Dissipative Dynamics in Classical and Quantum Conservative Systems

Whether it is classical (cf. Chaps. 2 and 3) or quantum context (cf. Chaps. 4 and 5) so far we have talked of the systems which are free from additional restrictions (called constraints) on the canonical variables of the system. For this reason only we have freely used the Poisson brackets in the classical context and the commutators in the quantum context as and when their discussion is required. In the presence of constraints, however, some modifications at the foundation level itself are necessary. In this chapter, we aim at a brief discussion of these modifications in the context of both classical and quantal ideas.

Different kinds of entropy associated with probability distribution functions characterizing the system state in classical and quantum domains are reviewed. Shannon entropy and Rényi entropy are discussed. The notion of tomographic entropy determined by the probability distribution in the phase space of the classical system and by the density operator of the quantum system is considered. Inequalities for the tomographic entropies in classical and quantum domains are studied, and a difference in the form of these inequalities in corresponding domains is suggested as a test to clarify the classicality and quantumness of the system state in quantum optics experiments. A new bound for tomographic entropy (ln πe)Φ(θ) depending on the local oscillator phase difference in homodyne photon detection experiments is discussed.

The CMOS-based sensor design technology has evolved with tremendous applications in MSP domain with enhanced characteristic of maximized S/N ratio. This feature has been obtained with the utilization of pixel-based amplification in addition to minimizing the presence of noise at the output of the device. It means that the noise present in the various active devices of the system has been suppressed with the help of SoC-based design technique, i.e., three-level pixel implementation. This system basically supports all the SoC components such as feedback amplifier, buffers, tapered reset device, etc. The CMOS-based smart system has both software and hardware modules which in turn behave as a complete system on chip (SoC). The main application of this system is in the computation and analysis of the mixed signal in the quantum domain. The author presents a novel approach to implementing a smart system using SoC in quantum context that enhances the performance of the system. The simulation of the system has been carried out on LabVIEW with optimum results desired. Some the result has been carried with mathematical modeling based approach using MATLAB software. The minimum S/N ratio of the system is 55 dB with 1920 × 1080 frame readout at 550 Hz. The author is also interested in carrying out the analysis of this system with QCA tool and then presents a comparative analysis of the two systems with better performance.

We employ two adaptive strategies for motion control of uncertain nonholonomic mechanical systems with chained form kinematics. Chained form systems possess sufficient structure to permit very effective control strategies to be developed for them. For example, these systems are differentially flat, which can be shown to be quite useful for motion control, and they are also easier to stabilize than some other classes of nonholonomic systems; both of these properties are exploited in the paper. As a consequence of the adaptive nature of the control laws and the properties of chained form systems, the employed schemes are simple and modular, are easily implemented with a wide range of mechanical systems, and provide a reasonable starting point for addressing such practical problems as collision avoidance and sensor-based planning and control. The first controller combines an efficient trajectory generation algorithm with an adaptive tracking strategy to obtain the desired motion control, while the second scheme achieves this control objective through the use of a stabilizing control law developed utilizing adaptation and homogeneous system theory. Each of the control strategies is computationally efficient, ensures accurate motion control, and is implementable in the presence of considerable uncertainty concerning the mechanical system model. The efficacy of the approach is illustrated through computer simulations with a single chain nonholonomic system.

Nonholonomic systems provide an important class of mechanical control systems. One reason for this importance is that nonintegrability is essential to both the mechanics and the control: Nonintegrable constraint distributions are the essence of nonholonomic systems, while a nonintegrable distribution of control vector fields is the key to controllability of nonlinear systems. We will learn how these two different types of nonintegrability work together when we study control of nonholonomic mechanical systems.

In previous studies, information dynamics methods such as Von Neumann entropy and Rényi entropy played an important role in many fields, covering both macroscopic and microscopic studies. They have a solid theoretical foundation, but there are few reports in the field of mechanical nonlinear systems. So, can we apply Von Neumann entropy and Rényi entropy to study and analyze the dynamic behavior of macroscopic nonlinear systems? In view of the current lack of suitable methods to characterize the dynamics behavior of mechanical systems from the perspective of nonlinear system correlation, we propose a new method to describe the nonlinear features and coupling relationship of mechanical systems. This manuscript verifies the above hypothesis by using a typical chaotic system and a real macroscopic physical nonlinear system through theory and practical methods. The nonlinear vibration correlation in multi-body mechanical systems is very complex. We propose a full-vector multi-scale Rényi entropy for exploring the chaos and correlation between the dynamic behaviors of mechanical nonlinear systems. The research results prove the effectiveness of the proposed method in modal identification, system dynamics evolution and fault diagnosis of nonlinear systems. It is of great significance to extend these studies to the field of mechanical nonlinear system dynamics.

The approximate representation of a quantum solid as an equivalent composite semiclassical solid is considered for insulating materials. The composite is comprised of point ions moving on a potential energy surface. In the classical bulk domain this potential energy is represented by potentials constructed to give the same structure and elastic properties as the underlying quantum solid. In a small local quantum domain the potential is determined from a detailed quantum calculation of the electronic structure. The new features of this well-studied problem are (1) a clearly stated theoretical context in which approximations leading to the model are introduced, (2) the representation of the classical domain by potentials focused on reproducing the specific quantum response being studied, (3) development of "pseudoatoms" for a realistic treatment of charge densities where bonds have been broken to define the environment of the quantum domain, and (4) inclusion of polarization effects on the quantum domain due to its distant bulk environment. This formal structure is illustrated in detail for a SiO(2) nanorod. More importantly, each component of the proposed modeling is tested quantitatively for this case, verifying its accuracy as a faithful multiscale model of the original quantum solid. To do so, the charge density of the entire nanorod is calculated quantum mechanically to provide the reference by which to judge the accuracy of the modeling. The construction of the classical potentials, the rod, the pseudoatoms, and the multipoles is discussed and tested in detail. It is then shown that the quantum rod, the rod constructed from the classical potentials, and the composite classical/quantum rod all have the same equilibrium structure and response to elastic strain. In more detail, the charge density and forces in the quantum subdomain are accurately reproduced by the proposed modeling of the environmental effects even for strains beyond the linear domain. The accuracy of the modeling is shown to apply for two quite different choices for the underlying quantum chemical method: transfer Hamiltonian and density functional methods.

In his works, V.V. Kozlov proposed a mathematical model for the dynamics of a mechanical system with nonintegrable constraints, which was called vakonomic. In contrast to the then conventional nonholonomic model, trajectories in the vakonomic model satisfy necessary conditions for a minimum in a variational problem with equality constraints. We consider the so-called irregular case of this variational problem, which was not covered by Kozlov, when the trajectory is a singular point of the constraints and the necessary minimum conditions based on the Lagrange principle make no sense. This situation is studied using the theory of abnormal problems developed by the first author. As a result, the classical necessary minimum conditions are strengthened and developed to this class of problems.

The motion planning problem of a nonholonomic multibody system is investigated. Nonholonomicity arises in many mechanical systems subject to nonintegrable velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the control problem of system can be converted to the motion planning problem for a driftless control system. In this paper, we propose an optimal control approach for nonholonomic motion planning. The genetic algorithm is used to optimize the performance of motion planning to connect the initial and final configurations and to generate a feasible trajectory for a nonholonomic system. The feasible trajectory and its control inputs are searched through a genetic algorithm. The effectiveness of the genetic algorithm is demonstrated by numerical simulation.

The numerical solution of multibody systems is not a straightforward problem. The formulation of the equations of motion is augmented with the constraint equations that lead to a set of differential algebraic equations (DAEs). These constraints govern the relative motion between the system’s components at the position level (geometric constraints) and may restrict the velocity of particular components (rolling constraints). There are several factors that determine the effectiveness of numerical integration methods and the extent of their applicability owing to the various motion circumstances. These factors include numerical stability throughout the integration and computation time, as well as allowable error percentage and the length of simulation time. In this regard, this research examines existing approaches for constraint stabilization during numerical integration and introduces a new methodology based on fuzzy control algorithm, whose coefficients are independent of the dynamic characteristics of different systems. Schematics of the new methodology are presented; two examples of spatial multibody systems with holonomic and nonholonomic constraints are solved to evaluate the effectiveness of the proposed method. It can be concluded that fuzzy control contributes an excellent solution for generic system configuration and is suitable for lengthy simulations with minimal computation time.

SummaryIn this paper, we present the modeling and local equilibrium controllability analysis of a spherical robot. The robot consists of a spherical shell that is internally actuated by a pendulum mechanism. The rolling motion of the sphere manifests itself as a nonholonomic constraint in the modeling. We derive the dynamic model of the system using Lagrangian reduction and the variational principle. We first compute the Lagrangian and identify the symmetry with respect to a group action. The system Lagrangian and the rolling constraint are invariant with respect to the group isotropy and hence permit a reduced dynamic formulation termed as the nonholonomic ‘Euler‐Poincaré’ equation with advected dynamics. Using Lie brackets and symmetric products of the potential and control vector fields, local configuration accessibility and local (fiber) equilibrium controllability are presented. Copyright © 2015 John Wiley & Sons, Ltd.

The concept of exceptional points-based optical amplifiers (EPOAs) has been recently proposed as a new paradigm for miniaturizing optical amplifiers while simultaneously enhancing their gain-bandwidth product. While the operation of this new family of amplifiers in the classical domain provides a clear advantage, their performance in the quantum domain has not yet been evaluated. Particularly, it is not clear how the quantum noise introduced by vacuum fluctuations will affect their operation. Here, we investigate this problem by considering three archetypal EPOA structures that rely either on unidirectional coupling, parity-time symmetry, or particle-hole symmetry for implementing the exceptional point. By using the Heisenberg-Langevin formalism, we calculate the added quantum noise in each of these devices and compare it with that of a quantum-limited amplifier scheme that does not involve any exceptional points. Our analysis reveals several interesting results: most notably that while the quantum noise of certain EPOAs can be comparable to those associated with conventional amplifier systems, in general the noise does not follow a universal scaling as a function of the exceptional point but rather varies from one implementation to another.

Many applications of quantum computers require the classical and quantum implementation of dynamic systems (DSs). These applications comprise interacting quantum and classical tasks. While quantum tasks evolve in the quantum domain, classical tasks behave in the classical domain. Besides tackling these kinds of tasks, the computational gap between these domains is covered by the current study. The quantum computing feature All at Once (A@O) executions is appropriate for static systems but less for DSs. The novelty of the proposed approach consists of using Distributed Quantum-Centric Petri Net (DQCPN) models composed of quantum and high-level Petri Nets for specification, design, verification, and implementation of classical–quantum applications. Quantum Processing Units (QPUs) are linked to classical components implementing the control and optimization operations in the proposed application. Many practical applications combine quantum and classical computing to address optimization problems. Quantum computers can be built with a combination of qubits and bosonic qumodes, leading to a new paradigm toward quantum computing. The optimizations are performed by some Evolutionary Algorithms (EAs), including Particle Swarm Optimization (PSO) methods and Genetic Algorithms (GAs). For experiments, an Urban Vehicle Traffic System (UVTS) is used as an open distributed system. The vehicle flows are implemented by discrete qubits, discrete vectors of qubits, or qumodes.

Observation of the magnetic vector potential in the classical macrodomain

The amplitude and phase noise spectra of the internal and external fields of a negative resistance oscillator are derived using electrical circuit theory. It is shown that the difference between internal and external fluctuations becomes important in the quantum domain ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h\omega &gt; kT</tex> ) where external noise sources are not negligible as they often are in the classical domain ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h\omega &lt; kT</tex> ). Using established methods to describe quantum fluctuations in electrical circuits, we obtain results in full agreement with the more exact but less general direct quantum mechanical analysis of a Fabry-Perot laser in Part I.