Volume 18, Issue 7, July 1977
Index of content:

Representation of the entropy functional for a grand canonical ensemble in classical statistical mechanics
View Description Hide DescriptionA representation theorem for the entropy functional of a system of an arbitrary but finite number of particles is proved. This theorem is a generalization of the main result of a previous paper of ours [J. Math. Phys. 16, 1453 (1975)], which gives a characterization of the entropy functional on the set of all probability densitiesf on R _{ n } s.t. flogf is integrable. As may be expected when n is a random variable, the expression for entropy consists of two parts; one which arises from the ignorance about n and another which is the average, over n, of the conditional entropy given the number of particles. As in the above‐mentioned paper, the expression includes the term corresponding to chemical reactions (with n replaced by the average number of particles) and the continuous analog of the Hartley entropy. It is conjectured that this last term might be of some significance in physics.

Stationary axially‐symmetric solutions of Einstein–Maxwell‐massless scalar field equations
View Description Hide DescriptionA procedure is presented which enables one to construct solutions to the stationary axially‐symmetric gravitational field coupled to massless scalar and Maxwell fields.

Fluid space–times including electromagnetic fields admitting symmetry mappings belonging to the family of contracted Ricci collineations
View Description Hide DescriptionThis paper investigates certain symmetry mappings belonging to the family of contracted Ricci collineations (FCRC) (satisfying g ^{ i j }LR _{ i j }=0) admitted by the general fluid space–times, including electromagnetic fields, that were classified and studied earlier by Stewart and Ellis (1967). Many of the results obtained are applicable to the perfect fluid models treated by Wainwright (1970) and Krasiński (1974,1975). A major part of this paper represents an extension of previous investigations (1976) of the Robertson–Walker metrics and more general perfect fluid space–times that admit FCRC symmetry mappings and concomitant conservation expressions. More specifically, these results provide a number of theorems relating to the more general fluid space–times that admit FCRC symmetry mappings (including both timelike and spacelike symmetry vectors) that lead to conservation expressions and specific conditions on the metric tensors for the given particular cases of these space–times. Also the form of the symmetry mappings induced on the electromagnetic fields (when they are present) is investigated in the case where specific symmetry mappings on the metric tensor are admitted. In particular, the results of Wainwright and Yaremovicz (1976) relating to homothetic motions admitted by given space–times, corresponding to perfect fluids including electromagnetic fields, are largely embraced by the more general results obtained in this paper.

Energy–momentum tensors in the theory of electromagnetic fields admitting electric and magnetic charge distributions
View Description Hide DescriptionWhen the field tensor of an electromagnetic field admitting both electric and magnetic charge distributions is expressed in terms of a Clebsch representation, the extended Maxwell equations in the presence of a given gravitational field are derivable from an invariant variational principle in which the Clebsch potentials play the role usually assumed by the classical 4‐potentials. The corresponding Lagrange density gives rise in a unique manner to a symmetric tensor density T ^{ h j }, which displays some of the properties normally associated with the energy–momentum tensor density of the electromagnetic field. However, this interpretation may be in conflict with the generally accepted expression for the modified Lorentz force. Accordingly an alternative energy–momentum tensor density ϑ^{ h j } is derived which does not suffer from this drawback. However, when a generalized variational principle for the simultaneous determination of the behavior of both the electromagnetic and the dynamical gravitational fields is introduced, the resulting Euler–Lagrange equations give rise to extended Einstein–Maxwell equations which involve the density T ^{ h j }. On the other hand, the alternative Einstein–Maxwell equations, obtained by the replacement of T ^{ h j } by ϑ^{ h j }, are not derivable from a variational principle. The solutions of the two Einstein–Maxwell equations, for the case of a spherically symmetric metric and static electromagnetic field, predict distinctly different effects of the magnetic charges on the gravitational field.

On the macroscopic equivalence of descriptions of fluctuations for chemical reactions
View Description Hide DescriptionThree descriptions of spontaneous fluctuations in macroscopic systems have been proposed: One uses generalized Fokker–Planck equations and treats fluctuations as a stochastic diffusion process; another uses a connection between fluctuations and dissipation and generalizes the Langevin method; the third is the master equationtheory which treats fluctuations as arising from a birth and death process. For a variety of systems it is known that the master equationtheory is identical to the fluctuation–dissipation theory in the macroscopic limit. For chemical reactions it is shown here that the appropriate diffusion process also becomes identical with the fluctuation–dissipation theory in the macroscopic limit.

Dispersive properties and observables at infinity for classical KMS systems
View Description Hide DescriptionFor infinite classical dynamical systems, satisfying the KMS condition, relations between asymptotic dispersive and cluster properties are proved. The local structure of the algebra of observables is explicitly characterized by the Poisson bracket commutant, and it is proved that the algebra of observables at infinity are constants of the motion.

Quantum system subject to random pulses
View Description Hide DescriptionWe ’’solve’’ the Schrödinger equation for a system subject to random pulses, and show how to compute ensemble averages of observables or their Laplace transforms.

Approximate symmetry groups of inhomogeneous metrics
View Description Hide DescriptionA useful step toward understanding inhomogeneous space–times would be to classify them, perhaps in a fashion analogous to that used for spatially homogeneous space–times. To that end, a technique for determining an approximate simply‐transitive three‐parameter symmetry group of a three‐dimensional positive‐definite Riemannian metric is developed. The technique employs a variational principle to find a set of three orthonormal vectors whose commutation coefficients are as close as possible to a set of structure constants. The Bianchi classification of the structure constants of three‐parameter groups is then used to classify these inhomogeneous metrics. Application of this technique to perturbed homogeneous metrics is discussed in detail. We find that only four types of symmetry groups can be considered generic in the space of all perturbed homogeneous metrics.

Localized solutions of a nonlinear scalar field with a scalar potential
View Description Hide DescriptionThe exact localized solutions for a nonlinear scalar field with a scalar potential are studied. In particular, we compare the stability of the above solutions and those obtained by Rosen in absence of the scalar potential.

Stationary localized solutions in nonlinear classical fields
View Description Hide DescriptionThe existence of stationary localized solutions for the Dirac field interacting with Maxwell and pseudoscalar fields is studied.

Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum
View Description Hide DescriptionFredholm theory is applied to the Lippmann–Schwinger equation for noncentral potentials. For a specified wide class of potentials it is proved that the modified Fredholm determinant cannot vanish for real k≠0. The point k=0 is examined and the analog of the distinction between zero‐energy bound states and zero‐energy resonances for central potentials is found. A generalized Levinson theorem is proved.

Group contraction in a fiber bundle with Cartan connection
View Description Hide DescriptionA contraction of the structural group with respect to the stability subgroup is performed in a fiber bundle with Cartan connection. The relation of the connections in the original and in the contracted bundle is examined. As an example interesting for physics the contraction of the SO(4,1) de Sitter bundle over space–time to the affine tangent bundle over space–time is discussed with the latter bundle possessing the Poincaré group as structural group.

Covariant inverse problem of the calculus of variations
View Description Hide DescriptionThe solution of the covariant inverse problem of the calculus of variations (that of finding a Lagrangian for a given set of dynamical equations) is presented as a generalization flat space formalism of Atherton and Homsy. Known Lagrangians such as those for the complkex scalar field, vector gauge fields, and Einstein’s equations are used as examples. Additional insight into the formalism is provided by new examples that include a two‐tensor theory, a method for obtaining conservation laws directly from dynamical equations, and a Hamiltonian formulation for higher order, nonlinear, differential equations.

Static stars : Some mathematical curiosities
View Description Hide DescriptionThe equations of structure of static Newtonian and general relativistic stars are investigated. By using Lie grouptheory, it is shown that, in each case, the condition that there should exist a simple ’’homologous’’ family of similar solutions necessitates precisely those equations of state for the stellar matter that are usually invoked by means of extraneous physical arguments. In the relativistic case, a diagram which depicts these families is drawn, using the qualitative theory of differential equations. This vividly exhibits the nature of the general soutions, and the exceptional character of the Misner–Zapolsky solution. This diagram is contrasted with similar ones obtained by Chandrasekhar in the Newtonian case.

Complete integrability conditions of the Einstein–Petrov equations, type I
View Description Hide DescriptionThe post‐Bianchi equations, defined as the integrability conditions of the Bianchi equations, are explicitly stated for the algebraically general (type I) Einstein–Petrov vacuum equations. A computer analysis of these equations has shown that they constitute a completely integrable set. Hence all conditions imposed by the Einstein equations of this type on the derivatives of the dependent variables are now known.

Orbits of the rotation group on the density matrices of spin‐1 particles
View Description Hide DescriptionThe SO(3) orbits of the spin‐one mixed states, contained in each SU(3) orbit are shown to be characterized by the s q u a r e s (defined by means of a symmetric bilinear form) of the eigenvectors of the density matrices. The orbits of matrices diagonalizable by a rotation in a spherical basis are deduced, and quadrupole matrices are considered as a special case.

Generating functions for the eigenvalues of the Casimir operators of the orthogonal and symplectic groups
View Description Hide DescriptionBy constructing the appropriate generating functions, the eigenvalues of the Casimir operators for the orthogonal and the symplectic groups are expressed in terms of p o w e r s u m s which are formally the same for the O(2n), Sp(2n), O(2n+1) groups as for the U(n) groups. The results for the O(2n), Sp(2n), and the O(2n+1) groups are written as the corresponding results for the U(n) groups plus very simple c o r r e c t i o n t e r m s. This approach unifies the treatment of the problem for the semisimple Lie groups. Explicit evaluation of the eigenvalues of the Casimir operators becomes very simple.

Hydrostatic density distribution of fluids in steep field gradients and near critical points
View Description Hide DescriptionA new perturbation expansion for the hydrostatic density distribution of a fluid in a steep field gradient is presented. It is a power series in the scale γ of an external potential ψ (γx), and is asymptotically valid as γ→0, although convergence is not proved. Significant corrections to the conventional hydrostaticequations are found near the critical point of a fluid, even in the gravitational field.

The class of continuous timelike curves determines the topology of spacetime
View Description Hide DescriptionThe title assertion is proven, and two corollaries are established. First, the topology of every past and future distinguishing spacetime is determined by its causal structure. Second, in e v e r yspacetime the path topology of Hawking, King, and McCarthy codes topological, differential, and conformal structure.

Equivalence transformations for nonlinear evolution equations
View Description Hide DescriptionA systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scatteringtheory) are all deduced from a single general equivalence condition.