Michael Nauenberg

Phase transition in four-dimensional compact QED

Abstract The energy and the specific heat of the four-dimensional U(1) lattice gauge model is evaluated by Monte Carlo simulations on lattices of size L 4 , where L = 4, 5 and 6, evidence is presented for the occurence of a second-order phase transition. A finite size scaling analysis of our results gives the critical value of the coupling constant e 2 c = 0.995 and a correlation length exponent v ≈ 1 3 .

Intermittency in the presence of noise: A renormalization group formulation

Abstract A renormalization group (RG) formulation of the transition to chaotic behavior via intermittency in one-dimesional maps is presented. The known scaling behavior of the length of the laminar regions in the presence of external noise is obtained from the leading relevant eigenvalues of the RG transformation. In addition, the complete spectrum of eigenvalues and corresponding eigenfunctions is found.

Critique of q-entropy for thermal statistics

During the past 12 years there have been numerous papers on a relation between entropy and probability which is nonadditive and has a parameter q that depends on the nature of the thermodynamic system under consideration. For q=1 this relation corresponds to the Boltzmann-Gibbs entropy, but for other values of q it is claimed that it leads to a formalism that is consistent with the laws of thermodynamics. However, it is shown here that the joint entropy for systems having different values of q is not defined in this formalism, and consequently fundamental thermodynamic concepts such as temperature and heat exchange cannot be considered for such systems. Moreover, for q not equal 1 the probability distribution for weakly interacting systems does not factor into the product of the probability distribution for the separate systems, leading to spurious correlations and other unphysical consequences, e.g., nonextensive energy, that have been ignored in various applications given in the literature.

Thermal effects of laser radiation in biological tissue

A theoretical model is presented that simulates the thermal effects of laser radiation incident on biological tissue. The multiple scattering and absorption of the laser beam and the thermal diffusion process in the tissue are evaluated by a numerical technique that is well suited for microcomputers. Results are compared with recent empirical observations.

Group integration for lattice gauge theory at large N and at small coupling

Abstract We consider the fundamental SU( N ) invariant integrals encountered in Wilsons lattice QCD with an eye to analytical results for N →∞ and approximations for small g 2 at fixed N . We develop a new semiclassical technique starting from the Schwinger-Dyson equations cast in differential form to give an exact solution to the single -link integral for N →∞. The third-order phase transition discovered by Gross and Witten for two-dimensional QCD occurs here for any dimension. Alternatively we parametrize directly the integral over the Haar measure and obtain approximate results for SU( N ) using stationary phase at small g 2 . Remarkably the single-loop correction gives the exact answer at N = ∞. We show that the naive lattice string of Weingarten is obtained from N →∞ QCD in the limit of dimensions d →∞. We discuss applications of our techniques to the 1/ N expansion.

Newton's early computational method for dynamics

On December 13, 1679Newton sent a letter toHooke on orbital motion for central forces, which contains a drawing showing an orbit for a constant value of the force. This letter is of great importance, because it reveals the state ofNewtons development of dynamics at that time. Since the first publication of this letter in 1929,Newtons method of constructing this orbit has remained a puzzle particularly because he apparently made a considerable error in the angle between successive apogees of this orbit. In fact, it is shown here thatNewtons implicitcomputation of this orbit is quite good, and that the error in the angle is due mainly toan error of drawing in joining two segments of the oribit, whichNewton related by areflection symmetry. In addition, in the letterNewton describes quite correctly the geometrical nature of orbits under the action of central forces (accelerations) which increase with decreasing distance from the center. An iterative computational method to evaluate orbits for central forces is described, which is based onNewtons mathematical development of the concept of curvature started in 1664. This method accounts very well for the orbit obtained byNewton for a constant central force, and it gives convergent results even for forces which diverge at the center, which are discussed correctly inNewtons letterwithout usingKeplers law of areas.Newton found the relation of this law to general central forces only after his correspondence withHooke. The curvature method leads to an equation of motion whichNewton could have solvedanalytically to find that motion on a conic section with a radial force directed towards a focus implies an inverse square force, and that motion on a logarithmic spiral implies an inverse cube force.

Stability and Eccentricity for Two Planets in a 1:1 Resonance, and Their Possible Occurrence in Extrasolar Planetary Systems

The nonlinear stability domain of Lagranges celebrated 1772 solution of the three-body problem is obtained numerically as a function of the masses of the bodies and the common eccentricity of their Keplerian orbits. This domain shows that this solution can be realized in extrasolar planetary systems similar to those that have been discovered recently with two Jupiter-size planets orbiting a solar-size star. For an exact 1 : 1 resonance, the Doppler shift variation in the emitted light would be the same as for stars that have only a single planetary companion. But it is more likely that in actual extrasolar planetary systems there are deviations from such a resonance, raising the interesting prospect that Lagranges solution can be identified by an analysis of the observations. The existence of another stable 1 : 1 resonance solution that would have a more unambiguous Doppler shift signature is also discussed.

Autocorrelation function and quantum recurrence of wavepackets

Quantum wavepackets follow classical orbits and spread initially with time except for the special case when the energy levels are equally spaced, e.g. the harmonic oscillator. However, for integrable systems bound localised wavepackets can exhibit at long times quantum recurrences which do not have a classical correspondence. For the Coulomb potential the recurrence time is approximately n/3 times the Kepler period 2 pi n3, where n is the mean value of the principal quantum numbers of the wavepacket.

Scaling representation for critical phenomena

A renormalization group method is applied to obtain a representation for the singular part of the free energy of systems described by scaling operators which exhibit a critical phase transition.

Hooke, orbital motion, and Newton’s Principia

A detailed analysis is given of a 1685 graphical construction by Robert Hooke for the polygonal path of a body moving in a periodically pulsed radial field of force. In this example the force varies linearly with the distance from the center. Hooke’s method is based directly on his original idea from the mid‐1660s that the orbital motion of a planet is determined by compounding its tangential velocity with a radial velocity impressed by the gravitational attraction of the sun at the center. This hypothesis corresponds to the second law of motion, as formulated two decades later by Newton, and its geometrical implementation constitutes the cornerstone of Newton’s Principia. Hooke’s diagram represents the first known accurate graphical evaluation of an orbit in a central field of force, and it gives evidence that he demonstrated that his resulting discrete orbit is an approximate ellipse centered at the origin of the field of force. A comparable calculation to obtain orbits for an inverse square force, whic...