Gauss Integral Research Articles

Helicity and the Călugăreanu invariant

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

Local knot model of entangled polymer chains. 1. Computer simulations of local knots and their collective motion

The local know (LK) theory recently proposed is confirmed by computer simulations of entangled ring polymer chains. The simple cubic lattice model chains (ring) of length L = 512 (volume fraction c = 0.5) is used. By tracing local maxima of Gauss integral along polymer chains, many {open_quotes}true{close_quotes} LKs (lifetime {tau}{sub true}={infinity}) and {open_quotes}temporary{close_quotes} LKs (lifetime {tau}{sub temp} = 2.3 Mu.t;u.t. = unit of time) are found. It is observed that the orders of true and temporary LKs along rings are conserved and that they perform a collective motion (reptation) as predicted by the theory. Various strange motions of true LKs, such as {open_quotes}merging effect{close_quotes}, {open_quotes}multipeak effect{close_quotes}, {open_quotes}ghost effect{close_quotes}, and {open_quotes}probe fluctuations{close_quotes}, are found. In this (part 1) and the following paper (part 2), we discuss in detail how to separate the true Markov motion of LKs and their collective motions from these non-Markov motions. The average number of true and temporary LKs per ring (L = 512) are estimated to be {bar n}{sub true} = 3.44 and {bar n}{sub temp} = 3.0{sub 6}. The average chain length per true LKs is L{sub true} = 149. The diffusion coefficient of single LK is estimated to be d{sub 0} = 0.0172more » bond{sup 2}/u.t. The mean-square displacement of LK coordinate {xi} along a ring, g(t) = ({xi}(t) - {xi}(0)){sup 2}, approaches the Markov line computed for the diffusion coefficient d{sub 0}/({bar n}{sub true} + {bar n}{sub temp}); this suggests that the temporary LKs join to the collective motion of LKs. The empirical entanglement spacing n{sub e} of this system is estimated to be 230 or slightly less; this n{sub e} = 120-133 estimated by Skolnick et al. 26 refs., 16 figs., 6 tabs.« less

Energy of a knot

IN THIS paper. we define a real valued functional. the unuryr E. which is well-behaved for embedded circles in the 3 dimensional Euclidean space, and which blows up for curves with self-intersections. Let EiB be the sum of the energy E and the total squared curvature functional. We show that for any real number z. there are only finitely many ambient isotopy clusscs of embeddinps (i.e. knor t)‘pc’s) with the value of E‘iR not greater than x. There have been studied the total curvature (Fary [I]. Fenchel [7]. Milnor [5]), the total squared curvature (Lanpcr ilnd Singer [-I]), and the Gauss integral of the linking numhcr for a single curve, which. with the total torsion, leads to the notion of the self linking number (Pohl [7]) as functionals on the space of closed curves in R.’ with suitable conditions. Hut these functionals do not have the above properties. They do not blow up for curves with self-intcrscctions. and we can not in general show the finiteness of knot types by them. though we can distinguish the trivial knot from non-trivial knots by the total curvature. and hence. by the total squared curvature ([I], [S]). “Energy” of polygonal knots which is something like electrostatic energy was studied by Fukuhara [3], and “energy” of geodesic links in S’ which is defined by the principal angles was studied by Sakuma [U]. This work was motivated by [8]. in #I. we define the energy E, show the continuity of E, give a lower bound of E and the formulation for E by the double integral, and state a fundamental property of E. In $2, we show the finiteness of the knot types under the bounded value of EAB. Throughout this paper, we always consider the embeddings from S * into R’ of class Cz such that the norm of the derivative is always one. We use the notation 1.1 for the standard norm of R’.

New model of polymer entanglement: Localized Gauss integral model. Plateau modulus G N, topological second virial coefficient Aθ2 and physical foundation of the tube model

The localized Gauss integral (LGI) model, which has been proposed recently by the authors [Macromolecules 21, 2901 (1988)], is discussed in detail. Plateau modulus GN is computed as a function of polymer concentration c and the topological interaction parameter γ̄. Parameter γ̄ is determined for various polymers using their bulk GN data; for polystyrene (PS), γ̄ determined from GN agrees well with that determined from the topological second virial coefficient, Aθ2. Using γ̄ determined from GN, Aθ2 of various polymers other than PS are computed numerically and it is found that Aθ2 of most polymers are much larger than that of PS. It is further shown that power α to the concentration dependence of GN is estimated to be 1.97–2.12 in agreement with experimental value 2.0–2.3 and that GN/c2lkBT is proportional to Aθ2, where cl is the number concentration of ‘‘localized chains.’’ Finally, LGI model is compared with the Doi–Edwards model and the two basic hypothesis of the latter model, the tensile force along the tube and the rubber-like expression of stress, are examined in view of the present model.

Some Applications of an L¹ Version of the Gauss Integral Theorem

Some Applications of an L¹ Version of the Gauss Integral Theorem