Statistical mechanics of two-dimensional vortices

Equilibrium statistics of a cluster of a large number of positive two-dimensional point vortices in an infinite region and the associated thermodynamic functions, exhibiting negative temperatures, are evaluated analytically and numerically from a microcanonical ensemble. Extensive numerical simulations of vortex motion are performed to verify the predicted equilibrium configurations. An application of Kubo's linear response theory is used to study the nonequilibrium situation that results from placing a cluster, of vortices in a weak external velocity field, such as that produced by a distant vortex cluster. The weak field causes the cluster to grow in size as if there were an effective positive eddy viscosity. When a number of clusters interact, the effect is for each to grow while the distances between them decrease with time. The latter effect is an exhibit of negative viscosity. The application of this to the motion of the atmosphere is discussed.Supported in part by National Science Foundation Grant GK-40263.Advanced Study Program, National Center for Atmospheric Research (sponsored by the National Science Foundation).     

Statistical mechanics of vortices in an inviscid two-dimensional fluid

In this paper we study rigorously the statistical mechanics of a gas of vortices in the thermodynamic limit. In this limit, no negative temperature states are found to exist.Work supported in part by NSF Grant MCS 78-02721     

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

We complement the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-Hückel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like ω ? ψ relationships depending on the sign of Ku ? 3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like ω ? ψ relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional.     

S-matrix elements from T-duality

Recently it has been speculated that the S-matrix elements satisfy the Ward identity associated with the T-duality. This indicates that a group of S-matrix elements is invariant under the linear T-duality transformations on the external states. If one evaluates one component of such T-dual multiplet, then all other components may be found by the simple use of the linear T-duality. The assumption that fields must be independent of the Killing coordinate, however, may cause, in some cases, the T-dual multiplet not to be gauge invariant. In those cases, the S-matrix elements contain more than one T-dual multiplet which are intertwined by the gauge symmetry.     

D-branes from matrix factorizations

B-type D-branes can be obtained from matrix factorizations of the Landau–Ginzburg superpotential. We here review this promising approach to learning about the spacetime superpotential of Calabi–Yau compactifications. We discuss the grading of the D-branes, and present applications in two examples: the two-dimensional torus, and the quintic. To cite this article: K. Hori, J. Walcher, C. R. Physique 5 (2004).

Résumé

Les D-branes de type B peuvent être décrites à partir de factorisations matricielles du super-potentiel de Landau–Ginzburg. On revoit ici cette approche prometteuse pour étudier le super-potentiel en espace-temps de compactifications de Calabi–Yau. On discute la graduation des D-branes, et présente deux exemples : le tore en deux dimensions, ainsi que la quintique. Pour citer cet article : K. Hori, J. Walcher, C. R. Physique 5 (2004).     

D-branes from classical macroscopic strings

This paper is a sequel to the series of papers dedicated to model independent analysis of brane-like extended objects in curved backgrounds. In particular, we study cylindrical membranes wrapped around the extra compact dimension of a (D + 1)-dimensional Riemann–Cartan spacetime. The world-sheet equations are obtained from the universally valid conservation equations of the membrane stress–energy and spin tensors. In the limit of small extra dimension, the dimensionally reduced theory is obtained. The narrow membrane becomes an effective string characterized not only by tension and spin, but also by electric and dilaton charges. The boundary of such an effective string has been shown to live in less spacetime dimensions than its interior. Precisely, the string endpoints are trapped by the surfaces orthogonal to the gradient of the effective dilaton field. The string dynamics has been shown to follow from an action functional subject to the Dirichlet boundary conditions. This way, we have succeeded in obtaining a macroscopic D-brane analogue.     

Statistical mechanics of viscoelasticity

A unified treatment of viscoelasticity is developed in the framework of non-equilibrium classical statistical mechanics. An exact correspondence between Mori's continued-fractions formalism and rheological circuits is shown.     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Statistical mechanics of interfaces

On the basis of van der Waals theory for interfaces we evaluate explicitly the small-wavevector behavior of the pair correlation function along an interface. A correction to the density profile is also found. The results obtained are in full accord with capillary wave theory when wave amplitudes are regarded as small.     

Statistical mechanics and population biology

This paper exploits the connection between statistical mechanics and stochastic processes in order to derive a class of macroscopic observables for populations. This review treats the dynamics of populations in both constant and variable environments and derives in each case the thermodynamic analogs of the population parameters.     

Statistical mechanics of systems nonlinearly displaced from equilibrium I

A graphical approach is used to extend the response theory expression for nonequilibrium averages to any arbitrry order in deviations from equilibrium. Various ways in which the perturbation series about equilibrium can be resummed into gradient or Chapman-Enskog like expansions are presented. As a matter of illustration, examples from the hydrodynamics of simple fluids and the motion of a Brownian particle are considered. Specifically the normal stresses, shear dependent viscosity and velocity dependent friction constant are examined.     

Interactions of strings and D-branes from M theory

We discuss the relation between M theory and type II string theories. We show that, assuming “natural” interactions between membranes and fivebranes in M theory, the known interactions between strings and D-branes in type II string theories arise in appropriate limits. Our discussion of the interactions is purely at the classical level. We remark on issues associated with the M theory approach to enhanced gauge symmetries, which deserve further investigation.     

D-branes and Spin structures

It was recently pointed out by E. Witten that for a D-brane to consistently wrap a submanifold of some manifold, the normal bundle must admit a Spin^c structure. We examine this constraint in the case of type II string compactifications with vanishing cosmological constant, and argue that in all such cases, the normal bundle to a supersymmetric cycle is automatically Spin^c.     

Statistical mechanics of steiner trees

The minimum weight Steiner tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows us to analyze the statistical mechanics properties of MST on random graphs of various types.