More is the Same; Phase Transitions and Mean Field Theories

This paper looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.Comment: 25 pages, 6 figure

Symmetry-breaking Effects for Polariton Condensates in Double-Well Potentials

We study the existence, stability, and dynamics of symmetric and anti-symmetric states of quasi-one-dimensional polariton condensates in double-well potentials, in the presence of nonresonant pumping and nonlinear damping. Some prototypical features of the system, such as the bifurcation of asymmetric solutions, are similar to the Hamiltonian analog of the double-well system considered in the realm of atomic condensates. Nevertheless, there are also some nontrivial differences including, e.g., the unstable nature of both the parent and the daughter branch emerging in the relevant pitchfork bifurcation for slightly larger values of atom numbers. Another interesting feature that does not appear in the atomic condensate case is that the bifurcation for attractive interactions is slightly sub-critical instead of supercritical. These conclusions of the bifurcation analysis are corroborated by direct numerical simulations examining the dynamics of the system in the unstable regime.MICINN (Spain) project FIS2008- 0484

Pulsed TeV Gamma Rays detected from Hercules X-l during X-ray source eclipse

Distinct 0.808-Hz pulsations were detected from Her X-1 in June 1985 using the atmospheric Cerenkov technique with the Whipple Observatory 10 m gamma-ray telescope, which had an effective gamma-ray energy threshold of about 600 GeV. The 2-hr observation spanned the transition into X-ray eclipse, at which the X-ray source Her X-1 is first occulted by the companion star HZ Herculis at orbital phase about 0.93 in the binary system. The pulsation persists for 1 hr or more after the beginning of the 6-hr eclipse, and it is concluded that the gamma-ray production site is in this case not coincident with the X-ray source

Observation of TeV Gamma-rays from the Crab Nebula using the Atmospheric Cherenkov Imaging Technique

The Whipple Observatory 10-m reflector, operating as a 37-pixel camera, has been used to observe the Crab Nebula in TeV gamma rays. By selecting gamma-ray images based on their predicted properties, more than 98 percent of the background is rejected; a detection is reported at the 9.0-sigma level, corresponding to a flux of 1.8 x 10 to the -11th photons sq cm/s above 0.7 TeV (with a factor of 1.5 uncertainty in both flux and energy). Less than 25 percent of the observed flux is pulsed at the period of PSR 0531. There is no evidence for variability on time scales from months to years. Although continuum emission from the pulsar cannot be ruled out, it seems more likely that the observed flux comes from the hard Compton synchrotron spectrum of the nebula