Complex Time Evolution Research Articles

Density relaxation in a vibrated granular material.

We report measurements of the density of a vibrated granular material as a function of time. The material studied consists of monodisperse spherical glass particles confined to a long, thin cylindrical tube. Vibrations cause the pile to evolve from a low density initial configuration toward a steady state with a final density that depends on the intensity of the vibrations. We find a complex time evolution that is incompatible with a single exponential relaxation. There appears to be a characteristic value of the acceleration that separates two regimes of packing behavior. The results are compared to current theories of compaction.

Accurate quantum mechanical calculation of thermally averaged reaction rate constants for polyatomic systems

A new approach to the calculation of quantum mechanical Boltzmann averaged reaction rate constants for polyatomic systems is described. The rate constant is obtained by integrating a set of coupled first order temperature-dependent differential equations, the number of which grows linearly with the size of the system. This is accomplished by (i) representing the complex time evolution operator in mixed position and operator form and (ii) introducing two-body correlations in the conventional time-dependent self-consistent field approximation, as suggested recently [Chem. Phys. Lett. 169, 541 (1990)]. The method is accurate and numerically stable; it is therefore expected to find considerable utility in the study of gas phase bimolecular reactions with the reaction path Hamiltonian formalism, as well as in the calculation of rate constants for reactive processes in condensed media.

Partial averaging approach to Fourier coefficient path integration

The recently introduced method of partial averaging is developed into a general formalism for computing simple Cartesian path integrals. Examples of its application to both harmonic and anharmonic systems are given. For harmonic systems, where analytical results can be derived, both imaginary and complex time evolution is discussed. For two representative anharmonic systems, Monte Carlo path integral simulations of the imaginary time propagator (statistical density matrix) are presented. Connections with other Cartesian path integral techniques are stressed.

A spatiotemporal model of civilization

A mathematical model of an urban civilization is presented. Both time- and space-dependence of the four main variables, population, rural commodities, urban commodities, and state power, are described by coupled partial differential equations. The interaction of the variables produces a complex time evolution of the whole living system. Numerical solution of the equations reveals a number of realistic predictions, including class stratification and destruction of states by conquest. The limitations of the very simple ideas used in constructing the model suggest avenues of further research.

Electron spin polarization during the photolysis of pivalophenone (phenyl t-butyl ketone)

Solution-phase photolysis of pivalophenone (phenyl t-butyl ketone) gives radicals with a spin polarization showing complex time-evolution. It is suggested that the polarization may be due to a process involving two rates of separation of the initial radical pair. One is separation from a triplet pair; the other, slower departure is from a singlet pair generated from the initial triplet pair by the strong perturbations arising from the t-butyl hyperfine interactions.