Class Of Initial Conditions Research Articles

Approximate solutions of the two-component Ginzburg-Landau equation

We present an approximate solution of the two-component Ginzburg-Landau equation for a broad class of initial conditions. Our method of solution is based on a novel singular perturbation expansion. Specifically, we consider the formation of vortex- antivortex pairs, from an initial condition consisting of small random fluctuations about zero. The analytic solution compares reasonably well with the results of a numerical integration of the two-component Ginzburg-Landau equation.

Power spectra in two-dimensional turbulence.

We discuss the small-scale energy spectrum E(k) of two-dimensional fully developed turbulence. As its scaling exponent is related to the fractal dimension of isovorticity lines, we suggest that it could be nonuniversal with a dependence on the particular class of initial conditions considered. This should explain the different values obtained in numerical experiments for decaying turbulence and for statistically stationary forced turbulence with particular kinds of forcing. Finally, the scaling E(k)\ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}3}$ is related to the most random situation where the vorticity can be considered as a scalar passively advected by the velocity field.

Optimal control of uncertain quantum systems.

The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions ${\mathit{H}}_{0}$(\ensuremath{\alpha}) indexed by the random variable \ensuremath{\alpha} taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions.The constrained optimization problem is reduced to an unconstrained optimization problem by the introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.

DETERMINISTIC STATISTICAL INDEPENDENCE, NONPERIODIC SYMBOL SEQUENCES AND GIBBS DISTRIBUTIONS

We use maps that generate a hierarchy of partitionings that can be organized onto a complete binary tree to ask the question when does the Gibbs distribution describe the frequency of distribution of iterates of a map over its natural partitioning. We are led to a generalization of Borel’s normal numbers, to symbol sequences where the blocks of symbols of a given length are distributed statistically independently, but with unequal frequencies. This result leads to a new universality principle based upon classes of initial conditions that limits the extent within which one can use an observed f(α) spectrum to infer the underlying dynamical system. An application to a turbulence experiment is discussed as an example. Our method is based upon the generation of symbol sequences by an algorithm and the consequent recovery of the corresponding initial condition by backward iteration of the map. Our analysis illustrates that measure one theorems in mathematics cannot serve as a guide to what should be the outcome of either computation or experiment.

On the relaxation of single hard-sphere gases

The nonlinear Boltzmann equation is solved to examine the Maxwellization of a spatially uniform hard-sphere gas using the Laguerre moment method. The computations are carried out for two different classes of initial conditions. Emphasis is layed on the characteristic times for the relaxation of the distribution function toward the equilibrium. As a result, in the thermal energy range the relaxation takes place within few mean collision times, regardless of the initial state. Depending on whether the high-energy tail is initially overpopulated or underpopulated the relaxation time in this part of the spectrum is a function of the molecular velocity via 1/v or log v, respectively. The solutions are compared with those of the linearized Boltzmann equation.

Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt = F(x, t, u, ux)

The aim of the paper is to give two theorems about existence and uniqueness of continuous solutions for hyperbolic nonlinear differential problems with nonlocal conditions in bounded and unbounded domains. The results obtained in this paper can be applied in the theory of elasticity with better effect than analogous known results with classical initial conditions.

Approximate asymptotic solutions to the d-dimensional fisher equation

We present approximate asymptotic solution to the Fisher equation in arbitrary dimensions for a broad class of initial conditions. Specifically, we consider the case of domain growth from a seed in a zero background. Our analytic solution for this case has an asymptotic velocity of 2 in all dimensions. In one dimension, our solution for this case compares well with a numerical simulations of the Fisher equation.

Estimation of flow duration of basaltic magma in fissures

We suggest a relationship between the thickness of chilled margins often observed in basaltic dikes and magma flow duration. For this purpose, a new thermal model of cooling basaltic dikes is proposed which takes into account flow duration in a fissure before solidification of the magma. The thermal problem is solved by adding a boundary condition (serving as a substitute for magma flow) to the classical initial conditions for dikes of Fourier's equation. We assume a bound temperature T o at the center of the dike as long as magma flows. At the beginning of the cooling process, the new model gives the same results as the classical one and, consequently, the same chilled margin thicknesses. But later on, when the classical model enforces temperature decreases, our temperatures increase due to the heat brought by magma flow. The chilled margin is reheated and partially or fully destroyed. Petrological observations on the chilled margins of dikes from Salagou and Bédarieux (South of France) and their country rock support this theoretical study. The chilled margins are always thinner than 10 mm; glass includes rare true phenocrysts (mainly olivine and Fe-Ti oxides) and branching quench crystallites of clinopyroxene. The conditions of formation and evolution of the chilled margins have been specified from experimental data: cooling rate = 800° C h −1; glass transition temperature ( T M) = 675° C and glass deformation temperature ( T D) = 705° C. The initial chilled margins predicted by the two models are always thicker than those actually observed; there is evidence for reduction due to reheating by devitrification and softening. In our new model, the chilled margin thickness which remains is directly dependent on the flow duration. For example, for a dike of Salagou (1-m wide and a chilled margin of 5 mm), a flow duration of six days has been computed. At Bédarieux, for a 1-m-wide dike and a thicker chilled margin (8.5 mm), flow duration was 4.6 days. These values must be regarded as upper limits because the effects of the latent heat on the temperature distribution during reheating have not been taken into account. The metamorphism of the country rock (pelites and bauxites) has been investigated; temperatures estimated from classical geothermometers are in agreement with those computed by our model and support its validity.

Non-stochastic Langevin equation and the arrow of time

The behavior of a Brownian particle coupled to a linear harmonic chain is studied using a non-stochastic analysis. We show how a non-stochastic time correlation function of the fluctuating force reproduces the essential properties of the stochastic correlation function under not too restrictive conditions. The width of our correlation function is ωm-1 in time and, for time scales ⪢ωm-1, behaves as a delta function. We discuss how irreversible-like behavior might occur in spite of the reversibility of our Langevin equation and how irreversibility depends upon a restricted narrow class of initial conditions. Our results thus help dispel the mystery of the “arrow of time”.

Heavy ion collisions and anisotropic hydrodynamics.

The study of the isotropization of momentum is important in heavy ion collisions. To do this we construct a generalized hydrodynamical equation system, in which the anisotropy of the momentum distribution is added as a new variable. These equations are derived from the moment equations of the relativistic Boltzmann equation where the closure of the set is achieved by assuming a particular class of initial conditions. The equations are then explicitly solved for two uniform interpenetrating hadron streams. The collision cross sections are the bare hadron cross sections; the presence of the other hadrons can be simulated by the use of a density- and energy-density-dependent temperature and mass, taken over from self-consistent calculations. The results are compared with other theoretical results. We find that the isotropization occurs sufficiently rapidly for medium energy head-on collisions to reach local thermal equilibrium.

Generalized Operator Riccati Equations

The Riccati equation \[\frac{{dU}}{{dt}} = AU + UB^* + UCU + D\]on the space $\mathcal {L}(X)$ of bounded linear operators on a reflexive Banach space X arises in control theory and transport theory. A more general problem is the following. Let A and B be closed linear operators in X and $X^* $ respectively and let $\mathcal {F}$ be a map from $[0,T) \times \mathcal {L}(X)$ into $\mathcal {L}(X)$ and consider the initial value problem \[\frac{{dU}}{{dt}} = \mathcal{L}\ell U[AU + UB^* ] + \mathcal{F}(t,U),\quad U(0) = U_0 ,\]on $\mathcal {L}(X)$. It is shown that for a certain class of initial conditions, determined by A, B and the geometry of X, there exist continuously differentiable solutions with respect to the uniform operator topology. It is also shown that if A and B have compact inverses, then there exist solutions with respect to the strong operator topology for arbitrary initial conditions.

A note on the Cauchy problem for differential difference equations

has a unique continuous solution on the interval [0, T] if functions x,, and g are continuous (see [ 1, 2,4]). However, the function y(t) which coincides with x(t) on [0, T] and with -u,(t) on ( -3, 0) will in general be discontinuous or have discontinuous derivatives at t = 0, unless x,(t) is carefully chosen. We endeavoured to investigate whether this situation could be improved by using the classical Cauchy type initial condition, x(0, S) = c instead of (2), and imposing some continuity conditions. This leads, however, to a non-uniqueness of the resulting solution even if we demand analyticity in t. For example, the equation,

On the Blow up Problem for Semilinear Heat Equations

This paper is concerned with the instability behaviour of a differential system arising from the theory of channel and cylindrical flow of viscous fluids with high heat generation. The spatial domain under consideration is the whole space $R^n $, since usually in this type of problem a bounded domain can be transformed into an unbounded domain. It is shown that if the physical parameter $g(x,t)$, which corresponds to the stress on the fluid is such that $g(x,t) \geq \lambda t^{n + \alpha - 1} H(x,t)$, where $H(x,t)$ is the fundamental solution of the heat equation and $\lambda $ and $\alpha $ are positive constants, then for certain classes of initial conditions the corresponding solution of the initial value problem grows unbounded in a finite time. We obtain an upper bound for the blow up time. We also explain how the method can be applied to a typical physical problem.

Mathematical problems of irreversible statistical mechanics for quantum systems. II: On the singularities of (Ψ(z)−z)−1 and the pseudo-Markovian equation. Application to Lee model

We study in the frame of the superspace of the Hilbert–Schmidt operators the contributions of some complex singularities of the analytic continuation of (Ψ(z)−z)−1 to the diagonal part of the solution of the Liouville–von Neumann equation. Under some conditions, the θ̄ operator of the pseudo-Markovian master equation can be explicitly constructed. It is necessary to specify the diagonal representation and the class of initial conditions having regularity properties. This Hilbertian structure does not allow the construction of a closed subspace which reduces the Liouville–von Neumann operator L, giving an exact irreversible subdynamics; more elaborated mathematical structures are therefore necessary. The above methods are illustrated in the case of the Lee model.

Spectral deformation techniques applied to the study of quantum statistical irreversible processes

A procedure of analytic continuation of the resolvent of Liouville operators for quantum statistical systems is discussed. When applied to the theory of irreversible processes of the Brussels School, this method supports the idea that the restriction to a class of initial conditions is necessary to obtain an irreversible behaviour. The general results are tested on the Friedrichs model.

Criteria for Pervasive Damping of Mechanical Systems with First Integrals of Motion

AbstractThis paper determines necessary and sufficient criteria for pervasive (or complete) damping of mechanical systems with first integrals of motion. The pervasiveness of the damping is essential when the stability of a system is investigated through its linearized set of equations of motion by Liapunov's method: the sufficient conditions for asymptotic stability then become necessary conditions. The existence of first integrals of motion restricts the asymptotic stability to a certain class of initial conditions. The pervasiveness of the damping will be considered in the same sense.

On a simple wave approximation

An approximation (the linear version of Burgers' equation with appropriate initial data) to a simple wave initial value problem for a set of two linear coupled dissipative partial differential equations is discussed. It has been shown that for the class of square integrable initial functions of which the spectra (Fourier-transforms) have bounded support 2Δ the approximation is valid for some finite interval of time [0, T(Δ)]. For some finite timeT 1 > T(Δ) the approximation may fail. However, fort→∞, it is asymptotically valid again. For the class of initial conditions mentioned above expansions in series of the two solutions, which for every finite interval of time [0, τ] are convergent, may be constructed.