Geometrical aspects and generalizations of Newton-Cartan mechanics

Cartan's geometrical approach to Newtonian mechanics is discussed and generalized. The approach is geometrical and use is made of the theory of holonomy groups. The concept of a degenerate metric connection is formulated and discussed.     

Distributions and channel capacities in generalized statistical mechanics

The purpose of this note is twofold. Firstly, we consider generalizations of Shannon's entropy and its applications to thermodynamics based on extensivity considerations. Secondly, we apply the generalized entropy formalism to deriving various generalised channel capacities. We arrive at some surprising conclusions of systems achieving “super-capacitance” or “sub-capacitance” depending on the circumstances. These results suggest the possibility of improving the conventional Shannon capacity by using physical systems obeying more generalized statistics, but also predict behaviour which is hard to reconcile with experience.     

Geometrical construction of a generalized Skyrme Lagrangian

A geometrical scheme is proposed that leads to a straightforward generalization of a Skyrme Lagrangian, comprising higher-order terms up to the eighth power in pion fields.Boursière DGRST.     

Geometrical setting of solid mechanics

The starting point in the geometrical setting of solid mechanics is to represent deformation process of a solid body as a trajectory in a convenient space with Riemannian geometry, and then to use the corresponding tools for its analysis. Based on virtual power of internal stresses, we show that such a configuration space is the (globally) symmetric space of symmetric positive-definite real matrices. From this unifying point of view, we shall analyse the logarithmic strain, the stress rate, as well as linearization and intrinsic integration of corresponding evolution equation.     

Geometrical aspects of the complex eikonal equations associated to the generalized Weierstrass representation

Certain geometrical aspects of a Laplace-Beltrami equation subjected to a complex eikonal equation are studied. It is shown that all principal curvatures of a solution surface are constant for this system. To illustrate the theoretical considerations, we look for a class of solutions of an overdetermined system composed of the sigma model and a system of complex eikonal equations. The connection between the sigma model and the generalized Weierstrass system for inducing constant mean curvature surfaces allows us to construct special classes of solutions for classical configurations of strings. Presented by A.M. Grundland at the DI-CRM Workshop held in Prague, 18–21 June 2000. This work was supported by a research grant from NSERC of Canada and the hospitality of the Doppler Institute of the Czech Republic.     

Towards a statistical mechanics of cities

Cities are some of the most complex dynamical systems in human societies and in nature. There is growing interest in producing more comprehensive quantitative theory, capable of describing many of the features now observable in urban environments, especially those that show empirical regularities across cities of different sizes, geographies, and levels of development. The principal challenge of achieving such a goal is our ability to build frameworks that include realistic but simple accounts of agents' choices and strategic behavior, beyond current approaches in statistical physics or economics. Here, I propose a general framework that integrates agents' behavior with their resource and information management towards seizing opportunities in their environment. I show how this approach integrates urban scaling theory with a statistical mechanics of open-ended (economic) growth. The framework is general and, with appropriate modifications and elaborations, can account for the statistical dynamics of other complex systems.     

Geometrical aspects of isoscaling

The property of isoscaling in nuclear fragmentation is studied using a simple bond percolation model with “isospin” added as an extra degree of freedom. It is shown, first, that with the assumption of fair sampling and with homogeneous probabilities it is possible to solve the problem analytically. Second, numerical percolations of hundreds of thousands of grids of different sizes and with different N to Z ratios confirm this prediction with remarkable agreement. It is thus concluded that isoscaling emerges even in the simple case of a classical non-interacting system such as two-species percolation under the assumption of fair sampling; if put in the nomenclature of the minimum information theory, isoscaling in percolation appears to require nothing more than the existence of equiprobable configurations in maximum entropy states.     

Cluster expansion and generalized transfer matrices for the statistical mechanics of linear chains

A cluster expansion of the statistical mechanical density operator for a general linear chain model with nearest-neighbor interactions is made. This expansion is then shown to lead to an expansion of a generalized transfer matrix, whose maximum eigenvalue is the per-site partition function. A number of computational features, as well as some illustrative examples, of this approach are described.Research supported in part by the Robert A. Welch Foundation, Houston, Texas.     

Fractional statistical mechanics

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.     

Geometrical aspects of Schlesinger''s equation

The equation (Schlesinger's equation) for the isomonodromic deformations of an (SL (2, C) connection with four simple poles on the projective line is shown to describe a holomorphic projective structure on a surface. The space of geodesics of this structure is, by a primitive version of twistor theory, a two-dimensional complex Poisson manifold containing complete rational curves. The Poisson structure degenerates on a divisor and it is shown that the complement of the divisor is a symplectic manifold which can be identified with the quotient of the moduli space of representations of a free group on three generators in SL (2, ) by the action of a braid group.     

Geometrical aspects of magnetic monopoles

The possible topological structures of elementary particles have been investigated to explore the possibility of the existence of magnetic monopoles. It is pointed out that when an elementary charged particle is depicted as an extended body such that the orientation of the internal space (internal helicity) defines the fermion number, the global conservation of this does not allow the existence of a magnetic monopole. Again it is argued that when anisotropy is introduced in the microlocal space-time depicting the internal space of hadrons, this gives rise to the internal symmetry algebra and no non-Abelian gauge fields and Higgs scalars are necessary to have a grand unified scheme of interactions. This avoids theSU ₂ and GUT monopoles. Besides, in this formalism, baryon number corresponds to the orientation or internal helicity of the composite system and the global conservation of this quantum number is found to be a consequence of Lorentz invariance. This forbids the existence of any sort of cosmological monopole in this Lorentz invariant Universe.     

Chiral models: Geometrical aspects

Two-dimensional classical chiral models of field theory are considered, the main attention being paid on geometrical aspects of such theories.A characteristic feature of these models is that the interaction is inserted not by adding the interaction Lagrangian to the free field Lagrangian, but has a purely geometrical origin and is related to the inner curvature of the manifold.These models are in many respects analogous to non-Abelian gauge theories and as became clear recently, they are also important for the superstring theory which nowadays is the most probable candidate for a truly unified theory of all interactions including gravitation.     

统计力学中的勒让德变换

勒让德变换是理论物理学中一个重要的工具,多变量函数的勒让德变换有明显的对称性.系综是统计力学教学中的核心内容.利用勒让德变换的无量纲性,从微正则系综的特性函数熵的勒让德变换出发,类比的定义了配分函数的勒让德变换.同时得到各种系综的特性函数和配分函数以及它们之间的关系和相应的热力学公式.     

Douglas Henderson: a life in statistical mechanics

In a previous study (Wang, L. S., Nicholas, J. B., Dupuis, M., Wu, H., and Colson, S. D., 1997, Phys. Rev. Lett., 78, 4450) of small silicon oxide clusters with stochiometry Si3Oy(y = 1-6), it was found that the molecule Si3O2 and its anion required special theoretical attention. This is due to a particular electronic structure feature of the neutral cluster: the s,p hybrid atomic orbitals of the two bonded silicon atoms give rise to a low lying empty π molecular orbital. As a result the ground state of Si3O2 has a triplet spin, and a state of singlet spin is a low lying excited electronic state at about 0.5 eV above the triplet state. Ab initio molecular orbital calculations show that photoelectron detachment proceeds from the 2B1 state of the anion to the 3B1 ground state of the neutral cluster. Detachment via the 1A1 state of the neutral cluster lies in the shoulder of the detachment via the triplet state.