H-Theorems from Autonomous Equations

The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution.Comment: 13 pages; v1 -> v2: Sec. 1-2 considerably rewritten, minor corrections in Sec. 3-

Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence $\{z_j\}$ in the unit disk there is always a Blaschke product with $\{z_j\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.Comment: 21 page

Harmonic functions on metric graphs under the anti-Kirchhoff law

When does an infinite metric graph allow nonconstant bounded harmonic functions under the anti-Kirchhoff transition law? We give a complete answer to this question in the cases where Liouville’s theorem holds, for trees, for graphs with finitely many essential ramification nodes and for generalized lattices. It turns out that the occurrence of nonconstant bounded harmonic functions under the anti-Kirchhoff law differs strongly from the one under the classical continuity condition combined with the Kirchhoff incident flow law.Peer ReviewedPostprint (author's final draft

The Statistical Mechanics for an Extended Nonlinear Epigenetic Dynamics. I

An extension of the control equations discussed by Goodwin is proposed which allows for arbitrary strong coupling and for arbitrary parallel coupling of metabolic pools and genetic loci. It is demonstrated that these generalized control equations can be put into canonical form and further that Liouville's theorem applies. In addition, it is demonstrated that after a suitable canonical transformation the resulting partition function can be solved in closed form, and this result, as well as that for the mean energy, is exhibited. Some remarks appropriate to additional extensions are presented

Nambu representation of an extended Lorenz model with viscous heating

We consider the Nambu and Hamiltonian representations of Rayleigh-Benard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec=0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivative of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.Comment: 15 pages, no figur

Influence of vane sweep on rotor-stator interaction noise

The influence of vane sweep in rotor-stator interaction noise is investigated. In an analytical approach, the interaction of a convected gust representing the rotor viscous wake, with a cascade of cascade of finite span swept airfoils, representing the stator, is analyzed. The analysis is based on the solution of the exact linearized equations of motion. High frequency convected gusts for which noise generation is concentrated near the leading edge of airfoils is considered. In a preliminary study, the problem of an isolated finite span swept airfoil interacting with a convected gust is analyzed. Results indicate that sweep can substantially reduce the farfield noise levels for a single airfoil. Using the single airfoil model, an approximate solution to the problem of noise radiation from a cascade of finite span swept airfoils interacting with a convected gust is derived. A parametric study of noise generated by gust cascade interaction is carried out to assess the effectiveness of vane sweep in reducing rotor-stator interaction noise. The results show that sweep is beneficial in reducing noise levels. Rotor wake twist or circumferential lean substantially influences the effectiveness of vane sweep. The orientation of vane sweep must be chosen to enhance the natural phase lag caused by wake lean, in which case rather small sweep angles substantially reduce the noise levels

Early Investigations in Conformal and Differential Geometry

The present article introduces fundamental notions of conformal and differential geometry, especially where such notions are useful in mathematical physics applications. Its primary achievement is a nontraditional proof of the classic result of Liouville that the only conformal transformations in Euclidean space of dimension greater than two are Möbius transformations. The proof is nontraditional in the sense that it uses the standard Dirac operator on Euclidean space and is based on a representation of Möbius transformations using 2x2 matrices over a Clifford algebra. Clifford algebras and the Dirac operator are important in other applications of pure mathematics and mathematical physics, such as the Atiyah-Singer Index Theorem and the Dirac equation in relativistic quantum mechanics. Therefore, after a brief introduction, the intuitive idea of a Clifford algebra is developed. The Clifford group, or Lipschitz group, is introduced and related to representations of orthogonal transformations composed with dilations; this exhausts Section 2. Differentiation and differentiable manifolds are discussed in Section 3. In Section 4 some points of differential geometry are reiterated, the Ahlfors-Vahlen representation of Möbius transformations using 2x2 matrices over a Clifford algebra is introduced, conformal mappings are explained, and the main result is proved