Principles of equilibrium statistical mechanics revisited: The idea of vortex energy

We show that the law of energy conservation with the fact of matter stability imply the existence of energy complementary to that given by the function of states of interacting systems and treated, with the environment, the function of states of interacting extended systems. The complementary energy, we called it vortex, is integral, not quantized, and causes trends contrary to that prescribed by equilibrium statistical mechanics. We formulate its principles and theorems, and question traditional insights in thermodynamics, entropy law, phase transitions, persistent currents, Brownian motion.Comment: 8 pages. Refined the title and wording, corrected typos, added the acknowledgement. arXiv admin note: substantial text overlap with arXiv:1109.2605; refined the title, clarified some phrases, refined concluding remarks, results unchange

On two conjectures concerning convex curves

We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex curve in RP^3 has 'degree' 4 and ii) construct an example of 4 tangent lines to a convex curve in RP^3 such that no real line intersects all four of them.Comment: AMSTEX, 15 pages, 3 eps pictures. to appear in Int. J. Mat

On moments of a polytope

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.Comment: 28 pages, 3 figure

Photoassociation adiabatic passage of ultracold Rb atoms to form ultracold Rb_2 molecules

We theoretically explore photoassociation by Adiabatic Passage of two colliding cold ^{85}Rb atoms in an atomic trap to form an ultracold Rb_2 molecule. We consider the incoherent thermal nature of the scattering process in a trap and show that coherent manipulations of the atomic ensemble, such as adiabatic passage, are feasible if performed within the coherence time window dictated by the temperature, which is relatively long for cold atoms. We show that a sequence of ~2*10^7 pulses of moderate intensities, each lasting ~750 ns, can photoassociate a large fraction of the atomic ensemble at temperature of 100 microkelvin and density of 10^{11} atoms/cm^3. Use of multiple pulse sequences makes it possible to populate the ground vibrational state. Employing spontaneous decay from a selected excited state, one can accumulate the molecules in a narrow distribution of vibrational states in the ground electronic potential. Alternatively, by removing the created molecules from the beam path between pulse sets, one can create a low-density ensemble of molecules in their ground ro-vibrational state.Comment: RevTex, 23 pages, 9 figure