Projection operator formalism and entropy

The entropy definition is deduced by means of (re)deriving the generalized non-linear Langevin equation using Zwanzig projector operator formalism. It is shown to be necessarily related to an invariant measure which, in classical mechanics, can always be taken to be the Liouville measure. It is not true that one is free to choose a ``relevant'' probability density independently as is done in other flavors of projection operator formalism. This observation induces an entropy expression which is valid also outside the thermodynamic limit and in far from equilibrium situations. The Zwanzig projection operator formalism therefore gives a deductive derivation of non-equilibrium, and equilibrium, thermodynamics. The entropy definition found is closely related to the (generalized) microcanonical Boltzmann-Planck definition but with some subtle differences. No ``shell thickness'' arguments are needed, nor desirable, for a rigorous definition. The entropy expression depends on the choice of macroscopic variables and does not exactly transform as a scalar quantity. The relation with expressions used in the GENERIC formalism are discussed

An integral turbulent kinetic energy analysis of free shear flows

Mixing of coaxial streams is analyzed by application of integral techniques. An integrated turbulent kinetic energy (TKE) equation is solved simultaneously with the integral equations for the mean flow. Normalized TKE profile shapes are obtained from incompressible jet and shear layer experiments and are assumed to be applicable to all free turbulent flows. The shear stress at the midpoint of the mixing zone is assumed to be directly proportional to the local TKE, and dissipation is treated with a generalization of the model developed for isotropic turbulence. Although the analysis was developed for ducted flows, constant-pressure flows were approximated with the duct much larger than the jet. The axisymmetric flows under consideration were predicted with reasonable accuracy. Fairly good results were also obtained for the fully developed two-dimensional shear layers, which were computed as thin layers at the boundary of a large circular jet

Atomic hydrogen maser active oscillator cavity and bulb design optimization

The performance characteristics and reliability of the active oscillator atomic hydrogen maser depend upon oscillation parameters which characterize the interaction region of the maser, the resonant cavity and atom storage bulb assembly. With particular attention to use of the cavity frequency switching servo (1) to reduce cavity pulling, it is important to maintain high oscillation level, high atomic beam flux utilization efficiency, small spin exchange parameter and high cavity quality factor. It is also desirable to have a small and rigid cavity and bulb structure and to minimize the cavity temperature sensitivity. Curves for a novel hydrogen maser cavity configuration which is partially loaded with a quartz dielectric cylinder and show the relationships between cavity length, cavity diameter, bulb size, dielectric thickness, cavity quality factor, filling factor and cavity frequency temperature coefficient are presented. The results are discussed in terms of improvement in maser performance resulting from particular design choices

Generalized (m,k)-Zipf law for fractional Brownian motion-like time series with or without effect of an additional linear trend

We have translated fractional Brownian motion (FBM) signals into a text based on two ''letters'', as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the $\zeta '$ exponent relating the word frequency and its rank on a log-log plot. We have studied the variation of the Zipf exponent(s) giving the relationship between the frequency of occurrence of words of length $m<8$ made of such two letters: $\zeta '$ is varying as a power law in terms of $m$. We have also searched how the $\zeta '$ exponent of the Zipf law is influenced by a linear trend and the resulting effect of its slope. We can distinguish finite size effects, and results depending whether the starting FBM is persistent or not, i.e. depending on the FBM Hurst exponent $H$. It seems then numerically proven that the Zipf exponent of a persistent signal is more influenced by the trend than that of an antipersistent signal. It appears that the conjectured law $\zeta ' = |2H-1|$ only holds near $H=0.5$. We have also introduced considerations based on the notion of a {\it time dependent Zipf law} along the signal.Comment: 24 pages, 12 figures; to appear in Int. J. Modern Phys

Measurement of spray combustion processes

A free jet configuration was chosen for measuring noncombusting spray fields and hydrocarbon-air spray flames in an effort to develop computational models of the dynamic interaction between droplets and the gas phase and to verify and refine numerical models of the entire spray combustion process. The development of a spray combustion facility is described including techniques for laser measurements in spray combustion environments and methods for data acquisition, processing, displaying, and interpretation