Free Energy as a Geometric Invariant

The free energy plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in the study of hyperbolic dynamical systems. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. This geometric free energy encodes a new type of marked length spectrum of closed geodesics, which lies between the well-known marked length spectrum (marked by the corresponding element of the fundamental group) and the unmarked length spectrum. We prove that the free energy parametrizes the boundary of the domain of convergence of a Poincaré series which also encodes this spectrum. We also show that this new length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity. The work of the second author was partially supported by a National Science Foundation grant DMS-0355180. This work was completed during a visit by the first author to Penn State as a Shapiro Fellow.     

Anomalous sensitivity to initial conditions and entropy production in standard maps: Nonextensive approach

We perform a throughout numerical study of the average sensitivity to initial conditions and entropy production for two symplectically coupled standard maps focusing on the control-parameter region close to regularity. Although the system is ultimately strongly chaotic (positive Lyapunov exponents), it first stays lengthily in weak-chaotic regions (zero Lyapunov exponents). We argue that the nonextensive generalization of the classical formalism is an adequate tool in order to get nontrivial information about the first stage of this crossover phenomenon. Within this context we analyze the relation between the power-law sensitivity to initial conditions and the entropy production.     

Photophysics of the Lanthanide Complexes with Conjugated Carboxylic Acids by Low Temperature Fluorescent Spectroscopy

In the context, some lanthanide (Eu³⁺, Tb³⁺ and Sm³⁺) complexes with conjugated carboxylic acids (pyridine-carboxylic acids derivatives) have been synthesized and characterized. The low temperature fluorescent spectra for these complexes have been measured at nitrogen atmosphere (77 K), indicating that the central Ln³⁺ ions locate in an equivalent coordination environment with low symmetry for most of these lanthanide complexes belonging to dimeric or polymeric structure. Therefore, the electronic dipole transition (supersensitive transition) (⁵D₀ → ⁷F₂ for Eu³⁺, ⁵D₄ → ⁷F₆ for Tb³⁺, ⁴G_5/2 → ⁶H_9/2 for Sm³⁺) and magnetic dipole transition (⁵D₀ → ⁷F₁ for Eu³⁺, ⁵D₄ → ⁷F₅ for Tb³⁺, ⁴G_5/2 → ⁶H_5/2 for Sm³⁺) show the regular change in the corresponding split number of fluorescent spectra, which can be realized to predict the fine structure of lanthanide complexes.     

Can a Circulating Light Beam Produce a Time Machine

In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC’s) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC’s occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett’s solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light.     

The electrostatics of Einstein’s unified field theory

When sources are added at their right-hand sides, and g_(ik) is a priori assumed to be the metric, the equations of Einstein’s Hermitian theory of relativity were shown to allow for an exact solution that describes the general electrostatic field of n point charges. Moreover, the injunction of spherical symmetry of g_(ik) in the infinitesimal neighbourhood of each of the charges was proved to yield the equilibrium conditions of the n charges in keeping with ordinary electrostatics. The tensor g_(ik), however, cannot be the metric of the theory, since it enters neither the eikonal equation nor the equation of motion of uncharged test particles. A physically correct metric that rules both the behaviour of wave fronts and of uncharged matter is the one indicated by Hély.In the present paper it is shown how the electrostatic solution predicts the structure of the n charged particles and their mutual positions of electrostatic equilibrium when Hély’s physically correct metric is adopted.     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).     

Cotunneling through a quantum dot coupled to ferromagnetic leads with noncollinear magnetizations

Spin-dependent electronic transport through a quantum dot has been analyzed theoretically in the cotunneling regime by means of the second-order perturbation theory. The system is described by the impurity Anderson Hamiltonian with arbitrary Coulomb correlation parameter U. It is assumed that the dot level is intrinsically spin-split due to an effective molecular field exerted by a magnetic substrate. The dot is coupled to two ferromagnetic leads whose magnetic moments are noncollinear. The angular dependence of electric current, tunnel magnetoresistance, and differential conductance are presented and discussed. The evolution of a cotunneling gap with the angle between magnetic moments and with the splitting of the dot level is also demonstrated.     

Hierarchy of the Selberg Zeta Functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.Mathematics Subject Classifications (2000). Primary 11M36, Secondary 33B15     

The Reality in Bohmian Quantum Mechanics or Can You Kill with an Empty Wave Bullet?

Several situations, in which an empty wave causes an observable effect, are reviewed. They include an experiment showing surrealistic trajectories proposed by Englert et al. and protective measurement of the density of the quantum state. Conditions for observable effects due to empty waves are derived. The possibility (in spite of the existence of these examples) of minimalistic interpretation of Bohmian quantum mechanics in which only Bohmian positions supervene on our experience is discussed.