Gerald L. Jones

Interaction of Elastic Waves in an Isotropic Solid

Nonlinear elasticity theory is used to investigate the scattering of two intersecting, plane, elastic waves in a homogeneous, isotropic medium. A criterion for the occurrence of a strong scattered wave is derived. The criterion is formulated as a relationship between the first‐order elastic constants of the material, the angle between the intersecting wave vectors, and the ratio of wave frequencies. The exact formulation also depends on the type of intersecting waves, i.e., longitudinal or transverse. The amplitude of the scattered wave is found to be proportional to the volume of interaction and dependent on the third‐order elastic constants of the material. Numerical results are given for wave scattering polystyrene.

Simulating convergent extension by way of anisotropic differential adhesion

Simulations using the Extended Potts Model suggest that anisotropic differential adhesion can account for convergent extension, as observed during embryonic development of the frog Xenopus laevis for example. During gastrulation in these frogs, convergent extension produces longitudinal tissue growth from latitudinal elongation and migration of aligned constituent cells. The Extended Potts Model employs clustered points on a grid to represent subdivided cells with probabilistic displacement of cell boundaries such that small changes in energy drive gradual tissue development. For modeling convergent extension, simulations include anisotropic differential adhesion: the degree of attachment between adjacent elongated cells depends on their relative orientation. Without considering additional mechanisms, simulations based on anisotropic differential adhesion reproduce the hallmark stages of convergent extension in the correct sequence, with random fluctuations as sufficient impetus for cell reorganization.

Symmetries of differential equations

The use of a symmetry to reduce the order of an nth-order differential equation is treated by considering the symmetry of an associated vector field. A particular choice of associated vector field leads to the usual extension of the Lie symmetry method. The possibility of other choices leads to a powerful generalisation. An algebraic classification of transformations arises naturally from the theory. It is shown to be equivalent to the geometric classification of transformations as contact and non-contact.

A density functional-variational treatment of the hard sphere transition

A density functional-variational version of the Ramakrishnan-Yussouff theory of freezing is used to reconsider the problem of the hard sphere transition. This calculation differs from previous ones in that the solid density and the lattice constant are included as independent variational parameters. Besides giving an unambiguous method for determining the lattice constant of the solid this method allows the computation of the average density of defects in the solid. In addition, we use real, rather than Fourier, space techniques in solving the resulting equations. We argue that real space techniques are numerically more accurate for the narrow distributions found by these methods. Our results for the densities of the coexisting solid and liquid phases are very close to those given by molecular dynamics studies. The width of the solid density peaks is too small as is the case with previous calculations. The average density of defects has the correct sign but is much too large (ρD ⋍ -0·1) for a realistic solid.

Complex Temperatures and Phase Transitions

The thermodynamic limit is considered for complex temperatures, and a picture of a phase transition, similar to the Yang‐Lee picture, is proposed. For certain cases a representation of the partition function as an infinite product is obtained. Some simple models are considered.

Critical angle for reflection at a liquid–solid interface in single crystals

Recent investigations have utilized the measurement of the critical angle for reflection from a liquid‐solid interface for determination of the elastic constants of the solid. For anisotropic media, this technique is appropriate only for certain special cases of the incident plane and reflecting surface. We discuss here the general condition for the critical angel in anisotropic media and show that for some planes in quartz, major errors may arise if one employs the usual statement of Snell’s law for definition of the critical angle.Subject Classification: [43]20.30; [43]35.26.

Irreversibility, entropy production, and thermal efficiency

The relationships between the entropy production per cycle and the thermal efficiency are investigated for a class of irreversible cyclic processes. Examples are given that pinpoint specific sources of irreversibility and their thermodynamic consequences. It is found that an increase (decrease) in an irreversible cycle’s thermal efficiency does not necessarily lead to a decrease (increase) in its entropy production even if the work done per cycle is held constant. Only for the case of a reversible Carnot cycle is it guaranteed that a change (negative for this case) in the efficiency is met by an entropy production change of opposite algebraic sign. Sufficiency conditions are found for which the entropy production and the efficiency η are inversely related for more general cyclic processes. For a given set of heat reservoirs and specified values of the work output W, the absolute minimum and maximum entropy productions are determined and are shown to be monotonically decreasing functions of η for fixed W. ...

Elastic constants in density-functional theory

A formalism for computing the elastic constants of a macroscopically uniformly strained solid within canonical density functional theory is developed, using a rather general characterization of the strained state as a constrained minimum of the density functional. This permits microscopic parameters characterizing the solid density to respond freely to the strain. Interesting extremal properties of some elastic constants are found. In particular it is shown that the exact value of any of the diagonal elastic constants is a lower bound to any variational approximation of them. The role of defects in the solid and the effects of the choice of the variational class of solid densities is discussed in some detail. The method is applied to a calculation of the three independent elastic constants of a hard sphere solid using the Ramakrishnan-Yussouf density functional. Numerical results are given and discussed.

Numerical and analytical studies of the long‐ranged solutions of the Yvon–Born–Green equation

In this paper we give a detailed description of our numerical studies of the critical behavior of the Yvon–Born–Green (YBG) equation in various dimensions and the conclusions which we believe these studies warrent. A central issue in these studies is the relationship between the convergence of the numerical methods used to solve the YBG equation, the long‐ranged nature of the solutions, and possible bifurcations in the solution structure of the equation. An understanding of these relationships is essential to the construction of reliable numerical solutions in the critical region of the YBG equation and is likely to be equally important for a large class of nonlinear integral equations with short‐ranged kernels. We give some analytic models and numerical techniques which are effective in exploring these questions. From these studies the following conclusions concerning the critical behavior of the YBG equation are drawn: (1) For d=3 there is a region of density and temperature in which the correlations ar...

Density functional theory of homogeneous states

We consider the weighted density approximation of the density functional description of systems in thermal equilibrium. We show that knowledge of the intermolecular potential puts constraints on the theory which take the form of a small number of nonlinear integral equations of unusual type. We show that for homogeneous states of systems with purely repulsive potentials these equations are sufficient to determine the free energy functional completely, at least at densities where the virial expansion of the theory converges. We have not been able to find either analytic or numerical solutions of these equations at arbitrary densities. We have solved the equations in the density expansion to the lowest order in which it disagrees with the exact virial expansion of the system. This extended weighted density approximation (EWDA) gives the exact virial expansion of the pressure to third order and the pair distribution function to first order in the density, as do the other standard integral equation theories. In the next order the EWDA is not exact, but it gives very good numerical results for the pressure and pair distribution and for both hard and soft repulsive potentials. In addition, the difference between the pressure and the compressibility equations of state is numerically very small, indicating a high degree of thermodynamic consistency. Were these properties to persist at higher densities, the EWDA would be clearly preferred to the usual integral approach, at least for repulsive potentials. For potentials with an attractive part the EWDA becomes singular at low temperatures in a way that suggests there is a structural flaw in the assumed form of the free energy functional.