Elementary excitations of biomembranes: Differential geometry of undulations in elastic surfaces

Biomembrane undulations are elementary excitations in the elastic surfaces of cells and vesicles. As such they can provide surprising insights into the mechanical processes that shape and stabilize biomembranes. We explain how naturally these undulations can be described by classical differential geometry. In particular, we apply the analytical formalism of differential-geometric calculus to the surfaces generated by a cell membrane and underlying cytoskeleton. After a short derivation of the energy due to a membrane's elasticity, we show how undulations arise as elementary excitations originating from the second derivative of an energy functional. Furthermore, we expound the efficiency of classical differential-geometric formalism to understand the effect of differential operators that characterize processes involved in membrane physics. As an introduction to concepts the paper is self-contained and rarely exceeds calculus level.     

A Proposal for a Bohmian Ontology of Quantum Gravity

The paper shows how the Bohmian approach to quantum physics can be applied to develop a clear and coherent ontology of non-perturbative quantum gravity. We suggest retaining discrete objects as the primitive ontology also when it comes to a quantum theory of space-time and therefore focus on loop quantum gravity. We conceive atoms of space, represented in terms of nodes linked by edges in a graph, as the primitive ontology of the theory and show how a non-local law in which a universal and stationary wave-function figures can provide an order of configurations of such atoms of space such that the classical space-time of general relativity is approximated. Although there is as yet no fully worked out physical theory of quantum gravity, we regard the Bohmian approach as setting up a standard that proposals for a serious ontology in this field should meet and as opening up a route for fruitful physical and mathematical investigations.     

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons. The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent. A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.     

Geometric calculus: A new computational tool for Riemannian geometry

We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus.     

Fractional power-law spatial dispersion in electrodynamics

Electric fields in non-local media with power-law spatial dispersion are discussed. Equations involving a fractional Laplacian in the Riesz form that describe the electric fields in such non-local media are studied. The generalizations of Coulomb’s law and Debye’s screening for power-law non-local media are characterized. We consider simple models with anomalous behavior of plasma-like media with power-law spatial dispersions. The suggested fractional differential models for these plasma-like media are discussed to describe non-local properties of power-law type.     

Synthetic Vector Analysis

Classical vector analysis is rife with geometric and physical ideas, but appears precarious from a modern viewpoint of pure mathematics. Modern vector analysis with differential forms is surely up to the contemporary standard of mathematical rigor, but geometric ideas are completely lost in the bulk of dull calculations. The main goal in this paper is to show that synthetic differential geometry, which has replenished differential geometry with nilpotent infinitesimals, can mathematically sanitize classical vector analysis by eradicating its total confusion between approximate calculations and infinitesimal calculations, thereby helping it retrieve mathematical rigor.     

Fractional Dynamics from Einstein Gravity,General Solutions,and Black Holes

We study the fractional gravity for spacetimes with non-integer fractional derivatives. Our constructions are based on a formalism with the fractional Caputo derivative and integral calculus adapted to nonholonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange–Finsler and Hamilton–Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, ellipsoid gravitational configurations and imbedding of such objects in solitonic backgrounds.     

The Fractional Kinetic Einstein-Vlasov System and its Implications in Bianchi Spacetimes Geometry

The main purpose of this work is to introduce the basic concepts and global properties of the fractional Einstein-Vlasov equation based on the fractional calculus of variations, mainly the fractional actionlike variational approach. We believe that kinetic theory in non-curved spacetimes is fundamental to a good understanding of kinetic theory in general relativity. Besides, the fractional calculus of variations has proved recently to be an important mathematical field of research which has been applied successfully to a broad range of physical and mathematical researches. We expect therefore that the merge of both fields will bring some new insights to general relativity and accordingly to its cosmological and astrophysical implications. Based on the new fractional settings, some cosmological applications are discussed in this work mainly within the aspects of Bianchi spacetimes geometry.     

A symmetry principle for topological quantum order

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev’s Toric code model and Wen’s plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.     

A 3D non-local density functional theory for any pore geometry

A general framework of classical non-local density functional theory (NLDFT) is presented, in order to consider the adsorption of spherical molecules in porous materials of any geometry. Fluid-fluid interactions and fluid-solid interactions can be repulsive or attractive. Some techniques that have been developed for the computation of weighted densities of hard-spheres are extended to attractive ones, in order to deal with an arbitrary pore geometry. This way, the computation method introduced in this work is validated by a comparison with analytical results for simple cases, and is directly applied to more complex systems. Density distributions depending on multi-dimensional effects are presented, and some radial distribution functions are recovered from NLDFT computations. Finally, the case of attractive continuous curved walls is detailed, which represents a large variety of real systems (e.g. micro and mesoporous silica, zeolites, carbonaceous nanoporous materials, etc.). With the new way of computation proposed, a general solution is presented, valid for any shape of continuous pore surface, by considering mathematical properties of discrete geometry due to the discretisation of the computational space with FFT computations.     

Quantization of contact manifolds and thermodynamics

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this ‘quantization’ describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.     

Quantum and braided group Riemannian geometry

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce. We then provide the corresponding quantum group and braided group formulations with the universal quantum differential calculus. We also give general constructions, for example, including quantum spheres and quantum planes.     

Dimensions and units in electrodynamics

We sketch the foundations of classical electrodynamics, in particular the transition that took place when Einstein, in 1915, succeeded to formulate general relativity. In 1916 Einstein demonstrated that, with a choice of suitable variables for the electromagnetic field, it is possible to put Maxwells equation into a form that is covariant under general coordinate transformations. This unfolded, by basic contributions of Kottler, Cartan, van Dantzig, Schouten & Dorgelo, Toupin & Truesdell, and Post, to what one may call premetric classical electrodynamics. This framework will be described shortly. An analysis is given of the physical dimensions involved in electrodynamics and subsequently the question of units addressed. It will be pointed out that these results are untouched by the generalization of classical to quantum electrodynamics (QED). We compare critically our results with those of L.B. Okun which he had presented at a recent conference.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.     

A hybrid method for the parallel computation of Green’s functions

Quantum transport models for nanodevices using the non-equilibrium Green’s function method require the repeated calculation of the block tridiagonal part of the Green’s and lesser Green’s function matrices. This problem is related to the calculation of the inverse of a sparse matrix. Because of the large number of times this calculation needs to be performed, this is computationally very expensive even on supercomputers. The classical approach is based on recurrence formulas which cannot be efficiently parallelized. This practically prevents the solution of large problems with hundreds of thousands of atoms. We propose new recurrences for a general class of sparse matrices to calculate Green’s and lesser Green’s function matrices which extend formulas derived by Takahashi and others. We show that these recurrences may lead to a dramatically reduced computational cost because they only require computing a small number of entries of the inverse matrix. Then, we propose a parallelization strategy for block tridiagonal matrices which involves a combination of Schur complement calculations and cyclic reduction. It achieves good scalability even on problems of modest size.     

Generalizing Planck’s distribution by using the Carati–Galgani model of molecular collisions

Classical systems of coupled harmonic oscillators are studied using the Carati–Galgani model. We investigate the consequences for Einstein’s conjecture by considering that the exchange of energy in molecular collisions follows the Lévy type statistics. We develop a generalization of Planck’s distribution admitting that there are analogous relations in the equilibrium quantum statistical mechanics of the relations found using the nonequilibrium classical statistical mechanics approach. The generalization of Planck’s law based on the nonextensive statistical mechanics formalism is compatible with our analysis.