Harry S. Robertson

Entropy, information theory, and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville functionρ for the infinite chain is reduced by integrating over the “outside” variables to a functionρN of the variables of theN-particle segment that is the thermodynamic system. The reduced Liouville functionρN which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta att = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms ofρN measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As ¦t ¦ → ∞, energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and ρN evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.

Ambipolar Diffusion in Radio‐Frequency‐Excited Magnetoplasmas

A theoretical analysis of ambipolar diffusion in electrodeless, weakly ionized, collision dominated, radio‐frequency‐excited magnetized plasmas reveals an essential dependence of the steady‐state density profile on the electron‐density dependence of the net volume‐ionization rate. In particular, if and only if the ionization rate is simply proportional to the electron density, the particle fluxes are ambipolar; otherwise, there exist divergenceless current cells for which the axial and radial current densities are not everywhere zero. Experimentally, it was found impossible to produce such a uniform plasma in argon, where the ionization process is two stage, and goes as n2, whereas the same method of excitation easily yielded a uniform cesium vapor plasma, for which the ionization process is proportional to n.

Collection of solar energy at specified output temperature

Abstract When a specified output temperature is required from a solar fluid-heating system we show that at constant irradiance there is no theoretical advantage in circulating the fluid through the collector more than once. Control of mass flow rate determines the output temperature. When the irradiance is not constant, however, efficiency can sometimes be imporved by multi-stage heating of the fluid. Theoretical calculations are based on the Hottel-Whillier equations. We examine several heating strategies, present a numerical comparison of two of these, and comment on collector design criteria for low-mass-flow-rate collectors.

Information Theory and the Approach to Equilibrium

After a general discussion of thermodynamic equilibrium and the information-theoretic formulation of equilibrium statistical mechanics, illustrative calculations are presented of the evolution to equilibrium of a finite segment (the system) of an infinite coupled harmonic-oscillator chain, most of which is regarded as the heat bath. The reduced Liouville function ρN is used to define the information-theoretic version of the Gibbs entropy as SN = −kB  ∫  ρN ln(hρNN)dΓN. This entropy evolves to a proper equilibrium value as |t| → ∞ from time-reversible dynamics, because ρN spreads from an initially sharp distribution to a diffuse one characteristic of the heat bath in equilibrium. The approach is regarded as generally valid, in principle, although the procedure is most easily carried out in the treatment of coupled harmonic-oscillator systems.

Alkali Vapor Plasmas

Alkali vapor positive colunms, using cesium, potassium, and rubidium, were examined at a current range of 0 to 300 ma and pressures of approximately 0.01 te 2.0 mm Hg. At the lowest possible pressures and currents of 1 ma or less, very faint standing striations were sometimes seen with spacings of approximately 1 ma. At higher currents and pressures, these striations were not presert. Cesium seems to be the most convenient alkali to use since it reaches an operable vapor pressure at a lower temperature than the others. (B.O.G.)

CPCs with segmented absorbers

One of the most promising means of improving the performance of solar thermal collectors is to reduce the energy lost by the hot absorber. One way to do this, not currently part of the technology, is to recognize that since the absorber is usually not irradiated uniformly, it is therefore possible to construct an absorber of thermally isolated segments, circulate the fluid in sequence from low to high irradiance segments, and reduce loss by improving effective concentration. This procedure works even for ideal concentrators, without violating Winstons theorem. Two equivalent CPC collectors with single and segmented absorber were constructed and compared under actual operating conditions. The results showed that the daily thermal efficiency of the collector with segmented absorber is higher (about 13%) than that of the collector with nonsegmented absorber.

Approach to equilibrium of coupled harmonic oscillator systems. II

The approach to equilibrium of a finite segment of an infinite chain of harmonically coupled masses is studied in several variations. The chain is taken as completely free, or it is bound atx0=0; ordinary coordinates and momenta or Schrödinger variables are used to treat the dynamics; and the inital distribution of heat-bath variables is chosen to be canonical or noncanonical. Equipartition of energy is found in all cases. Brownian drifts are obtained for the free chain with ordinary variables, but when this is excluded, the equilibrium entropy is found to be canonical and extensive when the initial heat bath is canonical, but less than canonical and slightly nonextensive when the initial heat bath is noncanonical. The modifications of the entropy do not contribute to the heat capacity of the system.

An inexpensive hybrid video imaging system

The core components of a low‐cost hybrid imaging system are an 80386‐based computer, a video digitizer board (frame grabber), and a consumer video camera/recorder. With the addition of a rotating mirror (or chopper), optical filters, and software from the Internet, this system becomes a powerful and versatile diagnostic tool. Although the system is bound by the NTSC or PAL video standards, an understanding of these limitations allows the development of techniques that fully utilize its characteristic spatiotemporal resolutions. The operation of the system is demonstrated via data acquisition from an ac neon Geissler tube.

ONSET OF THE HELICAL INSTABILITY IN WEAKLY IONIZED DISCHARGE PLASMAS.

The helical instability in a discharge plasma has been variously reported, sometimes as having an explosive onset, sometimes as reversible. Experiments in cesium are plasmas show that the onset is always explosive provided the ambient noise level is low enough; hysteresis is normally present; and plasma parameters change discontinuously at the onset and extinguishment of the instability. In the presence of noise, however, the onset is reversible and continuous.

Green's function solution of the time-dependent solar collector problem

Abstract The Greens function for the time-dependent solar collector problem is derived and examined. It is found that the influence of a point source of irradiance propagates with the speed of the heat transfer fluid. This result contradicts some earlier work. An analytical expression is derived for the output fluid temperature in terms of the irradiance as a function of position and time. Fluctuations in ambient and input fluid temperatures are also considered.