Excitonic-vibronic coupled dimers: A dynamic approach

The dynamical properties of exciton transfer coupled to polarization vibrations in a two site system are investigated in detail. A fixed point analysis of the full system of Bloch-oscillator equations representing the coupled excitonic-vibronic flow is performed. For overcritical polarization a bifurcation converting the stable bonding ground state to a hyperbolic unstable state which is basic to the dynamical properties of the model is obtained. The phase space of the system is generally of a mixed type: Above bifurcation chaos develops starting from the region of the hyperbolic state and spreading with increasing energy over the Bloch sphere leaving only islands of regular dynamics. The behaviour of the polarization oscillator accordingly changes from regular to chaotic.     

Multichannel pseudogap Kondo model: large-N solution and quantum-critical dynamics

We discuss a multichannel SU(N) Kondo model which displays nontrivial zero-temperature phase transitions due to a conduction electron density of states vanishing with a power law at the Fermi level. In a particular large- N limit, the system is described by coupled integral equations corresponding to a dynamic saddle point. We exactly determine the universal low-energy behavior of spectral densities at the scale-invariant fixed points, obtain anomalous exponents, and compute scaling functions describing the crossover near the quantum-critical points. We argue that our findings are relevant to recent experiments on impurity-doped d-wave superconductors.     

Deterministic equations of motion and phase ordering dynamics

We numerically solve microscopic deterministic equations of motion for the two-dimensional straight phi(4) theory with random initial states. Phase ordering dynamics is investigated. Dynamic scaling is found and it is dominated by a fixed point corresponding to the minimum energy of random initial states.     

Chaotic scattering on a double well: Periodic orbits,symbolic dynamics,and scaling

We investigate classical scattering of particles by a double-well potential. Irregularity in the scattering functions, such as scattering angle and escape time, appears when the collision energy is lowered below a threshold value. This threshold is closely related to the appearance of periodic orbits with energies above the potential maxima. We study the scattering as a function of the energy and impact parameter. In this initial parameter space the scattering functions consist of regular regions interlaced with chaotic rivers. A symbolic dynamics has been developed to organize these structures and used to reveal their scaling properties.     

Scaling properties of the regular dynamics for a dissipative bouncing ball model

Some scaling properties of the regular dynamics for a dissipative version of the one-dimensional Fermi accelerator model are studied. The dynamics of the model is given in terms of a two-dimensional nonlinear area contracting map. Our results show that the velocities of saddle fixed points (saddle velocities) can be described using scaling arguments for different values of the control parameter.     

Theory of the dynamics of first-order phase transitions: unstable fixed points, exponents, and dynamical scaling

Phase transitions are of great importance in a diversity of fields. They are usually classified into continuous phase transitions and first-order phase transitions (FOPTs). Whereas the former has a well-developed theoretical framework of the renormalization-group (RG) theory, no general theory has yet been developed for the latter that appear far more frequently. Focusing on the dynamics of a generic FOPT in the phi4 model below its critical point, we show by a field-theoretic RG method that it is governed by an unexpected unstable fixed point of the corresponding phi3 model. Accordingly, it exhibits a distinct scaling and universality behavior with unstable exponents different from the critical ones.     

Measurement of the gluon fragmentation function and a comparison of the scaling violation in gluon and quark jets

The fragmentation functions of quarks and gluons are measured in various three-jet topologies in Z decays from the full data set collected with the Delphi detector at the Z resonance between 1992 and 1995. The results at different values of transverse momentum-like scales are compared. A parameterization of the quark and gluon fragmentation functions at a fixed reference scale is given. The quark and gluon fragmentation functions show the predicted pattern of scaling violations. The scaling violation for quark jets as a function of a transverse momentum-like scale is in a good agreement with that observed in lower energy annihilation experiments. For gluon jets it appears to be significantly stronger. The scale dependences of the gluon and quark fragmentation functions agree with the prediction of the DGLAP evolution equations from which the colour factor ratio is measured to be: Received: 5 November 1999 / Published online: 25 February 2000     

Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) modifications to DPSM for ultrasonic field modeling

In the last few years, Distributed Point Source Method (DPSM) a mesh-free semi-analytical technique has been developed. In spite of its many advantages, one shortcoming of the conventional DPSM method is that the field obtained by conventional DPSM method needs to be scaled to match the theoretical solutions. Two modification techniques called Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) are developed here to avoid the scaling need. G-DPSM technique introduces additional fictitious point sources around every parent point source. Gaussian weight functions determine the strength of these additional fictitious point sources that are denoted as child point sources. ESM replaces discrete point sources used in the conventional DPSM by continuous sources. In the ESM formulation individual point sources are denoted as nodes. Special elements are formed on the boundary by connecting these nodes. The source strength inside the element can vary linearly or non-linearly depending on the order of the interpolation function used inside the element. Results generated by both these methods are compared with the conventional DPSM solution and analytical solution. It is shown that the ultrasonic field in front of the transducer computed by G-DPSM and ESM matches very well with the theory without using any scaling factor.     

Chaos,fractional kinetics,and anomalous transport

Chaotic dynamics can be considered as a physical phenomenon that bridges the regular evolution of systems with the random one. These two alternative states of physical processes are, typically, described by the corresponding alternative methods: quasiperiodic or other regular functions in the first case, and kinetic or other probabilistic equations in the second case. What kind of kinetics should be for chaotic dynamics that is intermediate between completely regular (integrable) and completely random (noisy) cases? What features of the dynamics and in what way should they be represented in the kinetics of chaos? These are the subjects of this paper, where the new concept of fractional kinetics is reviewed for systems with Hamiltonian chaos. Particularly, we show how the notions of dynamical quasi-traps, Poincaré recurrences, Lévy flights, exit time distributions, phase space topology prove to be important in the construction of kinetics. The concept of fractional kinetics enters a different area of applications, such as particle dynamics in different potentials, particle advection in fluids, plasma physics and fusion devices, quantum optics, and many others. New characteristics of the kinetics are involved to fractional kinetics and the most important are anomalous transport, superdiffusion, weak mixing, and others. The fractional kinetics does not look as the usual one since some moments of the distribution function are infinite and fluctuations from the equilibrium state do not have any finite time of relaxation. Different important physical phenomena: cooling of particles and signals, particle and wave traps, Maxwell's Demon, etc. represent some domains where fractional kinetics proves to be valuable.     

Critical Dynamics in Thin Films

Critical dynamics in film geometry is analyzed within the field-theoretical approach. In particular we consider the case of purely relaxational dynamics (Model A) and Dirichlet boundary conditions, corresponding to the so-called ordinary surface universality class on both confining boundaries. The general scaling properties for the linear response and correlation functions and for dynamic Casimir forces are discussed. Within the Gaussian approximation we determine the analytic expressions for the associated universal scaling functions and study quantitatively in detail their qualitative features as well as their various limiting behaviors close to the bulk critical point. In addition we consider the effects of time-dependent fields on the fluctuation-induced dynamic Casimir force and determine analytically the corresponding universal scaling functions and their asymptotic behaviors for two specific instances of instantaneous perturbations. The universal aspects of nonlinear relaxation from an initially ordered state are also discussed emphasizing the different crossovers occurring during this evolution. The model considered is relevant to the critical dynamics of actual uniaxial ferromagnetic films with symmetry-preserving conditions at the confining surfaces and for Monte Carlo simulations of spin system with Glauber dynamics and free boundary conditions.     

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

Power-Law Behavior in Geometric Characteristics of Full Binary Trees

Natural river networks exhibit regular scaling laws in their topological organization. Here, we investigate whether these scaling laws are unique characteristics of river networks or can be applicable to general binary tree networks. We generate numerous binary trees, ranging from purely ordered trees to completely random trees. For each generated binary tree, we analyze whether the tree exhibits any scaling property found in river networks, i.e., the power-laws in the size distribution, the length distribution, the distance-load relationship, and the power spectrum of width function. We found that partially random trees generated on the basis of two distinct types of deterministic trees, i.e., deterministic critical and supercritical trees, show contrasting characteristics. Partially random trees generated on the basis of deterministic critical trees exhibit all power-law characteristics investigated in this study with their fitted exponents close to the values observed in natural river networks over a wide range of random-degree. On the other hand, partially random trees generated on the basis of deterministic supercritical trees rarely follow scaling laws of river networks.     

Quasiclassical green function in an external field and small-angle scattering processes

A representation is obtained for the quasiclassical Green functions of the Dirac and Klein-Gordon equations allowing for the first nonvanishing correction in an arbitrary localized potential which generally possesses no spherical symmetry. This is used to obtain a solution of these equations in an approximation similar to the Furry-Sommerfeld-Maue approximation. It is shown that the quasiclassical Green function does not reduce to the Green function obtained in the eikonal approximation and has a wider range of validity. This is illustrated by calculating the amplitude of small-angle scattering of a charged particle and the amplitude of Delbrück forward scattering. A correction proportional to the scattering angle was obtained for the amplitude of charged particle scattering in a potential possessing no spherical symmetry. The real part of the Delbrück forward scattering amplitude was calculated in a screened Coulomb potential.     

One-parameter scaling at the dirac point in graphene

We numerically calculate the conductivity sigma of an undoped graphene sheet (size L) in the limit of a vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function beta(sigma)=dlnsigma/dlnL. Contrary to a recent prediction, the scaling flow has no fixed point (beta>0) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data support an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering--without reaching a scale-invariant limit.     

Parameter renormalization of maps based on potential function

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.     

Three Lemmas on Dynamic Cavity Method

We study the dynamic cavity method for dilute kinetic Ising models with synchronous update rules. For the parallel update rule we find for fully asymmetric models that the dynamic cavity equations reduce to a Markovian dynamics of the (time-dependent) marginal probabilities. For the random sequential update rule, also an instantiation of a synchronous update rule, we find on the other hand that the dynamic cavity equations do not reduce to a Markovian dynamics, unless an additional assumption of time factorization is introduced. For symmetric models we show that a fixed point of ordinary Belief propagation is also a fixed point of the dynamic cavity equations in the time factorized approximation. For clarity, the conclusions of the paper are formulated as three lemmas.     

A multidimensional pendulum in a nonconservative force field under the presence of linear damping

A nonconservative force field in the dynamics of a multidimensional solid is constructed according to the results from the dynamics of real solids occurring in the force field of the action of the medium. In this case, it becomes possible to generalize the equations of motion of a multidimensional solid in a similarly constructed field of forces and to obtain a complete list of, generally speaking, transcendental first integrals expressed through a finite combination of elementary functions. In the study, the integrability in elementary functions is shown for the simultaneous equations of motion of a dynamically symmetric fixed multidimensional solid under the action of a nonconservative pair of forces in the presence of the linear damping moment (the additional dependence of the force field on the tensor of angular velocity of the solid).     

Discretized diffusion processes

We study the properties of the "rigid Laplacian" operator; that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the presence of a metastable set of numerical solutions, whose presence can be related to granularity. We provide some scaling analysis in order to determine the value of the exponents characterizing the process. We believe that this prototype model is also suitable to provide an explanation of the widespread presence of power law in a social and economic system where information and decision diffuse, with errors and delay from agent to agent.