Initial Unstable State Research Articles

The collapse of the cores of slowly rotating isothermal clouds

A generalized model which accounts for the effects of initially uniform and slow rotation is defined for the spherical collapse of a singular isothermal sphere such as protosolar and binary nebulae. An initial unstable equilibrium state is described for a sound speed of 0.35 km/sec and a rotation rate of 10 to the -14th/sec for the molecular cloud surrounding the accreting core. The total angular momentum and mass of the inner cloud is set equal to solar system values. The evolution of the collapse is traced by applying a perturbation analysis to the similarity solution for a nonrotating condition, and matched asymptotic expansions solve the hydrodynamic equations. The model is concluded a valid tool for studying star and nebular disk formation.

On the linearization of nonlinear langevin-type stochastic differential equations

We show that a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required. The scheme is equivalent to solving a few coupled equations. The method is applied to models which represent (a) a single steady state, (b) passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state, and (c) a bivariate case of passage from an unstable state to a final stable state. The results are shown to be in very good agreement with the Monte Carlo calculations carried out for these cases. The method should be of much value in multidimensional cases.

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.

Ersaks's regeneration hypothesis and deviations from the exponential decay law

Recent formal arguments have related deviations from the exponential decay law of an unstable system to the regeneration of the initial unstable state. We investigate this relationship using simple models drawn from quantum optics.

Lagrangian description of phase space flow: Turbulent heating

Anomalous electrostatic dissipation in a collisionless plasma is caused by an induced charge separation which, in the homogeneous case, is particularly large when high local concentrations of the two species occur close together. If a turbulent plasma state has evolved from an unstable initial state which is only slightly perturbed, particle bunching (with maximum number density proportional to T0−1/4 at low initial temperature T0) takes place at points of overtaking - reversal of the ordering of particles with the same initial velocity. Overtaking is governed by a differential equation which involves the applied field only implicitly. Use of this equation in a simple case shows that a bunch of particles of one species causes bunching in the other species, and that the sense of the resulting turbulent electric field is such as to oppose the laminar motion. The characteristic time of formation of an ion bunch by an electron bunch of incremental number density Δn is (4πe2Δn/mi)−1/2. When the charge separation is expressed in terms of Lagrangian coordinates, a possible example of this phenomenon is revealed by the appearance of islands in the relevant curve. The islands represent adjacent bunches of electrons and ions, and preliminary investigation indicates that they might provide a major portion of the dissipation. A computer study to amplify these findings is proposed.

Ignition of simulated propellants based on ammonium perchlorate.

GNITION of a solid propellant is a complex process that involves initial attainment of a thermally unstable state, growth of chemical reaction, appearance of a visible flame, and spreading of the flame over the exposed surface of the propellant. The initial unstable state usually is reached by the application of energy from an external source. The nature of that state then depends on the rate of energy supply, time of exposure, the ambient conditions, and the chemical reactions responsible for ignition. Experimental studies (Refs. 1 and 2 include reviews of such studies) aimed at a detailed knowledge of the ignition of propellants based on NH4C104 have differed mainly in the means of energy supply. The principal means employed have been: 1) conductive heat transfer from an electrically heated wire in contact with the sample3?4; 2) conductive and convective heat transfer from a hot atmosphere68; 3) absorption of radiant energy911; and, 4) heat release resulting from reaction of the sample or vapors resulting from degradation of the sample with a chemically reactive atmosphere in contact with the sample. The experimental conditions of these studies have differed considerably, particularly in the rate of energy supply, and in the pressure, composition, temperature, and chemical reactivity of the atmosphere in contact with the propellant. The conclusions drawn from these studies also have differed: ignition has been attributed to heat release within the propellant,3'9 to attack of the binder by the decomposition products of NH 4C104,12 and to unidentified vapor phase reactions. 10>n The multiplicity of conclusions regarding the reactions responsible for ignition is not surprising in view of the chemical complexity of composite propellants based on NH 4C104. The propellant may contain, besides NH4C104, an organic binder, a burning rate catalyst, and powdered aluminum. In such mixtures the possible reactions are many. The NH4C104, itself, is unstable at moderate temperatures and can decompose with the evolution of heat and chemically reactive gases, and its decomposition is affected by burningrate catalysts such as copper chromite. The NH4C104 also can dissociate13 to NH 3 and HC104, the latter a reactive,

A method of measuring the velocity of very rapid chemical reactions

In devising methods for determining the velocity of any chemical reaction there are two experimental problems which invariably arise : (1) To arrange that the chemical system under investigation be made initially unstable in a period of time that is negligibly short in comparison with that taken by the chemical reaction. (2) To record from time to time the stages reached by the system (during its passage from the initial unstable state to the final stable condition wherein the several reacting substances are in chemical equilibrium) by means of methods which take a negligibly short time in comparison with that taken by the chemical reaction. A perusal of the literature shows that previous investigators have, in the main, restricted themselves to the study of slow reactions, such as may require many minutes or even hours to reach completion. In such cases, both requirements which we have mentioned can be easily met. For the production of the initially unstable condition can be achieved without difficulty by merely mixing the several reacting substances together in proportions far removed from those which prevail when equilibrium has been attained. The time required by the mixing operation can be reduced to a few seconds, and can therefore be neglected when dealing with a process which may last many minutes or even hours. The slow reactions possess a further attraction, in that the procedure for estimating the concentrations of the several reactants at different stages during the progress of the reaction need not be a hurried one. This permits the use of a wide variety of methods,e. g., polarimetry as in the study of the inversion of sucrose, ordinary titration as in the saponification of esters, and separation of one of the constituents as a gas phase as in the decomposition of diazo-acetic ester by water,i. e., N2. CHCOOC2H5+H2O→OHCH2. COOC2H5+N2(gas phase).