Evidence of many-particle interactions in two-dimensional charge-stabilized colloidal crystals

Electrostatic interaction in two-dimensional colloidal crystals obeying the nonlinear Poisson-Boltzmann equation is studied numerically. The force constants which are the coefficients of the energy quadratic form of the crystal are obtained. Significant discrepancy between the results of computer experiments and predictions of the harmonic crystal theory based on the idea of pairwise interaction has been detected. Two simple parameters are introduced for uniform quantitative estimation of many-particle effects and validity of the approximation of nearest-neighbor interaction. It is shown that the contribution of many-particle interaction into the total electrostatic interaction in colloidal crystals is significant for a broad range of crystal lattice constants and particles' sizes.     

Osmotic stress tensor in charge stabilized colloidal crystals

Osmotic stress tensor is introduced to describe the osmotic pressure in colloidal crystals within the framework of the theory of the Poisson-Boltzmann equation. The osmotic stress tensor is related to the fundamental stress tensor, which is associated with the Poisson-Boltzmann equation. It is shown that the osmotic stress tensor can be determined for colloidal crystals with arbitrary structures, as well as for media that are described by cell models. The general results are exemplified by spherical and cylindrical cell models.     

Force constants and many-body electrostatic interactions in two-dimensional colloidal crystal

A model of a two-dimensional colloidal crystal with a hexagonal lattice, the electrostatic interactions in which are described by the nonlinear Poisson-Boltzmann equation, is considered. The calculation procedure for force constants of this crystal is treated in detail. Properties of system symmetry, which make it possible to significantly decrease the volume of calculations and to classify force constants, are analyzed. Numerical data for force constants of a crystal as functions of lattice parameters at different particle sizes are reported. A method that allows us to disclose the presence of many-body interactions in a system by the behavior of force constants at some interval of the values of lattice parameters is proposed. The application of this method to the system under consideration demonstrated that electrostatic interparticle interactions in the system cannot be reduced to simply a pair interaction of any kind; the introduction of many-body potentials is required for the adequate representation of the elastic properties of a crystal.     

Two-dimensional colloidal crystal in nonlinear Poisson-Boltzmann model

A two-dimensional hexagonal colloidal crystal of charged particles obeying the general nonlinear Poisson-Boltzmann equation is studied by the numerical method. Force constants and pressure in a system, as well as elastic constants of a crystal, are calculated on the basis of the solutions of the equation. Calculation procedures are described briefly and numerical data are reported. The effect of nonlinearity of charge distribution on the manifestation of many-body interactions and on the validity of the approximation of interaction of the nearest neighbors is considered.     

HARVESTING ECONOMIC MODELS AND CATCH‐TO‐BIOMASS DEPENDENCE: THE CASE OF SMALL PELAGIC FISH

Abstract In the case of small pelagic fish, it seems reasonable to consider harvest functions depending nonlinearly on fishing effort and fish stock. Indeed, empirical evidence about these fish species suggests that marginal catch does not necessarily react in a linear way neither to changes in fishing effort nor in fish stock levels. This is in contradiction with traditional fishery economic models where catch‐to‐input marginal productivities are normally assumed to be constant. While allowing for nonlinearities in both catch‐to‐effort and catch‐to‐stock parameters, this paper extends the traditional single‐stock harvesting economic model by focusing on the dependence of the stationary solutions upon the nonlinear catch‐to‐stock parameter. Thus, we analyze equilibrium responses to changes in this parameter, which in turn may be triggered either by climatic or technological change. Given the focus in this study on the case of small pelagic fish, the analysis considers positive but small values for the catch‐to‐stock parameter.