under investigation to gain more information about the SRCs. The description of the exclusive 3He( e, e'NN)N reaction has been done using the formalism of the nuclear decay function, which is a practically unexplored quantity and is related to the conventional spectral function through the integration of the phase space of the recoil nucleons. Detailed investigation showed that the decay function clearly exhibits the main features of two- and three-nucleon correlations. Four highly practical types of SRCs in 3He nucleus were discussed in great detail for different orders of the final state re-interactions using the decay function as an unique identifying tool. ^ The overall conclusion in this dissertation suggests that the investigation of the decay function opens up a completely new venue in studies of short range nuclear properties. ^

High energy three-body breakup of three-nucleon systems

The main goal of this dissertation was to study two- and three-nucleon Short Range Correlations (SRCs) in high energy three-body breakup of 3He nucleus in 3He(e, e'NN)N reaction. SRCs are characterized by quantum fluctuations in nuclei during which constituent nucleons partially overlap with each other. A theoretical framework is developed within the Generalized Eikonal Approximation (GEA) which upgrades existing medium-energy methods that are inapplicable for high momentum and energy transfer reactions. High momentum and energy transfer is required to provide sufficient resolution for probing SRCs. GEA is a covariant theory which is formulated through the effective Feynman diagrammatic rules. It allows self-consistent calculation of single and double re-scatterings amplitudes which are present in three-body breakup processes. The calculations were carried out in detail and the analytical result for the differential cross section of 3He(e, e'NN)Nreaction was derived in a form applicable for programming and numerical calculations. The corresponding computer code has been developed and the results of computation were compared to the published experimental data, showing satisfactory agreement for a wide range of values of missing momenta. In addition to the high energy approximation this study exploited the exclusive nature of the process under investigation to gain more information about the SRCs. The description of the exclusive 3He(e, e'NN)N reaction has been done using the formalism of the nuclear decay function, which is a practically unexplored quantity and is related to the conventional spectral function through the integration of the phase space of the recoil nucleons. Detailed investigation showed that the decay function clearly exhibits the main features of two- and three-nucleon correlations. Four highly practical types of SRCs in 3He nucleus were discussed in great detail for different orders of the final state re-interactions using the decay function as an unique identifying tool. The overall conclusion in this dissertation suggests that the investigation of the decay function opens up a completely new venue in studies of short range nuclear properties.

On Multi- Server Queues With Consultation With Main Server

Queueing Theory is the mathematical study of queues or waiting lines. Queues abound in every day life - in computer networks, in tra c islands, in communication of electro-magnetic signals, in telephone exchange, in bank counters, in super market checkouts, in doctor's clinics, in petrol pumps, in o ces where paper works to be processed and many other places. Originated with the published work of A. K. Erlang in 1909 [16] on congestion in telephone tra c, Queueing Theory has grown tremendously in a century. Its wide range applications includes Operations Research, Computer Science, Telecommunications, Tra c Engineering, Reliability Theory, etc.

The Quantum Tetrahedron in 3 and 4 Dimensions

Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions.