arXiv subjects

J. H. Hannay

Publications and source records attributed to J. H. Hannay.

10 recordsLinked to original sources

arXiv2024

Double well ground state energy splitting (or instanton flipping rate); rendering the implicit explicit

A prime example of quantum tunnelling is the semiclassical 'energy splitting' of the levels of a symmetrical double well potential, or equivalently the flipping rate of an instanton. Curiously the accepted expression for the ground state energy splitting in terms of the (smooth) potential function has not been pursued to the full explicitness available from classical mechanics. This implicitness is rectified here.

quant-ph↗

arXiv2023

Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

Manipulation ('shaking') of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a 'perfect', or Euler liquid, Kelvin's theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations snd rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. However these are rarely tractable, and their combined consequences are obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, leads (except for a sphere) to an infinitely rich variety of area preserving re-configurations of the liquid. Even for a sphere, any chosen point of the liquid can be moved, in the shell container frame, to any other point, and one would expect the same in general. However, for torus with a small enough hole (diameter<0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside.

physics.flu-dyn↗

arXiv2022

Mean square winding angle of Brownian motion around an impenetrable cylinder

An exact formula is derived, as an integral, for the mean square winding angle of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius a, having started at radius R(>a) from the axis. Strikingly, for the simpler problem with a=0, the mean square winding angle around a straight line, is long known to be instantly infinite however far away the starting point lies. the fractally small, fast, random walk steps of mathematical Brownian motion allow unbounded windings around the zero thickness of the straight line. A remedy if it is required, is to accord the line non-zero thickness, an impenetrable cylinder, as analysed here. The problem straight away reduces to a 2D one of winding around a disc in a plane since the axial component of the 3D Brownian motion is independent of the others. After deriving the exact mean square winding angle, the integral is evaluated in the limit of a narrow cylinder a<<R, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.

cond-mat.stat-mech↗

arXiv2019

The correlated linking numbers of a Brownian loop with two arbitrary curves

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to (plus or minus) infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying curves in magnetostatics, a double integral, but dependent on a single extra parameter, the duration t of the path. The result for the equivalent two-dimensional problem was given previously [Hannay 2018].

cond-mat.stat-mech↗

arXiv2019

Winding number correlation for a Brownian loop in a plane

A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or negative, around that point. Indeed from the (long known) probability distribution, the mean square winding number is infinite, so all statistical moments - averages of powers of the winding number - are infinity (even powers) or zero (odd powers, by symmetry). If an additional marked point is introduced at some distance from the origin, there are now two winding numbers, which are correlated. That correlation, the average of the product of the two winding numbers, is finite and is calculated here. The result takes the form of a single well-convergent integral that depends on a single parameter - the suitably scaled separation of the marked points. The integrals of the correlation weighted by powers of the separation are simple factorial expressions. Explicit limits of the correlation for small and large separation of the marked points are found.

cond-mat.stat-mech↗

arXiv2017

Vortex reconnection rate, and loop birth rate, for a random wavefield

A time dependent, complex scalar wavefield in three dimensions contains curved zero lines, wave 'vortices', that move around. From time to time pairs of these lines contact each other and 'reconnect' in a well studied manner, and at other times tiny loops of new line appear from nowhere (births) and grow, or the reverse, existing loops shrink and disappear (deaths). These three types are known to be the only generic events. Here the average rate of their occurrences per unit volume, R, B, and D is calculated exactly for a Gaussian random wavefield that has isotropic stationary statistics, arising from a superposition of an infinity of plane waves in different directions. A simplifying 'axis fixing' technique is used to achieve this. The resulting formulas are expressed in terms of the power spectrum of the ensemble plane waves: R=WSqrt[K4^3/(12 Pi^4 K2(K4-K2^2))], and B=D=(R/2)-(W/2)Sqrt[9 K2^3/(16 Pi^4)] where W is the standard deviation of angular frequencies, and K2 and K4 are the second and fourth moments of a wave vector component (say the x one). Thus reconnections are always more common than births and deaths combined. As an expository preliminary, the case of two dimensions, where the vortices are points, is studied and the average rate of pair creation (and likewise destruction) per unit area is calculated to be WSqrt[(K4-K2^2)/(4 Pi^4)].

math-ph↗

arXiv2014

An experiment on the shifts of reflected C-lines

An experiment is described that tests theoretical predictions on how C-lines incident obliquely on a surface behave on reflection. C-lines in a polarised wave are the analogues of the optical vortices carried by a complex scalar wave, which is the usual model for describing light and other electromagnetic waves. The centre of a laser beam that carries a (degenerate) C-line is shifted on reflection by the well-known Goos-Hänchen and Imbert-Fedorov effects, but the C-line itself splits into two, both of which are shifted longitudinally and laterally; their shifts are different from that of the beam centre. To maximise the effect to be measured, internal reflection in a glass prism close to the critical angle was used. In a simple situation like this two recently published independent theories of C-line reflection overlap and it is shown that their predictions are identical. The measured differences in the lateral shifts of the two reflected C-lines are compared with theoretical expectations over a range of incidence angles.

physics.optics↗

arXiv2005

Geometry of Calugareanu's theorem

A central result in the space geometry of closed twisted ribbons is Calugareanu's theorem (also known as White's formula, or the Calugareanu-White-Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately) - the `local' crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve - the `writhe framing' ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.

math-ph↗

arXiv2003

Exact scattering theory for any straight reflectors in two dimensions

The exact Green function for the scalar wave equation in a plane with any set of perfectly reflecting straight mirrors, which may be joined to form corners, is given as a diffraction scattering series. Instances would be slit diffraction in optics, or the Schrodinger equation inside (or outside) a general polygonal enclosure ('quantum polygon billiards'). The method is based on the seminal 1896 Riemann helicoid surface solution by Sommerfeld for optical diffraction by a single corner. It is generalised to account for multiple scatter by adapting the analysis of Stovicek for a closely related problem: a collection of magnetic flux lines (points) in a plane, the multi-flux Aharonov-Bohm effect. The short wavelength limit is shown to yield the 'geometrical theory of diffraction'. For slit diffraction the exact series is shown to coincide with that of Schwarzschild in 1902.

physics.optics↗

arXiv2002

Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realising the logarithmic derivative of these infinite polynomials as the electric field of a distribution of coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as $π^{-1}|z|^{-4}$ from the origin. This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel quadratically.

nlin.CD↗