Exact General Solutions to Extraordinary N-body Problems

We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If the potential is supplemented by any function that scales as the inverse square of the radius there is still no damping of the fundamental breathing mode. For such systems a remarkable new statistical equilibrium is found for the other coordinates and momenta, which persists even as the radius changes continually.Comment: 15 pages, LaTeX. Accepted for publication in Proc. Roy. Soc.

Relaxation to a Perpetually Pulsating Equilibrium

Paper in honour of Freeman Dyson on the occasion of his 80th birthday. Normal N-body systems relax to equilibrium distributions in which classical kinetic energy components are 1/2 kT, but, when inter-particle forces are an inverse cubic repulsion together with a linear (simple harmonic) attraction, the system pulsates for ever. In spite of this pulsation in scale, r(t), other degrees of freedom relax to an ever-changing Maxwellian distribution. With a new time, tau, defined so that r^2d/dt =d/d tau it is shown that the remaining degrees of freedom evolve with an unchanging reduced Hamiltonian. The distribution predicted by equilibrium statistical mechanics applied to the reduced Hamiltonian is an ever-pulsating Maxwellian in which the temperature pulsates like r^-2. Numerical simulation with 1000 particles demonstrate a rapid relaxation to this pulsating equilibrium.Comment: 9 pages including 4 figure

Exact Quantum Solutions of Extraordinary N-body Problems

The wave functions of Boson and Fermion gases are known even when the particles have harmonic interactions. Here we generalise these results by solving exactly the N-body Schrodinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in 3 dimensions. For the harmonic case that function is linear in r^2. Explicit N-body solutions are given when U(r) = -2M \hbar^{-2} V(r) = \zeta r^{-1} - \zeta_2 r^{-2}. Here M is the sum of the masses and r^2 = 1/2 M^{-2} Sigma Sigma m_I m_J ({\bf x}_I - {\bf x}_J)^2. For general U(r) the solution is given in terms of the one or two body problem with potential U(r) in 3 dimensions. The degeneracies of the levels are derived for distinguishable particles, for Bosons of spin zero and for spin 1/2 Fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the Fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N-body wave function. The N-body forces involved in these extraordinary N-body problems are not the usual sums of two body interactions, but nor are forces between quarks or molecules. Bose-Einstein condensation of particles in 3 dimensions interacting via these strange potentials can be treated by this method.Comment: 24 pages, Latex. Accepted for publication in Proceedings of the Royal Societ

Coupling/decoupling between translational and rotational dynamics in a supercooled molecular liquid

We use molecular dynamics computer simulations to investigate the coupling/decoupling between translational and rotational dynamics in a glass-forming liquid of dumbbells. This is done via a careful analysis of the $\alpha$-relaxation time $\tau_{q^{*}}^{\rm C}$ of the incoherent center-of-mass density correlator at the structure factor peak, the $\alpha$-relaxation time $\tau_{2}$ of the reorientational correlator, and the translational ($D_{t}$) and rotational ($D_{r}$) diffusion constants. We find that the coupling between the relaxation times $\tau_{q^{*}}^{\rm C}$ and $\tau_{2}$ increases with decreasing temperature $T$, whereas the coupling decreases between the diffusivities $D_{t}$ and $D_{r}$. In addition, the $T$-dependence of $D_{t}$ decouples from that of $1/\tau_{2}$, which is consistent with previous experiments and has been interpreted as a signature of the "translation-rotation decoupling." We trace back these apparently contradicting observations to the dynamical heterogeneities in the system. We show that the decreasing coupling in the diffusivities $D_{t}$ and $D_{r}$ is only apparent due to the inadequacy of the concept of the rotational diffusion constant for describing the reorientational dynamics in the supercooled state. We also argue that the coupling between $\tau_{q^{*}}^{\rm C}$ and $\tau_{2}$ and the decoupling between $D_{t}$ and $1/\tau_{2}$, both of which strengthen upon cooling, can be consistently understood in terms of the growing dynamic length scale.Comment: revised manuscript, to appear in Phys. Rev. Let

17O Nuclear Magnetic Resonance Chemical Shift in Oxyhaemoglobin

The 170 chemical shift of oxygen in oxyhaemoglobin is calculated for two models, one corresponding to the Griffith structure and the other to the Pauling structure. In both cases the oxygen resonance is predicted to be several thousand ppm to low field of the oxygen resonance in water. The shift between the oxygen nuclei in the Pauling structure is predicted to be at least one thousand ppm. This large deshielding arises from the local environment of the oxygen molecule and depends critically on the splitting of the degenerate it orbitals on complexing

17O Nuclear Magnetic Resonance Chemical Shift in Oxyhaemoglobin

The 170 chemical shift of oxygen in oxyhaemoglobin is calculated for two models, one corresponding to the Griffith structure and the other to the Pauling structure. In both cases the oxygen resonance is predicted to be several thousand ppm to low field of the oxygen resonance in water. The shift between the oxygen nuclei in the Pauling structure is predicted to be at least one thousand ppm. This large deshielding arises from the local environment of the oxygen molecule and depends critically on the splitting of the degenerate it orbitals on complexing

Do Rotations Beyond the Cosmological Horizon Affect the Local Inertial Frame?

If perturbations beyond the horizon have the velocities prescribed everywhere then the dragging of inertial frames near the origin is suppressed by an exponential factor. However if perturbations are prescribed in terms of their angular momenta there is no such suppression. We resolve this paradox and in doing so give new explicit results on the dragging of inertial frames in closed, flat and open universe with and without a cosmological constant.Comment: 12 page

Relativistic Poynting Jets from Accretion Disks

A model is developed for relativistic Poynting jets from the inner region of a disk around a rotating black hole. The disk is initially threaded by a dipole-like magnetic field. The model is derived from the special relativistic equation for a force-free electromagnetic field. The ``head'' of the Poynting jet is found to propagate outward with a velocity which may be relativistic. The Lorentz factor of the head (Gamma) is found to be dependent on the magnetic field strength close to the black hole, B_0, the density of the external medium n_ext, and on the ratio R=r_0/r_g >1, where r_g is the gravitational radius of the black hole, and r_0 is the radius of the O-point of the initial dipole field threading the disk. For conditions pertinent to an active galactic nuclei, Gamma is approximately equal to 8 (10/R)^(1/3) (B_0/10^3 Gauss)^(1/3) (1/cm^3/n_ext)^(1/6). This model offers an explanation for the observed Lorentz factors which are of the order of 10 for the parsec-scale radio jets measured with very long baseline interferometry.Comment: 4 pages, 1 figur

From infall to rotation around young stellar objects: A transitional phase with a 2000 AU radius contracting disk?

Evidence for a transitional stage in the formation of a low-mass star is reported, intermediate between the fully embedded and the T Tauri phases. Millimeter aperture synthesis observations in the HCO+ J=1-0 and 3-2, HCN 1-0, 13CO 1-0, and C18O 1-0 transitions reveal distinctly different velocity fields around two embedded, low-mass young stellar objects. The 0.6 M(sun) of material around TMC 1 (IRAS 04381+2517) closely follows inside-out collapse in the presence of a small amount of rotation (~3 km/s/pc), while L1489 IRS (IRAS 04016+2610) is surrounded by a 2000 AU radius, flared disk containing 0.02 M(sun). This disk shows Keplerian rotation around a ~0.65 M(sun) star and infall at 1.3 (r/100 AU)^-0.5 km/s, or, equivalently, sub-Keplerian motions around a central object between 0.65 and 1.4 M(sun). Its density is characterized by a radial power law and an exponential vertical scale height. The different relative importance of infall and rotation around these two objects suggests that rotationally supported structures grow from collapsing envelopes over a few times 10^5 yr to sizes of a few thousand AU, and then decrease over a few times 10^4 yr to several hundred AU typical for T Tauri disks. In this scenario, L1489 IRS represents a transitional phase between embedded YSOs and T Tauri stars with disks. The expected duration of this phase of ~5% of the embedded stage is consistent with the current lack of other known objects like L1489 IRS. Alternative explanations cannot explain L1489 IRS's large disk, such as formation from a cloud core with an unusually large velocity gradient or a binary companion that prevents mass accretion onto small scales. It follows that the transfer and dissipation of angular momentum is key to understanding the formation of disks from infalling envelopes.Comment: Accepted ApJ. 33 pages, including 10 B/W figures and 1 color figure. Uses AASTe

Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems

We study, using both theory and molecular dynamics simulations, the relaxation dynamics of a microcanonical two dimensional self-gravitating system. After a sufficiently large time, a gravitational cluster of N particles relaxes to the Maxwell-Boltzmann distribution. The time to reach the thermodynamic equilibrium, however, scales with the number of particles. In the thermodynamic limit, $N\to\infty$ at fixed total mass, equilibrium state is never reached and the system becomes trapped in a non-ergodic stationary state. An analytical theory is presented which allows us to quantitatively described this final stationary state, without any adjustable parameters