We present a general discussion of the isoenthalpic–isostress molecular dynamics theories of Andersen and Parrinello–Rahman. The Parrinello–Rahman theory is shown to be applicable to the case of nonlinear elasticity if the reference state of zero strain is taken to be the state under zero stress; this brings the theory into accord with the thermodynamics of anisotropic solids for arbitrary values of the strain. For the isoenthalpic–isostress ensemble there is a microcanonical counterpart for which we present fluctuation formulas involving the constant strain specific heat, temperature coefficients of thermodynamic tension, and stiffness coefficients. The use of these various ensembles for the molecular dynamic study of polymorphic transitions in crystals is discussed.
Exact coherent states for the time-dependent harmonic oscillator are constructed.These new coherent states have most, but not all, of the properties of the coherent states for the time-independent oscillator. For example, these coherent states give the exact classical Inotion, but they are not minimum-uncertainty states.
A previously discussed variational principle for a perfect fluid in general relativity was restricted to irrotational, isentropic motions of the fluid. It is proven that these restrictions can be dropped, and the original variational principle can be generalized to general motions of the perfect fluid. The form of the basic Lagrangian density is unchanged by these generalizations. An Eulerian fluid description is used throughout. As a by-product of our variational principle, the 4-velocity is required to have the generalized Clebsch form.
We report several important additions to our original discussion of Ermakov systems. First, we show how to derive the Ermakov system from more general equations of motion. Second, we show that there is a general nonlinear superposition law for Ermakov systems. Also, we give explicit examples of the nonlinear superposition law. Finally, we point out that any ordinary differential equation can be included in many Ermakov systems.
We have recently discussed how the Parrinello–Rahman theory can be brought into accord with the theory of the elastic and thermodynamic behavior of anisotropic media. This involves the isoenthalpic–isotension ensemble of statistical mechanics. Nosé has developed a canonical ensemble form of molecular dynamics. We combine Nosé’s ideas with the Parrinello–Rahman theory to obtain a canonical form of molecular dynamics appropriate to the study of anisotropic media subjected to arbitrary external stress. We employ this isothermal–isotension ensemble in a study of a fcc→ close-packed structural phase transformation in a Lennard-Jones solid subjected to uniaxial compression. Our interpretation of the Nosé theory does not involve a scaling of the time variable. This latter fact leads to simplifications when studying the time dependence of quantities.
An Eulerian variational principle for a spinning fluid in the Einstein-Cartan metric-torsion theory is presented. The variational principle yields the complete set of field equations for the system. The symmetric energy-momentum tensor is a sum of a perfect-fluid term and a spin term.
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