Dynamical systems theory for music dynamics

We show that, when music pieces are cast in the form of time series of pitch variations, the concepts and tools of dynamical systems theory can be applied to the analysis of temporal dynamics in music. (i) Phase space portraits are constructed from the time series wherefrom the dimensionality is evaluated as a measure of the global dynamics of each piece. (ii) Spectral analysis of the time series yields power spectra ( approximately f(-nu)) close to red noise (nu approximately 2) in the low frequency range. (iii) We define an information entropy which provides a measure of the local dynamics in the musical piece; the entropy can be interpreted as an evaluation of the degree of complexity in the music, but there is no evidence of an analytical relation between local and global dynamics. These findings are based on computations performed on eighty sequences sampled in the music literature from the 18th to the 20th century. (c) 1995 American Institute of Physics.     

Dynamical modelling of superstatistical complex systems

We show how to construct the optimum superstatistical dynamical model for a given experimentally measured time series. For this purpose we generalise the superstatistics concept and study a Langevin equation with a memory kernel whose parameters fluctuate on a large time scale. We show how to construct a synthetic dynamical model with the same invariant density and correlation function as the experimental data. As a main example we apply our method to velocity time series measured in high-Reynolds-number turbulent Taylor-Couette flow, but the method can be applied to many other complex systems in a similar way.     

Dynamical process of complex systems and fractional differential equations

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.     

Dynamical Green's function theory to study the optical phenomena related to metamaterials

We combine a space–time Lorentz transformation with a dyadic Green's function technique to establish a general and rigorous dynamical theory, which can be employed to study the optical phenomena occurring in a moving environment. As the applications of this method, we studied the Doppler effects of a source moving near a metamaterial slab and the super imaging effect achieved by a moving metamaterial lens. Many interesting anomalous phenomena were discovered, induced typically by the interplays between the source and the surface waves of the metamaterial slab.     

Dynamical systems study of chameleon scalar field

The present work is an extensive study of the viable stable solutions of chameleon scalar field models leading to possibilities of an accelerated expansion of the universe. It is found that for various combinations of the chameleon field potential V(?)$V\left(?\right)$ and the coupling f(?)$f\left(?\right)$ of the chameleon field with matter, a stable solution for an accelerated expansion is quite possible. The investigation provides a diagnostics for the stability criteria for all sorts of combinations of V(?)$V\left(?\right)$ and f(?)$f\left(?\right)$.     

布朗运动理论及其在复杂气候系统研究中的应用

文章将从非平衡态统计物理发展和应用的角度,介绍德国科学家哈塞尔曼荣获2021年诺贝尔物理学奖的研究工作——基于布朗运动理论,建立了描述气象(天气)影响气候长期演化的随机气候学模型,并建立了寻求影响气候主因的最优指纹方法,从而能够分辨出人类活动和自然界局部改变对气候这一复杂系统的影响。哈塞尔曼的工作本质上是理论物理在实际复杂系统领域的成功应用,他采用的基础物理方法——布朗运动理论是我国杰出女物理学家王明贞和其导师乌伦贝克在20世纪40年代基于爱因斯坦的工作发展起来的^[1,2]。文章将介绍布朗运动理论的发展及其相关的非平衡统计物理思想的当代发展,以展示哈塞尔曼如何把相关的物理理论巧妙地用于气候长期预测的实际应用研究：(1)建立了快变的局部“气象”变量涨落通过耗散涨落关系影响缓变的整体气候变量的基础理论;(2)通过最优指纹方法,寻找局部“噪音”和外驱动力影响气候演化的关键要素。     

Time-dependent random walks and the theory of complex adaptive systems

Motivated by novel results in the theory of complex adaptive systems, we analyze the dynamics of random walks in which the jumping probabilities are time dependent. We determine the survival probability in the presence of an absorbing boundary. For an unbiased walk, the survival probability is maximized in the case of large temporal oscillations in the jumping probabilities. On the other hand, a random walker who is drifted towards the absorbing boundary performs best with a constant jumping probability. We use the results to reveal the underlying dynamics responsible for the phenomenon of self-segregation and clustering observed in the evolutionary minority game.     

Dynamical systems in the theory of solitons in the presence of nonlocal interactions

The behavior of solitons in models which take into account complex dispersion or nonlocal interaction of nonlinear waves is examined. A method is proposed to reduce this problem to one involving special trajectories (homoclinic and heteroclinic) of the dynamic system. This method involves replacing the nonlinear integrodifferential equation with the differential equations which link the original nonlinear field with the auxiliary linear fields. The interaction of fields in such a model is a local interaction. The number of introduced linear fields is determined by the Laplace transform of the integral operator kernel of the basic integrodifferential equation. The problem involving topological solitons for the nonlocal generalization of the Klein-Gordon equation is considered. Nonlocal interactions are found to lead to a number of singularities (unrestricted increase in the slope of the topological soliton front, break in the solutions, and other singularities).     

Simulation of unsteady combustion in a solid-propellant rocket motor

Various regimes of combustion in end-burning-grain solid-propellant rocket motors were examined within the framework of the phenomenological theory of unsteady combustion. A system of equations capable of describing the interaction between the process of burning and acoustic waves was derived. A specific feature of the problem is that its formulation involves two characteristic times: the acoustic time and oscillation amplitude variation time. These characteristic times differ by about three orders of magnitude, a circumstance that requires a high accuracy of calculations. Based on the quadratic approximation in oscillation amplitude, a simpler method for solving the problem was proposed, according to which only the effects associated with the oscillation amplitude variation time are taken into account. Numerical results were obtained for the simplest model of propellant burning, which contains the minimum number of parameters and disregards entropy waves in the combustion products. The steady and unsteady regimes of burning were identified. In the latter case, nonlinear effects may generate shock waves in the combustion chamber.     

Dynamical theory of topological defects

A method is presented to obtain stochastic equations of motion for topological defects from the underlying TDGL-like stochastic dissipative field equations. The method makes use of virtual displacements of the Goldstone coordinates of topological defects. Effects of kinematical constraints among Goldstone coordinates are studied. The method is applied to modulated systems and we obtain stochastic equations of motion for interfaces (domain walls) and vortex lines (dislocation or defect lines). The driving force for a vortex line is found to include besides the usual surface tension force a new force due to misfit, which is an analogue of the Magnus force on a quantized vortex line and the Peach-Kochler force on a dislocation. A general expression for interactions between parts of interfaces is obtained in terms of asymptotic forms of field variables far from interfaces.     

Dynamical theory of polariton amplifiers

We present the theory of the dynamics of the polariton amplifier in the region of small polariton densities. We give an analytical solution for the polariton condensate density matrix and show that the formation of a coherent quantum state is possible. Once the condensate is formed, the coherence becomes macroscopically long living. Polariton amplifier represents, therefore, an optical memory element, where the input weak coherent signal can be amplified and kept.     

Dynamical invariants and time-dependent harmonic systems

A simplified algebraic derivation of the dynamical (Lewis) invariant for the time-dependent harmonic oscillator is presented. The treatment is based on the concept of the dynamical algebra.     

Dynamical renormalization group approach to relaxation in quantum field theory

The real time evolution and relaxation of expectation values of quantum fields and of quantum states are computed as initial value problems by implementing the dynamical renormalization group (DRG). Linear response is invoked to set up the renormalized initial value problem to study the dynamics of the expectation value of quantum fields. The perturbative solution of the equations of motion for the field expectation values of quantum fields as well as the evolution of quantum states features secular terms, namely terms that grow in time and invalidate the perturbative expansion for late times. The DRG provides a consistent framework to resum these secular terms and yields a uniform asymptotic expansion at long times. Several relevant cases are studied in detail, including those of threshold infrared divergences which appear in gauge theories at finite temperature and lead to anomalous relaxation. In these cases the DRG is shown to provide a resummation akin to Bloch-Nordsieck but directly in real time and that goes beyond the scope of Bloch-Nordsieck and Dyson resummations. The nature of the resummation program is discussed in several examples. The DRG provides a framework that is consistent, systematic, and easy to implement to study the non-equilibrium relaxational dynamics directly in real time that does not rely on the concept of quasiparticle widths.     

Dynamical linked cluster expansions with applications to disordered systems

Dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics. We discuss physical applications to systems with annealed and quenched disorder. Examples are the bond-diluted Ising model and the Sherrington-Kirkpatrick spin glass. We derive the rules and identify the full set of graphs that contribute to the series in the quenched case. This way it becomes possible to avoid the vague extrapolation from positive integer n to n = 0, that usually goes along with an application of the replica trick. Received 13 December 2001 Published online 25 June 2002     

Dynamical symmetries in relativistic kinetic theory

This paper is part of a program investigating symmetries that are defined at a physical or observational level rather than purely geometrically. Here we generalize previous work on dynamical matter symmetries of relativistic gases. If the matter symmetry vector is surface-forming with the dynamical Liouville vector, then Einstein's equations reduce it to a Killing symmetry of the metric. We show that this conclusion is unaltered if the gas particles are subject to a nongravitational force (including the electromagnetic force on charged particles) or if the gravitational field obeys higher-order field equations. In the Brans-Dicke theory, the matter symmetry reduces to a homothetic symmetry of the metric. This is also the case for a generalized conformal symmetry in Einstein's theory. We consider the problem of relaxing the surface-forming assumption in an attempt to determine whether there are dynamical symmetries that do not necessarily reduce to geometrical symmetries of the metric.     

Dynamical tunneling in macroscopic systems

We investigate macroscopic dynamical quantum tunneling (MDQT) in the driven Duffing oscillator, characteristic for Josephson junction physics and nanomechanics. Under resonant conditions between stable coexisting states of such systems we calculate the tunneling rate. In macroscopic systems coupled to a heat bath, MDQT can be masked by driving-induced activation. We compare both processes, identify conditions under which tunneling can be detected with present day experimental means and suggest a protocol for its observation.     

Dynamical theory of polymer melt morphology

A general kinetic equation of the monomer density variables for polymer blends and block copolymer melts is obtained which describes slow morphology variations. The general theory is applied to a polymer blend adopting the biased reptation model of a polymer chain under mean field. We obtain an equation of motion of interfaces in a phase-separated polymer blend, which contains an interface reaction term for length scales shorter than l_c ≡ R²_G/ξ, where R_G is the gyration radius of a polymer chain and ξ the interfacial width. We also discuss some problems associated with the incompressibility requirement for phase separation kinetics of binary systems not limited to polymers. For length scales greater than l_c the interface dynamics involves diffusion in bulk pure phases even in the strong segregation limit in a way different from that for the usual time-dependent Ginzburg-Landau equation for the conserved order parameter. Implications of the existence of the new term on the late stage phase separation kinetics of polymer blend are discussed.A phenomenological model to study morphology dynamics not relying on the reptation model is also proposed.     

Dynamical mean-field theory of nickelate superlattices

Dynamical mean-field methods are used to calculate the phase diagram, many-body density of states, relative orbital occupancy, and Fermi-surface shape for a realistic model of LaNiO(3)-based superlattices. The model is derived from density-functional band calculations and includes oxygen orbitals. The combination of the on-site Hunds interaction and charge transfer between the transition metal and the oxygen orbitals is found to reduce the orbital polarization far below the levels predicted either by band-structure calculations or by many-body analyses of Hubbard-type models which do not explicitly include the oxygen orbitals. The findings indicate that heterostructuring is unlikely to produce one band-model physics and demonstrate the fundamental inadequacy of modeling the physics of late transition-metal oxides with Hubbard-like models.