Radiative orbital electron capture by the atomic nucleus

The rate for the photon emission accompanying orbital 1S electron capture by the atomic nucleus is recalculated. While a photon can be emitted by the electron or by the nucleus, the use of the length gauge significantly suppresses the nuclear contribution. Our calculations resolve the long standing discrepancy of theoretical predictions with experimental data for $\Delta J=2$ forbidden transitions. We illustrate the results by comparison with the data established experimentally for the first forbidden unique decays of $^{41}$Ca and $^{204}$Tl.Comment: 18 pages, 2 figures, submitted to Phys. Rev.

Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases

The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium. The calculation of the largest Lyapunov exponent makes interesting connections with the theory of propagation of hydrodynamic fronts. Calculations are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and three dimensions, which are considerably simpler than the corresponding calculations for hard-ball gases. The article concludes with a brief discussion of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source. (Fig.7 doesn't print well on some printers) This revised paper will appear in "Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of Mathematical Sciences, Springe

An Elementary Proof of Lyapunov Exponent Pairing for Hard-Sphere Systems at Constant Kinetic Energy

The conjugate pairing of Lyapunov exponents for a field-driven system with smooth inter-particle interaction at constant total kinetic energy was first proved by Dettmann and Morriss [Phys. Rev. E {\bf 53}, R5545 (1996)] using simple methods of geometry. Their proof was extended to systems interacting via hard-core inter-particle potentials by Wojtkowski and Liverani [Comm. Math. Phys. {\bf 194}, 47 (1998)], using more sophisticated methods. Another, and somewhat more direct version of the proof for hard-sphere systems has been provided by Ruelle [J. Stat. Phys. {\bf 95}, 393 (1999)]. However, these approaches for hard-sphere systems are somewhat difficult to follow. In this paper, a proof of the pairing of Lyapunov exponents for hard-sphere systems at constant kinetic energy is presented, based on a very simple explicit geometric construction, similar to that of Ruelle. Generalizations of this construction to higher dimensions and arbitrary shapes of scatterers or particles are trivial. This construction also works for hard-sphere systems in an external field with a Nos\'e-Hoover thermostat. However, there are situations of physical interest, where these proofs of conjugate pairing rule for systems interacting via hard-core inter-particle potentials break down.Comment: 16 pages, 4 figures, to appear in J. Stat. Phy

Front propagation techniques to calculate the largest Lyapunov exponent of dilute hard disk gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.Comment: 26 pages REVTeX, 1 eps figure. Added remarks, a reference and corrected some typo

Non-equilibrium Thermodynamics and Fluctuations

In the last ten years, a number of ``Conventional Fluctuation Theorems'' have been derived for systems with deterministic or stochastic dynamics, in a transient or in a non-equilibrium stationary state. These theorems gave explicit expressions for the ratio of the probability to find the system with a certain value of entropy (or heat) production to that of finding the opposite value. A similar theorem for the fluctuations of the work done on a system has recently been demonstrated experimentally for a simple system in a transient state, consisting of a Brownian particle in water, confined by a moving harmonic potential. In this paper we show that because of the interaction between the stochastic motion of the particle in water and its deterministic motion in the potential, very different new heat theorems are found than in the conventional case. One of the consequences of these new heat Fluctuation Theorems is that the ratio of the probability for the Brownian particle to absorb heat from rather than supply heat to the water is much larger than in the Conventional Fluctuation Theorem. This could be of relevance for micro/nano-technology.Comment: 10 pages, 6 figures. Some corrections in the text were made. Submitted to Physica

An allosteric model of KaiC phosphorylation

In a recent series of ground-breaking experiments, Nakajima et al. [Science 308, 414-415 (2005)] showed that the three cyanobacterial clock proteins KaiA, KaiB, and KaiC are sufficient in vitro to generate circadian phosphorylation of KaiC. Here, we present a mathematical model of the Kai system. At its heart is the assumption that KaiC can exist in two conformational states, one favoring phosphorylation and the other dephosphorylation. Each individual KaiC hexamer then has a propensity to be phosphorylated in a cyclic manner. To generate macroscopic oscillations, however, the phosphorylation cycles of the different hexamers must be synchronized. We propose a novel synchronisation mechanism based on differential affinity: KaiA stimulates KaiC phosphorylation, but the limited supply of KaiA dimers binds preferentially to those KaiC hexamers that are falling behind in the oscillation. KaiB sequesters KaiA and stabilizes the dephosphorylating KaiC state. We show that our model can reproduce a wide range of published data, including the observed insensitivity of the oscillation period to variations in temperature, and that it makes nontrivial predictions about the effects of varying the concentrations of the Kai proteins.Comment: 8 pages, 6 figures. Accepted for publication in PNA

Radiative electron capture in the first forbidden unique decay of 81Kr

The photon spectrum accompanying the orbital K-electron capture in the first forbidden unique decay of 81Kr was measured. The total radiation intensity for the photon energies larger than 50 keV was found to be 1.47(6) x 10^{-4} per K-capture. Both the shape of the spectrum and its intensity relative to the ordinary, non-radiative capture rate, are compared to theoretical predictions. The best agreement is found for the recently developed model which employs the length gauge for the electromagnetic field.Comment: 7 pages, 6 figure