Time-independent States Research Articles

On the Derivation of the Time-Dependent Equation of Schrödinger

Few have done more than Martin Gutzwiller to clarify the connection between classical time-dependent motion and the time-independent states of quantum systems. Hence it seems appropriate to include the following discussion of the origins of the time-dependent Schrodinger equation in this volume dedicated to him.

Non-Connected Branches of Alternative Stationary States — An Experimental Illustration

The steady-state and dynamic behavior of a partial glycolytic reaction sequence are investigated in cell-free extracts of yeast. Pyruvate kinase, adenylate kinase and glucose 6-phosphate isomerase cooperate to a multi-enzyme system centered around the 6-phosphofructose 1-kinase (6-PFK) and fructose 1,6-bisphosphatase (FBPase) cycle. The reaction system operates under thermodynamically open conditions maintained by a continuous supply of substrates, i.e. glucose 6-phosphate (Glc6P), ATP and phospho-enolpyruvate (PPrv) in a flow-through reaction chamber. Appropriate parametric conditions lead to the occurence of (two) coexisting and markedly different time-independent states in the metabolite concentrations and fluxes. It was already shown that for particular experimental conditions, changes of the influx adenylate energy charge, [AEC] IN , may cause transitions between these alternative steady states which are either reversible as it occurs in classical hysteresis phenomena, or irreversible (irreversible transitions) where the system is not able to switch back to its previous state even when the perturbation is reverted. It was also shown numerically from the same qualitative model, that changes in the [AEC] IN , could also lead to the occurence of non-connected branches of alternative steady states [27]. In the present article, we propose the first experimental demonstration for the existence of such dynamic structures related to bistability and where no transition between the two coexisting stable branches of the bistable are physically possible and defining thus functionally non-equivalent states with distinct metabolite pattern.

Irreversible Metabolic Transitions: The Glucose 6-Phosphate Metabolism in Yeast Cell-Free Extracts

The steady-state and dynamic behavior of a partial glycolytic reaction sequence are investigated in cell-free extracts of yeast. Pyruvate kinase, adenylate kinase and glucose 6-phosphate isomerase cooperate to a multienzyme system centered around the 6-phosphofructokinase (6-PFK) and fructose 1,6-bisphosphatase (FBPase) cycle. The reaction system operates under thermodynamically open conditions maintained by a continuous supply of substrates, i.e., glucose 6-phosphate (Glc6P), ATP and phosphoenolpyruvate (PPrv) in a flow-through reaction chamber. Appropriate conditions lead to the occurence of (two) coexisting and markedly different time-independent states in the metabolite concentrations and fluxes. For particular experimental conditions, changes in the influx adenylic energy charge, [AEC]IN, may cause transitions between these alternative steady states which are either reversible as it occurs in classical hysteresis phenomena, or, more importantly, irreversible (irreversible transitions, IT) where the system is not able to switch back to its previous state even when the perturbation is reverted. The emergence of these irreversible transitions do not result from artificial or non-realistic experimental constraints, but are a potential intrinsic property of any non-linear dynamic system exhibiting bi- or multistability. These one-way transitions may well have important biological implications with respect to switching, adaptation and memory phenomena.

Stability of Cellular States of the Kuramoto–Sivashinsky Equation

This paper is concerned with the instability property of a particular type of solution to the Kuramoto–Sivashinsky equation, namely, the symmetric, time-independent cellular states. Attention is focused on the dimension of their unstable manifolds. We show that as a control parameter varies, the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation. Further change in dimension is shown to be related to Hopf bifurcations from the cellular states. An asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur.

Influence of trapped electrons on the time-independent states of a negatively biased single-ended Q machine

Measurements of basic plasma parameters in a low-density single-ended Q machine with a negatively biased cold plate are compared with pertinent predictions of time-independent collisionless plane-diode theory. As was also found in particle simulations [Phys. Fluids B 3, 244 (1991)] and previous experiments, theory and experiment agree well if space-charge conditions near the hot-plate surface are electron rich, in which case the potential distributions are monotonically decreasing. For ion-rich conditions, however, where the potential distributions exhibit a single maximum located in front of the hot plate, the plasma properties are found to depend very sensitively on the amount and distribution of the electrons trapped in the region of positive potential. In this regime, the experimental findings agree very well with a theoretical model in which a certain amount of trapped electrons is assumed.

Multistability in networks of weakly coupled bistable units

We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a ‘propagation failure’ phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations

Trapped-electron effects on time-independent negative-bias states of a collisionless single-emitter plasma device: Theory and simulation

Time-average values from particle simulations of a collisionless, single-emitter plasma device modeling single-ended Q machines or thermionic converters with a negatively biased collector are presented. These results quantitatively confirm the predictions of collisionless, kinetic plane-diode theory for spatial potential profiles that decrease monotonically. However, simulations of negative-bias potential profiles with a single internal maximum differ significantly from previous theoretical predictions which assumed electron phase space to have either (i) no trapped electrons or (ii) trapped electrons isothermal with the passing electrons. A more general class of trapped-electron model distributions is introduced from which new equilibrium potential values can be recovered that closely match the simulations. These simulations clearly demonstrate the sensitive role that trapped electrons play in shaping the potential profiles of the equilibrium (or slowly evolving) states of the simulated systems. The trapped-electron distributions in these simulations are themselves shown to be controlled critically by fluctuations whose levels are varied by the choice of particle injection scheme. These effects, although found and discussed here in the context of a particular model, are believed to be important in many bounded plasma systems where electrons can be trapped in potential wells.

Symplectic manifolds, coadjoint orbits, and mean field theory

Mean field theory is given a geometrical interpretation as a Hamiltonian dynamical system. The Hartree-Fock phase space is the Grassmann manifold, a symplectic submanifold of the projective space of the full many-fermion Hilbert space. The integral curves of the Hartree-Fock vector field are the time-dependent Hartree-Fock solutions, while the critical points of the energy function are the time-independent states. The mean field theory is generalized beyond determinants to coadjoint orbit spaces of the unitary group; the Grassmann variety is the minimal coadjoint orbit.

Nonlinear resistive helical states in reversed-field pinches

Time-independent nonlinear states resulting from resistive modes with helical symmetry are shown to exist in reversed-field pinch configurations. The importance of flow effects in such states is demonstrated, and the nonlinear redistribution of magnetic flux caused by the presence of the deformation is discussed.

A plasma model with energy transport and dynamo for sustained reversed field pinches

A method is presented to calculate the hypothesized dynamo mean electric field required to sustain reversed field pinch (RFP) plasmas in time-independent states such as have been observed in recent RFP experiments. It is also shown how to simultaneously calculate energy transport in terms of a phenomenological radial thermal conductivity κ. It was found that the dynamo extracts energy from the central plasma in order to drive the large azimuthal current needed by the RFP. The quantitative results are only weakly sensitive to the choice of parameters. The principal predictions are the distribution of the dynamo mean electric field in the plasma and the ratio between average and peak electrical conductivity. The theory is compared with experimental RFP data, and it is concluded that the mean plasma resistivity is not much different from the classical resistivity of Spitzer.

Doppler Imaging of the Onset of Turbulent Convection

We demonstrate that large rectangular fluid layers have several essentially time-independent convective states above the critical Rayleigh number ${R}_{c}$ in which the rolls align perpendicular to all lateral boundaries. The pattern is hence splayed and contains defects. Using real-time laser Doppler imaging, we show that the onset of time dependence at ${R}_{1}\ensuremath{\cong}5{R}_{c}$ arises from a structural instability of the rolls which can be suppressed by a local heat source.

The Branching Hierarchy of Multiple Solutions in a Model of Moist Convection

Abstract The dynamics of two-dimensional, shallow moist convection is examined with the use of a six-component spectral model. Latent heating effects are incorporated by assuming that upward motion is moist adiabatic and that downward motion is dry adiabatic. The resulting nondimensional system of equations has the same form as that for Benard convection, with the moist effects included by replacing the Rayleigh number with a modified form. The six-coefficient model contains a wide variety of multiple solutions with as many as 12 time-independent convective states and one conductive state occurring simultaneously. Temporally periodic solutions are also indicated, and some are found numerically that branch from stationary solutions at critical values of the external parameters. Only some of the solutions are linearly stable and hence observable, and we give a summary of the possible branching orders of these solutions. We find that the development of moist convection proceeds in the model via one of severa...

Soluble model for the analysis of stability in an imploding compressible liner

A soluble model of the development of the linear pertubations about a time-varying state of a compressible medium is presented. A Lagrangian description is employed to rederive the equations for the self-similar motion of an ideal fluid and to obtain the linearized equations of motion for pertubations about a general time-varying basic state. The resulting formalism is applied in cylindrical geometry to calculate the growth of flute-like modes associated with a similarity solution modeling the implosion and expansion of a fluid liner. A complete solution is obtained for the perturbed motion. The only modes for which the perturbation amplitudes grow faster than the unperturbed liner radius during both implosion and expansion are divergence- and curl-free. Numerical and analytical results are obtained for these and shown to reduce, in the short-wavelength limit, to the Rayleigh–Taylor instability found previously for incompressible time-independent basic states. In addition, a new kind of instability is found: a class of overstable internal modes (sound waves), which are ’’pumped up’’ in amplitude during implosion, but decay during the expansion phase.

Systemanalyse der Silizium-Diatomeen-Wechselwirkung in einem see der DDR

Predictions of and actions against blooms of plankton di-atoms in lakes and slowly flowing rivers are special and important problems of water quality management. In the paper we use methods of mathematical systems analysis, computer simulation and parameter identification in a simple lake-ecosystem, consisting only of a plankton diatom population an soluted silicone (SiO 2 ). Especially we deal with the problems of existence and stability of time independent steady states of the system on the basis of the Ljapounoff-stability theory.