A new approach to the relaxation process in gravitational systems

A reformulation of the kinetic theory of N-body gravitational systems, retaining periodic trajectories in the mean field, leads to an intrinsically non-markovian evolution equation, and a √N dependence of the relaxation time. Explicit calculations are carried out for one-dimensional systems.     

New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures

A method is presented that performs the exact electromagnetic analysis of 2D periodic dielectric structures of arbitrary profile or index distribution and possibly large period. The generalized source method is used to formulate the problem of light diffraction in the form of a volume integral equation reduced to a linear equation system, which is solvable by known fast algorithms. The calculation time and required memory are linearly proportional to the total number N_o of considered diffraction orders instead of N_o³ typical for conventional methods. Numerical examples are provided to demonstrate the potential of the method for the analysis of complex periodic structures.     

Compact optimal quadratic spline collocation methods for the Helmholtz equation

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N² log N) on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition.     

Evolution of cooperation in well-mixed N-person snowdrift games

We study the time evolution of cooperation in a recently proposed N-person evolutionary snowdrift game, by focusing on the details of the evolutionary dynamics. It is found that the analytic solution for the equilibrium fraction of cooperators as given previously by the replicator dynamics stems from a balance between the terms: the cost to contribute to a common task and the risk in refusing to participate in a common task. Analytic expressions for these two terms are given, and their magnitudes are studied over the whole range of parameter space. Away from equilibrium, it is the imbalance between these terms that drives the system to equilibrium. A continuous time first-order differential equation for the degree of cooperation is derived, for arbitrary interacting group size N and cost-to-benefit ratio. Analytic solutions to the time evolution of cooperation for the cases of N=2 and N=3 are obtained, with results in good agreement with those obtained by numerical simulations. For arbitrary N, numerical solutions to the equation give the time evolution of cooperation, with the long time limit giving the equilibrium fraction of cooperators.     

A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTS_N) methods applied to the radiative transfer equation in the discrete ordinates (S_N) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” S_N problems by using the LTS_N method. We show some numerical experiments for a typical model problem.     

Collective quantum decoherence induced by qubit-electron interaction

We investigate a model of quantum register composed of N qubits coupling with itinerant electrons by adopting the Born-Markov master equation. Decoherence induced by this coupling is studied for various initial states. By solving the master equation for N=4 with the numerical integration, we obtain time evolution of fidelity and linear entropy of the register. The decoherence rate of this model is proportional to ²|J^′| with J^′ being the exchange coupling strength of electrons and qubits. We also investigate the decoherence free subspace which provides a possible routine of applications in quantum computation.     

Homoclinic phenomena in the gravitational collapse

A class of Bianchi IX cosmological models is shown to have chaotic gravitational collapse, due to Poincaré's homoclinic phenomena. We can program such models so that for any given positive integer N (N=∞ included) the universe undergoes N non-periodic oscillations (each oscillation requiring a long time) before collapsing. For N=∞ the universe undergoes periodic oscillations.     

Coherence and chaos in the driven damped sine-gordon equation: Measurement of the soliton spectrum

A numerical procedure is developed which measures the sine-Gordon soliton and radiation content of any field (φ, φ_t) which is periodic in space. The procedure is applied to the field generated by a damped, driven sine-Gordon equation. This field can be either temporally periodic (locked to the driver) or chaotic. In either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized (soliton wave-train) modes. The numerical procedure quantitatively identifies the presence, number and properties of these soliton wave-trains. For example, an increase of spatial symmetry is accompanied by the injection of additional solitons into the field.     

Multiple dark soliton solutions of the nonlinear Schrödinger equation

We show that the Zakarov-Shabat inverse scattering equations for the stable nonlinear Schrödinger equation can be reduced from 2N to N in the radiationless case. The two-soliton formula is derived. We also present some numerical simulations which contain radiation. Finally we make some comments on the application to optical communications.     

Decreasing of the finite-size effects through the twisted boundary conditions II

In the large-N limit it is shown that a model with twisted boundary conditions becomes equivalent to the U(N) invariant theory which has a volume N² times larger than the theory with periodic boundary conditions. Even for finite N, it is confirmed that the finite-size effects in the models with twisted boundary conditions rather decrease, compared with the ones with periodic boundary conditions, by performing a Monte Carlo simulation for the two-dimensional SU(3) chiral models.     

A Lagrangian fractional step method for the incompressible navier-stokes equations on a periodic domain

In the Lagrangian fractional step method introduced in this paper, the fluid velocity and pressure are defined on a collection of N fluid markers. At each time step, these markers are used to generate a Voronoi diagram, and this diagram is used to construct finite-difference operators corresponding to the divergence, gradient, and Laplacian. The splitting of the Navier-Stokes equations leads to discrete Helmholtz and Poisson problems, which we solve using a two-grid method. The nonlinear convection terms are modeled simply by the displacement of the fluid markers. We have implemented this method on a periodic domain in the planee. We describe an efficient algorithm for the numerical construction of periodic Voronoi diagrams, and we report on numerical results which indicate that the fractional step method is convergent of first order. The overall work per time step is proportional to N log N.     

Solitary wave solutions for the regularized long-wave equation

The regularized long-wave equation has been solved numerically using the collocation method based on the Adams-Moulton method for the time integration and quintic B-spline functions for the space integration. The method is tested on the problems of propagation of a solitary wave and interaction of two solitary waves. The three conserved quantities of motion are calculated to determine the conservation properties of the proposed algorithm. The L _?? error norm is used to measure the difference between exact and numerical solutions. A comparison with the previously published numerical methods is performed.     

The zone boundary mode in periodic nonlinear electrical lattices

We study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of N inductors and N voltage-dependent capacitors with periodic boundary conditions. The inductances are allowed to alternate, while the capacitors are identical and each have a quadratic dependence on voltage. By block-diagonalizing a 2N×2N Floquet problem, we reduce the question of the stability of the mode to a single Hill’s equation that is analyzed using methods of perturbation theory and averaging. We show that periodicity of the lattice inductances degrades stability, and also show that the instability threshold is proportional to N⁻². Numerical computations validate the perturbative results.     

On the dynamical structure of the Dicke maser model

For suitable states of the Dicke maser model we study the time evolution of the mean photon number n(t) in the limit N → ∞. Here N is the number of maser active atoms. Our starting point is a well-tuned cavity with only one mode of the radiation field excited. Introducing new dynamical variables we are able to exploit fully the conservation laws so as to get a simple but completely rigorous solution. We find that n(t) is periodic, and given by a Weierstrassian elliptic function, provided a net polarization is present in the cavity. No approximation is involved.     

Modal prediction of atmospheric turbulence wavefront for open-loop liquid-crystal adaptive optics system with recursive least-squares algorithm

For an open-loop liquid-crystal adaptive optics system, its performance is mainly limited by the time delay. We propose a new modal recursive least-squares (RLS) predictive algorithm to overcome this problem. The RLS algorithm has a simple architecture, low computational complexity and a high converging speed. The impact of the number of the foreword prediction frame N and the number of the predictor order M of the RLS predictor is analyzed in detail. The results show that the best foreword prediction frame N must be equal to the system loop delay frame SLDF. The appropriate predictor order M of the RLS predictor is equal to 2 or 3 when there is no measurement noise and it depends on noise ratio N_R when the measurement error cannot be neglected. We present numerical simulations to show the significant improvements brought by the RLS predictor.     

Quantization of higher-derivative field theories

A free-scalar field satisfying a wave equation with any number of time derivatives is expanded in terms of creation and annihilation operators that are quantized by replacing the classical Poisson brackets of Ostrogradski by the commutator. Regardless of the coefficients in the wave equation, various algebraic identities make it possible to explicitly carry out the quantization, construct the Hamiltonian, and evaluate the propagator. There are always states of negative norm. A wave equation whose highest time derivative is order 2N can have N single-particle states with positive, real energy. Of these, the number of negative-norm states will be N/2 if N is even and (N−1)/2 if N is odd.     

Derivation and simulation of Langevin equations for the Wilson loop in the SU(N) lattice gauge theory

Within the framework of the stochastic quantization method, a general Langevin equation is derived for any function of link variables in the SU(N) lattice gauge theory. The Langevin equation for the Wilson loop is applied to numerical calculations of its expectation value. The same kind of simulation for the thermal Wilson loop is also carried out.     

An equation of oscillations for the arrangement of attracting solids on the equilibrium line of a balance with account for nonlinearity of up to the seventh power

The accuracy of calculating the gravitational constant, G, using a method based on the numerical integration of an equation of oscillations and a method based on the nonlinear oscillation theory is analyzed. Taking account of a higher (the seventh) power at an amplitude of 80 mrad reduces the error of calculating the moment of attraction forces by 47 times. This reduces the error of calculating G from 15 to 0.3 ppm.     

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.