Diffusion processes and their sample paths

The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) we estimate the probability of a trivial knot formation on the lattice using the Kauffman algebraic invariants and show the connection of this problem with the thermodynamic properties of 2D disordered Potts model; (ii) we investigate the limit behavior of random walks in multi-connected spaces and on non-commutative groups related to the knot theory. We discuss the application of the above mentioned problems in statistical physics of polymer chains. On the basis of non-commutative probability theory we derive some new results in statistical physics of entangled polymer chains which unite rigorous mathematical facts with more intuitive physical arguments.

https://arxiv.org/pdf/cond-mat/9812205.pdf

Statistics of Knots and Entangled Random Walks

First results from a series of experiments aimed at the study of the interaction of slow very highly charged ions with conductor and insulator surfaces are reported. Charge state dependences of secondary electron, x-ray and secondary ion emission are measured. In addition, microscopic studies are performed using an ‘Atomic Force Microscope’ to investigate surface defects, produced through the stored electrostatic potential of the incident ion. The ions that have been used for these studies range from O7+ to Th80+, with low kinetic energies (generally 1–3 keV/amu). Overall, enhancement of the low energy secondary electron, sputter ion and photon emission are observed with increasing charge states. Saturation in the electron emission yield at very low velocities confirm present models for image charge acceleration effects. X-ray emission spectra are found to be consistent with the formation of so called ‘hollow’ atoms near the surface during the neutralization processes. The microscopic studies rev...

Ion/surface interaction studies with 1-3 keV/amu ions up to Th80+

The authors compute the joint probability for a closed Brownian curve to wind n times around a prescribed point and to enclose a given algebraic area. An estimate from below of the arithmetic area is obtained.

https://iopscience.iop.org/article/10.1088/0305-4470/23/15/027/pdf

Winding of planar Brownian curves

Addresses the problem of the algebraic area enclosed by a Brownian curve in two dimensions, recently reconsidered by Khandekar and Wiegel (1988). The author recalls the principal results actually first obtained by Paul Levy in 1950. Another derivation by functional integrals is given for several probability distributions: that of the area of a Brownian ring, the area between an open Brownian arc and its chord and, finally, the area swept with respect to the mean position of the Brownian path, as a result which seems to be new.

Areas of planar Brownian curves

The authors derive an exact expression for the distribution function for the (algebraic) area enclosed by a plane closed random walk, using a continuous model from the start. Their result has a different analytical form but is numerically close to an expression for the same quantity of a discrete model, derived by Brereton and Butler (1987).

Distribution of the area enclosed by a plane random walk

Continuous area-preserving models for self-interacting polymers

https://ris.utwente.nl/ws/files/6435716/Khandekar88on.pdf

On a path integral with a topological constraint

We derive an exact expression for the distribution of the area which is enclosed by a closed plane polymer loop in a uniform electric field. We show that for strong fields the distributions is a Gaussian whereas for weak electric fields it has a more general exponential form.

Distribution of the area enclosed by a closed polymer loop in an external electric field

The statistical mechanical properties of plane polymer loops enclosing a constant area are investigated, using a continuous model from the start. For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit.

/pdf/statistical-mechanical-properties-of-polymer-configurations-29qekh1ih0.pdf

F.W. Wiegel

Papers

Distribution of the area enclosed by a plane random walk

Continuous area-preserving models for self-interacting polymers

On a path integral with a topological constraint

Distribution of the area enclosed by a closed polymer loop in an external electric field

Statistical Mechanical Properties of Polymer Configurations which Enclose a Constant Area