Are there more than five linearly-independent collision invariants for the Boltzmann equation?

The problem of finding the summational collision invariants for the Boltzmann equation is tackled with the aim of proving that the most general solution of the problem is not different from the standard one even when the equation defining a collision invariant is only satisfied almost everywhere inR ³×R ³×S ². The collision invariant is assumed to be in the Hilbert spaceH of the functions which are square integrable with respect to a Maxwellian weight.     

A reply to “a comment on generalized electromagnetism and Dirac algebra”

Issues raised by W. A. Rodrigues, Jr. are discussed.1. This is not a new result; see,e.g., Rohrlich.⁽³⁾ 2. A typographical error in Eq. (77) is corrected here: The productj A in the right-hand parentheses was erroneously transcribed in Ref. (2) as A.3. I define electromagnetic fieldF = A to be that generated by electric charges and the magnetoelectric fieldG = M to be that generated by magnetic monopoles:F F +₅ G. 4. Rodrigues, on the other hand, takes the position that the importance of the Lagrangian formulation should be downgraded if not discarded altogether: ... it is redundant to look for Lagrangians.⁽¹⁾ 5. In fact, he reformulates it using the language of differential forms.6. It is interesting to observe that this bilinear form has the additional virtue of being appropriate for dealing with the monopolecharge parity question, which was pointed out long ago.⁽¹⁴⁾ 7. In fact, even mathematics looks to Nature for its authority.⁽¹⁶⁾ There is evidence that Rodrigues does not understand this concept.⁽¹⁷⁾     

Front propagation in certain one-dimensional exclusion models

Recent results on two interacting particle systems on are summarized, the asymmetric simple exclusion process and the branching exclusion process.     

Turbulence without strange attractor

It is shown that pipe-flow turbulence consists of transients. The fractal dimensions of the dynamical process are thus all zero. Nevertheless, this is compatible with Grassberger-Procaccia analyses suggesting the existence of a high-dimensional strange attractor. The usefulness of the Grassberger-Procaccia method to detect the aging of transients is demonstrated.     

Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

The time-local view of nonequilibrium statistical mechanics. I. Linear theory of transport and relaxation

The various approaches to nonequilibrium statistical mechanics may be subdivided into convolution and convolutionless (time-local) ones. While the former, put forward by Zwanzig, Mori, and others, are used most commonly, the latter are less well developed, but have proven very useful in recent applications. The aim of the present series of papers is to develop the time-local picture (TLP) of nonequilibrium statistical mechanics on a new footing and to consider its physical implications for topics such as the formulation of irreversible thermodynamics. The most natural approach to TLP is seen to derive from the Fourier-Laplace transform ) of pertinent time correlation functions, which on the physical sheet typically displays an essential singularity at z= and a number of macroscopic and microscopic poles in the lower half-plane corresponding to long- and short-lived modes, respectively, the former giving rise to the autonomous macrodynamics, whereas the latter are interpreted as doorway modes mediating the transfer of information from relevant to irrelevant channels. Possible implications of this doorway mode concept for socalled extended irreversible thermodynamics are briefly discussed. The pole structure is used for deriving new kinds of generalized Green-Kubo relations expressing macroscopic quantities, transport coefficients, e.g., by contour integrals over current-current correlation functions obeying Hamiltonian dynamics, the contour integration replacing projection. The conventional Green-Kubo relations valid for conserved quantities only are rederived for illustration. Moreover, may be expressed by a Laurent series expansion in positive and negative powers ofz, from which a rigorous, general, and straightforward method is developed for extracting all macroscopic quantities from so-called secularly divergent expansions of as obtained from the application of conventional many-body techniques to the calculation of . The expressions are formulated as time scale expansions, which should rapidly converge if macroscopic and microscopic time scales are sufficiently well separated, i.e., if lifetime (memory) effects are not too large.     

Fields as Kolmogzrov flows

It is widely recognized that for highly unstable dynamical systems there exists a fundamental limitation on predictability and determinism. An important class of such highly unstable systems is the class of K-flow, which is further characterized by the existence of time-asymmetric objects in the form of K-partition. Our recent approach to the problem of irreversibility has shown that when the implications of the limitation on determinism arising from strong form of instability and those of the existence of K-partition are consistently taken into account, one is naturally led from the physically unrealizable deterministic evolution of phase points to an entropy-increasing stochastic Markovian evolution. Furthermore, this transition is not the result of extraneously imposed coarse graining and/or approximation schemes, but can be brought about by an invertible transformation whose existence and construction are determined by the nature of the instability of the dynamical system itself. After a brief review of this theory which also contains some relatively new remarks, we prove that classical Klein-Gordon field (both massive and massless) possess the structure of K-flow. This seems to provide the first examples of relativistic systems that are K-flows. Some of the implications of this result are briefly discussed. From a mathematical point of view, this seems to be a first step toward an ergodic theory of partial differential equations. In the process, we also provide an independant group-theoretic proof of the existence of incoming and outgoing subspaces of the scattering theory of Lax and Phillips for the wave equation.     

Diffusion coefficients from stopped-flow gas chromatography

Summary Mutual diffusion coefficients of two gases A and B can be determined in an empty gas chromatographic column by letting component B enter at an intermediate position of the column and continuously flow through a part only of it, as a carrier gas. The other component A is injected in a small amount instantaneously at the closed end of the column with the detector placed at the other end. By repeatedly stopping and then restoring after a short time the flow of B, narrow extra peaks are produced on the chromatographic elution curve, owing to diffusion of A into B. An equation is derived giving the area under the curve of each stop-peak as a function of time of the corresponding stop. Plotting the experimental data according to this equation permits the determination of the diffusion coefficient of A into B. Some results obtained by this method show negligible variations with changes in the experimental parameters.     

Mono- and diboronates derived from tridentate ONO ligands and arylboronic acids

The syntheses of eight [4.3.0] heterobicyclic boronates containing a N → B coordinative bond are described. The monomeric compounds were prepared by reaction of arylboronic acids with a tridentate ligand having the ONO donor set of atoms. It was shown that substituents at the para-position of the B-phenyl moiety transmit electronic effects to the CN bond which in turn is polarized by formation of the N → B coordination bond. At the same time, related tridentate ligands were also reacted with 1,4-benzenediboronic acid in order to prepare benzene diboron complexes. The structure of this type of compounds was confirmed by X-ray analysis for one of the derivatives.     

General constructions of biquandles and their symmetries

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.