Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal.The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.     

Sum rules and dualities for generalized parton distributions: is there a holographic principle?

To leading order approximation, the physical content of generalized parton distributions (GPDs) that is accessible in deep virtual electroproduction of photons or mesons is contained in their value on the cross-over trajectory. This trajectory separates the t-channel and s-channel dominated GPD regions. The underlying Lorentz covariance implies correspondence between these two regions through their relation to GPDs on the cross-over trajectory. This point of view leads to a family of GPD sum rules which are a quark analogue of finite energy sum rules and it guides us to a new phenomenological GPD concept. As an example, we discuss the constraints from the JLab/Hall A data on the dominant u-quark GPD H. The question arises whether GPDs are governed by some kind of holographic principle.     

Two theorems on energy-momentum conservation

The mathematical meaning of the law of conservation of energy-momentum is examined. A distinction is made between the intrinsic properties of the metric tensor (i.e., those properties that are independent of the coordinate system), and the nonintrinsic properties of this tensor (i.e., those properties that depend upon the coordinate system). The covariance of the energy-momentum law is used to demonstrate that if one is given (a) any analytic contravariant energy-momentum tensor density in a given coordinate systemx and (b) an analytic specification of the intrinsic properties of the metric tensor, no matter what these properties may be, one can always choose the nonintrinsic properties of the metric tensor in such manner as to satisfy the law of conservation of energy-momentum in the coordinate systemx and thereby in every coordinate system. This result is proved only in the case where the contravariant components of the energy-momentum tensor density are given. Neither the covariant, nor the mixed energy-momentum tensor densities are considered. Other theorems similar to that described above are also derived. Many of the results obtained are nontrivial even when space-time is flat.     

Gauge Fields in General Parametrical Approach and Their Finslerian Reduction

The covariance principle of general relativity is extended to internal space. Associated gauge fields and tensors are systematically described, whereupon the variational principle is set up for all gauge fields by applying a Palatini-type method, thereby giving rise to an attractive self-contained theory in which the Einstein equations are intrinsically synthesized with the generalized Yang-Mills equations. General gauge-covariant physical field equations are formulated, showing that currents, external + internal spin tensors, and energy-momentum tensors can be introduced unambiguously under these general conditions and that the associated conservation laws can be derived. The electromagnetic field finds its gauge-geometric origin as the gauge field related to internal densities. To be operative with the tensor indices of external and internal types, this general theory must be bimetric. The assumptions that the gauge-covariant derivatives of metric tensors should vanish simplify the theory to the level of a Finslerian gauge approach.     

A YANG–MILLS ELECTRODYNAMICS THEORY ON THE HOLOMORPHIC TANGENT BUNDLE

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained.

We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written.

A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T′M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T′M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.     

Feature fusion of palmprint and face via tensor analysis and curvelet transform

In order to improve the recognition accuracy of the unimodal biometric system and to address the problem of the small samples recognition, a multimodal biometric recognition approach based on feature fusion level and curve tensor is proposed in this paper. The curve tensor approach is an extension of the tensor analysis method based on curvelet coefficients space. We use two kinds of biometrics: palmprint recognition and face recognition. All image features are extracted by using the curve tensor algorithm and then the normalized features are combined at the feature fusion level by using several fusion strategies. The k-nearest neighbour (KNN) classifier is used to determine the final biometric classification. The experimental results demonstrate that the proposed approach outperforms the unimodal solution and the proposed nearly Gaussian fusion (NGF) strategy has a better performance than other fusion rules.     

Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space-time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.     

Algebraic classification of a bivector in the Kaluza flat space and the magnetic charge

An algebraic classification of a bivector (understood as a generalized electromagnetic field tensor) in the Kaluza flat space-time is presented. It is assumed that a cylindricity condition holds with respect to the fifth coordinate. The concept of a dual rotation that depends on a selected four-dimensional section is introduced in five-dimensional space. By means of this rotation, we are able to relate the stationary gradient magnetic field with the fifth component of the five-dimensional vector potential which, in turn, is associated with a magnetic charge that cannot be observed in four-dimensional space-time. Krasnoyarsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 114–119, January, 1997.     

Relativistic theory of superpotentials for a nonhomogeneous spatially isotropic medium

This work deals with the relativistic, coordinate invariant theory of electromagnetic wave propagation in a nonhomogeneous medium that is spatially isotropic with respect to the generalized 4-velocity. The existence of a new superpotential for the electromagnetic field is established. The governing field equation for the superpotential is derived via a generalized Lorentz gauge condition for the electromagnetic potential and two new tensor identities involving the curvature tensor. The new field equation takes a simple form which reveals the effect of non-rigidity, rotation, acceleration, and incomplete material isotropy with respect to the Fermi frames as well as curvature and nonhomogeneous material properties.     

Free energy from molecular dynamics with multiple constraints

In molecular dynamics simulations of reacting systems, the key step to determining the equilibrium constant and the reaction rate is the calculation of the free energy as a function of the reaction coordinate. Intuitively the derivative of the free energy is equal to the average force needed to constrain the reaction coordinate to a constant value, but the metric tensor effect of the constraint on the sampled phase space distribution complicates this relation. The appropriately corrected expression for the potential of mean constraint force method (PMCF) for systems in which only the reaction coordinate is constrained was published recently. Here we will consider the general case of a system with multiple constraints. This situation arises when both the reaction coordinate and the ‘hard’ coordinates are constrained, and also in systems with several reaction coordinates. The obvious advantage of this method over the established thermodynamic integration and free energy perturbation methods is that it avoids the cumbersome introduction of a full set of generalized coordinates complementing the constrained coordinates. Simulations of n-butane and n-pentane in vacuum illustrate the method.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

Applications of periodic-orbit theory

The periodic-orbit theory of Gutzwiller is applied in various ways by using generalized periodic-orbit sum rules. Numerical evaluations are carried out for the hyperbola billiard, a strongly chaotic system. The most efficient semiclassical determination of quantum energies is achieved by a quantization condition, which is formulated in terms of a zeta function by using a functional equation.     

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

Because of the long range nature of hydrodynamic interactions, the problem of boundary conditions on a finite simulation cell of a hydrodynamically dense suspension of particles in Brownian motion is quite as complicated as the analogous problem in simulation of the statistical mechanics of charged and dipolar systems. One resolution of this problem is to use periodic boundary conditions and to view them as a way of describing a physical system composed of a large spherical array of periodic replicas of the simulation cell. The hydrodynamic interactions are calculated using the quasi-static linearized Navier-Stokes equation. This requires that the suspending fluid velocity remains small throughout the array. That the sum of the particle velocities in the simulation cell be zero is insufficient to force boundedness of the fluid velocity as the array becomes large. Boundedness in the array of the suspending fluid velocity is achieved if a rigid wall boundary condition is applied at the outer edge of the array as the array becomes large. With this condition the net particle velocity equals zero condition is not needed. The condition allows lattice sum representations for the suspending fluid velocity to be derived. These lattice sums are absolutely and rapidly convergent and periodic. Representations of the velocity in the array with boundary condition allow calculation of mobility tensors which are also periodic and can be evaluated numerically in tolerable amounts of computer time. A major effect of these calculations is to identify the physical model system corresponding to a truly periodic fluid velocity and mobility tensor as a large array with rigid wall boundary condition.     

Nanoscale ab-initio calculations of optical and electronic properties of LaCrO3 in cubic and rhombohedral phases

We report nanoscale ab-initio calculations of the linear optical and electronic properties of LaCrO₃ in nonmagnetic cubic and rhombohedral phases using the full potential linear augmented plane wave (FP-LAPW) method. In this work the generalized gradient approximation is used for exchange-correlation potential. The dielectric tensor is derived within random-phase approximation. We present results for the band structure, density of states, imaginary and real parts of dielectric tensor, electron energy loss spectroscopy, sum rules, reflectivity, refractive index and extinction coefficient. The regions of transparent, absorption and reflection are discussed. We are not aware of any published experimental or theoretical data for these phases, so our calculations can be used to cover this lack of data for these phases.     

An invariant theory of the constitutive parameters of a plasma and a ferrite

Under the influence of a constant magnetic field, the electric property of a plasma and the magnetic property of a ferrite are anisotropic. In this paper, the general coordinatefree invariant forms of the dielectric tensor of a plasma and the permeability tensor of a ferrite are obtained. The tensors are expressed explicitly as a sum of three tensors: a unit tensor, a symmetric tensor and an antisymmetric tensor, each of which is weighted by different constants. The symmetric and antisymmetric tensors are related to the unit vector of the constant magnetic field. The invariant forms in terms of the sum of the projectors of the tensors are also derived. When a Cartesian coordinate system is introduced, the invariant forms are easily reduced to the commonly used matrix representations. The invariant forms clearly show the effects of the constant magnetic field on the anisotropies of the media. Moreover, they effectuate and simplify the deduction of the general solutions of problems involving wave propagation and excitation in plasma and ferrite and thus facilitate interpretations of the final results.     

Decentered elliptical flattened Gaussian beam

A decentered flattened Gaussian beam (DEFGB), is defined by a tensor method. The propagation formula for a DEFGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a generalized decentered elliptical flattened Gaussian beam under certain condition. As an example application of the derived formula, the propagation characteristics of a DEFGB in free space are calculated and discussed. As another example we have studied the properties of superposition with radial array consisted by DEFGB.