STABILITY FOR SOLUTIONS OF A STATIONARY EULER–POISSON PROBLEM

We study the stability of the solutions of a stationary Euler–Poisson problem modeling a plasma diode. The model was presented in a previous paper, where we proved the existence of multiple solutions. Since, physically speaking, we expect only one solution to be present, we propose here two strategies for selecting the physical stationary regime. On the one hand, we study the energy functional associated with the stationary problem and order the different solutions in terms of their energy level and as stationary points for the functional. On the other hand, starting from the remark that a stable stationary state is the state which is reached after an evolution, we solve numerically the corresponding time-dependent Euler–Poisson problem and show that only one of the possible solutions is indeed stable in this sense. These two independent approaches select the same solution that we label therefore as stable.

Response of Systems to Pulse Perturbations

Consider a chemical reaction system with many chemical species; it may be in a transient state but it is easier to think of it in a stable stationary state, not necessarily but usually away from equilibrium. We wish to probe the responses of the concentrations of the chemical species to a pulse perturbation of one of the chemical species. The pulse need not be small; it can be of arbitrary magnitude. This is analogous to providing a given input to one variable of an electronic system and measuring the outputs of the other variables. The method presented in this chapter gives causal connectivities of one reacting species with another as well as regulatory features of a reaction network. Much more will be said about the responses of chemical and other systems to pulses and other perturbations in chapter 12. The effects of small perturbations on reacting systems have been investigated in a number of studies, to which we return in chapters 9 and 13. Let us begin simply: Consider a series of first-order reactions as in fig. 5.1, which shows an unbranched chain of reversible reactions. We shall not be restricted to first-order reactions but can learn a lot from this example. Let there be an influx of k0 molecules of X1 and an outflow of k8 molecules of X8 per unit time. We assume that the reaction proceeds from left to right and hence the Gibbs free energy change for each step and for the overall reaction in that direction is negative. The mass action law for the kinetic equations, say that of X2, is . . . dX2/dt = k1X1 + k−2X3 − (k−1 + k2) X2 (5.1) . . . If all the time derivatives of the concentrations are zero, then the system is in a stationary state. Suppose we perturb that stationary state with an increase in X1 by an arbitrary amount and solve the kinetic equations numerically for the variations of the concentrations as a function of time, as the system returns to the stationary state. A plot of such a relaxation is shown in fig. 5.2.

Lifetime and Transit Time Distributions and Response Experiments in Chemical Kinetics

We discussed some aspects of the responses of chemical systems, linear or nonlinear, to perturbations on several earlier occasions. The first was the responses of the chemical species in a reaction mechanism (a network) in a nonequilibrium stable stationary state to a pulse in concentration of one species. We referred to this approach as the “pulse method” (see chapter 5 for theory and chapter 6 for experiments). Second, we studied the time series of the responses of concentrations to repeated random perturbations, the formulation of correlation functions from such measurements, and the construction of the correlation metric (see chapter 7 for theory and chapter 8 for experiments). Third, in the investigation of oscillatory chemical reactions we showed that the responses of a chemical system in a stable stationary state close to a Hopf bifurcation are related to the category of the oscillatory reaction and to the role of the essential species in the system (see chapter 11 for theory and experiments). In each of these cases the responses yield important information about the reaction pathway and the reaction mechanism. In this chapter we focus on the design of simple types of response experiments that make it possible to extract mechanistic and kinetic information from complex nonlinear reaction systems. The main idea is to use “neutral” labeled compounds (tracers), which have the same kinetic and transport properties as the unlabeled compounds. In our previous work we have shown that by using neutral tracers a class of response experiments can be described by linear response laws, even though the underlying kinetic equations are highly nonlinear. The linear response is not the result of a linearization procedure, but it is due to the use of neutral tracers. As a result the response is linear even for large perturbations, making it possible to investigate global nonlinear kinetics by making use of linear mathematical techniques. Moreover, the susceptibility functions from the response law are related to the probability densities of the lifetimes and transit times of the various chemical species, making it easy to establish a connection between the response data and the mechanism and kinetics of the process.

HAMILTONIAN DYNAMICS OF THE COMPLEX FROEHLICH DIMER

A classical Hamiltonian system modelling dynamics of dipole momenta of the complex Froehlich dimer is proposed and analyzed. Formally, the classical system is a system of two quartic oscillators with three different coupling constants, and all formal parameters in the Hamiltonian's function are expressed via only two parameters with microscopic physical interpretation. The classification of stable configurations of the dimer in terms of stationary states of its classical model is given. Their stability, in the linear approximation as well as for the full nonlinear dynamics, is analyzed with respect to the variations of the physical parameters. For example, it is shown that for the medium values of the parameter related to the rate of the energy supplied to the dimer, the stable stationary state is not with the minimal energy, but corresponds to the deformed dimer, with parallel dipole momenta of the monomers.

Effects of Statistical Fluctuations in a Model of Tropospheric Chemistry

We present numerical simulations of the ozone concentration in the troposphere based on a six-variable model of tropospheric chemistry. Although the model is not meant to describe the detailed mechanism of tropospheric chemistry it contains nonlinear steps leading to oscillatory behaviour. Statistical noise imposed upon the emission intensity of nitric oxide may lead to the interesting phenomenon of "internal signal stochastic resonance". Here the imposed noise amplifies oscillatory dynamics of the system under conditions which lead to a stable stationary state in the absence of noise. A combination of statistical perturbations with a slow modulation of the rates of photosensitive steps allows to qualitatively reproduce observed time series of the ozone concentration.

THE ROLE OF MEMBRANE ELECTROSTATICS IN THE REGULATION OF CELL VOLUME AND ION CONCENTRATIONS

We present a model to study how membrane surface negative charges can affect the electro-osmotic regulation properties of a cell. This model is based on the cellular analog proposed by Jakobsson, which includes passive and active ion transports; we further introduce the effect of membrane surface charges, using a generalized formulation of the Gouy–Chapman theory. We derive a system of nonlinear differential-algebraic equations (DAEs) which describes the dynamics of the cellular analog. The system admits a unique asymptotically stable stationary state, in which the Na-pump rate, which is crucial for electro-osmotic regulation, is inversely related to the Ca 2+ level in the extracellular milieu; numerical integration shows that this apparent inhibition of the Na-pump by external Ca 2+ results from a decrease in the electrostatic field produced by surface charges at the external side of the membrane. Furthermore, the degree of stability of the stationary state dramatically depends on the amount of negative charges on the membrane; a maximal stability is obtained for densities around - e /500 Å2, where the Na-pump is maximally activated by an increase in the Na content of the cytoplasm.

Entropy Production in a Chaotic Chemical System

The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.

Dynamics of Discovery and Exploitation: The Case of the Scotian Shelf Groundfish Fisheries

The bases and shortcomings of the current models used in fisheries management are briefly examined. An alternative set of models in part based on the Volterra–Lotka equations are developed which incorporate recent advances in our understanding of the evolution of complex systems. Simulations based on a dynamic model of a Nova Scotia fishery reveal that human responses amplify rapid random fluctuations in recruitment and excite strong Volterra–Lotka type oscillations in a system that would normally repose in a stable stationary state. A dynamic, multispecies, muitifleet spatial model calibrated to the Nova Scotian groundfish fisheries is presented and used to explore the concepts of "discovery" and "exploitation." Two types of fishermen are identified, "stochasts" and "cartesians," characterized respectively as hunters, or high risk takers, and followers, or low risk takers. Significant results include the importance of calibration in providing models of relevance to the real world; the "out of phase" relationship between abundance and the ease with which fishermen locate a highly sought species and its converse; the importance of information exchange in defining the attractivity of a particular fishing zone to different fleets and the ability of the model to take into account coded information, misinformation, spying and lying; and the fact that models based on global principles, such as "optimal efficiency" or "maximum profit," are clearly of dubious relevance to the real world.