Patrick L. Nash

On the Kramers-Kronig relation for the phase spectrum

Abstract The accepted result for the Kramers-Kronig relation for the phase spectrum ψ(ω) of the complex reflection coefficient at normal incidence r(ω) is shown to contain an error. Previously, phase shifts computed from reflectance data using Kramers-Kronig analysis have in some cases been shifted by ± π with no basic justification other than a claim that this provides agreement with accepted values and consistency with the incomplete Stokes treatment. The theoretical basis for this shift is provided in this short note. We derive the correct formula, and show that well-known model dielectrics are consistent with our result and not the accepted result.

On the exceptional equivalence of complex Dirac spinors and complex space‐time vectors

It is well known that there exists an equivalence of R8 vectors and spinors, which has its roots in the rotational symmetry of the Dynkin diagram for D4. We endow R8 with a real action of ∼(SO(4,4)), and restrict to ∼(SO(3,1)). Under this restriction the R8 spinor decomposes into the direct sum of two real Dirac spinors, while the R8 vector decomposes into a real space‐time vector plus four real scalars. The equivalence is preserved under this restriction; it is shown that it is realized in the (exceptional) equivalence of a complex Dirac spinor and a complex space‐time vector.

Transition path sampling with a one-point boundary scheme

Studying the motion of Lennard-Jones clusters in an external potential having a very narrow channel passage at the saddle point, we present a one-point boundary scheme to numerically sample transition (reaction) paths. This scheme does not require knowledge of the transition states (saddle points) or that of the final states. A transition path within a given time interval (0,tf) consists of an activation path during (0,tM) and a deactivation path during (tM,tf) (0<tM<tf) joined at an intermediate time tM. The activation path is a solution to a Langevin equation with negative friction, while the deactivation path is that to a regular Langevin equation with positive friction. Each transition path so generated carries a determined statistical weight. Typical transition paths are found for two-particle and three-particle clusters. A two-particle cluster adjusts its orientation to the direction of the narrow channel and then slides through it. A three-particle cluster completes a transition by openning one of ...

Animated Pseudocolor Activity Maps (Pam’s): Scientific Visualization of Brain Electrical Activity

Major advances in neuroscience have often followed directly from the application of new and more powerful methodological approaches to the study of brain structure and function (Clarke & Jacyna, 1987). Within the last decade a new technique has been developed that allows both brain structure and function to be studied in a closely integrated and highly complimentary fashion. This technique is multiple-site optical recording of membrane potential, or more simply, optical recording. Optical recording is based upon the ability of certain vital dyes (potentiometric probes) to optically signal changes in intracellular membrane potential. By viewing brain tissue stained with a voltage-sensitive dye with a suitable light detector system, changes in neuronal activity can be monitored simultaneously from a 100 or more contiguous anatomical regions (cf. Grinvald et al., 1988).

Chebyshev polynomials and quadratic path integrals

A simple method for the evaluation of path integrals associated with quadratic Lagrangians is discussed. This approach makes use of a relationship between the Van Vleck–Morette determinant and a limit that involves the Chebyshev polynomials of the second kind.

Efficient transition path sampling for systems with multiple reaction pathways

A new algorithm is developed for sampling transition paths and computing reaction rates. To illustrate the use of this method, we study a two-dimensional system that has two reaction pathways: one pathway is straight with a relatively high barrier and the other is roundabout with a lower barrier. The transition rate and the ratio between the numbers of the straight and roundabout transition paths are computed for a wide range of temperatures. Our study shows that the harmonic approximation for fluctuations about the steepest-descent paths is not valid even at relatively low temperatures and, furthermore, that factors related to entropy have to be determined by the global geometry of the potential-energy surface (rather than just the local curvatures alone) for complex reaction systems. It is reasonable to expect that this algorithm is also applicable to higher dimensional systems.

On the structure of the split octonion algebra

SummaryThe known equivalence of special spinors and vector-scalar sets is discussed within the context of the algebra of the split octonions. One implication of this equivalence is that the usual Dirac spinor field can be recast as a vector-scalar field, and this construction is outlined. A process of structure constant factorization is illustrated by the realization of the split octonion multiplication constants (with respect to a spinor basis) as products of certain matrix generators and an arbitrary normalized spinor.RiassuntoSi discute la nota equivalenza di set vettoriali-scalari e spinoriali speciali nel contesto dell’algebra degli ottonioni separati. Un’implicazione di questa equivalenza è che il campo spinoriale di Dirac consueto può essere rimodellato come campo vettoriale-scalare e si sottolinea questa costruzione. Si illustra un processo di fattorizzazione costante di struttura con la realizzazione delle costanti di moltiplicazione degli ottonioni separati (rispetto ad una base spinoriale) come prodotti di certi generatori di matrici e uno spinore normalizzato arbitrario.РезюмеОбсуждается известная эквивалентность специальных спиноров и системы векторов-скаляров в контексте алгебры расщепленных октонионов. Одно из применений этой эквивалентности заключается в том, что обычно дираковское спинорное поле может быть преобразовано, как векторно-скалярное поле. Предлагается указанная конструкция. Иллюстрируется процесс факторизации структурной постоянной с помощью представления мультипликационных постоянных расщепленных октонионов (по отношению с спинорному базису), как произведений некоторых матричных генераторов и произвольного нормированного спинора.

Associated Bessel functions and the discrete approximation of the free-particle time evolution operator in cylindrical coordinates

A central finite difference approximation for the radial contribution Δr to the Laplacian ∇2=Δr+Δr⊥(r) is considered in a three-dimensional cylindrical coordinate system (r,θ,z). A free-particle Schrodinger time evolution operator is constructed by exponentiation, e(i/2)ξ∇2=⋯e−(1/2)ξ2[Δr,Δr⊥(r)] e(i/2)ξΔr⊥(r) e(i/2)ξΔr→⋯e(i/2)ξΔr. Denoting the central finite difference approximation of Δr by (1/Δr2) T, the matrix S≡e(i/2)λT, with λ=ξ/Δr2, is shown to be similar to a particular unitary representation UVK of the group of motions on Euclidean three-space that has been described by Vilenkin and Klimyk. The matrix elements of UVK generalize the Bessel function and provide an approximation of the leading term in the radial contribution to the evolution operator.

Transient response of a Brownian particle with general damping

We study the transient response of a Brownian particle with general damping in a system of metastable potential well. The escape rate is evaluated as a function of time after an infinite wall is removed from the potential barrier. It takes a relaxation time for the rate to reach its limit value and this rate relaxation time differs from the relaxation time of the majority of the probability around the bottom of the potential well. The rate relaxation time is found to depend on the temperature as well as the damping constant. It involves the diffusion time and the instanton time, in general agreement with recent studies of the overdamped case by Bier et al. [Phys. Rev. E 59, 6422 (1999)].

Path integral approach to Brownian motion driven with an ac force

Brownian motion in a periodic potential driven by an ac (oscillatory) force is investigated for the full range of damping constant from the overdamped limit to the underdamped limit. The path (functional) integral approach is advanced to produce formulas for the probability distribution function and for the current of the Brownian particle in response to an ac driving force. The negative friction Langevin dynamics technique is employed to evaluate the dc current for various parameters without invoking the overdamped or the underdamped approximation. The dc current is found to have nonlinear dependence upon the damping constant, the potential parameter, and the ac force magnitude and frequency.