Random Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005     

Genesis of Dark Energy: Dark Energy as Consequence of Release and Two-Stage Tracking of Cosmological Nuclear Energy

Recent observations on Type-Ia supernovae and low density (Ω _m =0.3) measurement of matter including dark matter suggest that the present-day universe consists mainly of repulsive-gravity type ‘exotic matter’ with negative-pressure often said ‘dark energy’ (Ω _x =0.7). But the nature of dark energy is mysterious and its puzzling questions, such as why, how, where and when about the dark energy, are intriguing. In the present paper the authors attempt to answer these questions while making an effort to reveal the genesis of dark energy and suggest that ‘the cosmological nuclear binding energy liberated during primordial nucleo-synthesis remains trapped for a long time and then is released free which manifests itself as dark energy in the universe’. It is also explained why for dark energy the parameter w=-\frac23w=-\frac{2}{3} . Noting that w=1 for stiff matter and w=\frac13w=\frac{1}{3} for radiation; w=-\frac23w=-\frac{2}{3} is for dark energy because “−1” is due to ‘deficiency of stiff-nuclear-matter’ and that this binding energy is ultimately released as ‘radiation’ contributing “ +\frac13+\frac{1}{3} ”, making w=-1+\frac13=-\frac23w=-1+\frac{1}{3}=-\frac{2}{3} . When dark energy is released free at Z=80, w=-\frac23w=-\frac{2}{3} . But as on present day at Z=0 when the radiation-strength-fraction (δ), has diminished to δ→0, the w=-1+d\frac13=-1w=-1+\delta\frac{1}{3}=-1 . This, almost solves the dark-energy mystery of negative pressure and repulsive-gravity. The proposed theory makes several estimates/predictions which agree reasonably well with the astrophysical constraints and observations. Though there are many candidate-theories, the proposed model of this paper presents an entirely new approach (cosmological nuclear energy) as a possible candidate for dark energy.     

On the collective excitations and the existence of the gap in the degenerate Bose gas spectrum

Themodel of degenerate weakly nonideal Bose gas is considered without the assumption on the C-number nature of creation and annihilation operators of particles in the zero-momentum state. It was shown that the “density-density” Green’s function pole defines the Bogolyubov spectrum of phonon-roton-type collective excitations. The single-particle excitation spectrum is characterized by the gap whose value is defined by the particle density in the “condensate”.     

Discrete Spectrum of the Deficit Angle and the Differential Structure of a Cosmic String

Differential properties of Klein-Gordon and electromagnetic fields on the space-time of a straight cosmic string are studied with the help of methods of the differential space theory. It is shown that these fields are smooth in the interior of the cosmic string space-time and that they loose this property at the singular boundary except for the cosmic string space-times with the following deficit angles: Δ=2π(1−1/n), n=1,2,… . A connection between smoothness of fields at the conical singularity and the scalar and electromagnetic conical bremsstrahlung is discussed. It is also argued that the smoothness assumption of fields at the singularity is equivalent to the Aliev and Gal’tsov “quantization” condition leading to the above mentioned discrete spectrum of the deficit angle.     

Early Universe in Scalar-Tensor Theory

We consider the flat Robertson–Walker model in scalar-tensor theory proposed by Lau and Prokhovnik. In this model, the field equations are solved by using “gamma-law” form of equation of state p=(γ−1)ρ, where the adiabatic parameter ‘gamma’ (γ) varies continuously as the universe expands. Our aim is to study how the adiabatic parameter γ should vary so that in the course of its evolution the universe goes through a transition from an inflationary to a radiation-dominated phase. A unified one parameter function of γ has been considered to describe the two early phases of evolution of universe. The solutions show the power-law expansion and cosmological constant is found to be positive and decreasing function of cosmic time. The solutions are compatible with the Dirac’s large number hypothesis. The deceleration parameter has been presented in a unified manner in terms of scale factor, which describes the inflation of the model. The nature of singularity and the physical properties have been discussed in details.     

Against ‘Realism’

We examine the prevalent use of the phrase “local realism” in the context of Bell’s Theorem and associated experiments, with a focus on the question: what exactly is the ‘realism’ in ‘local realism’ supposed to mean? Carefully surveying several possible meanings, we argue that all of them are flawed in one way or another as attempts to point out a second premise (in addition to locality) on which the Bell inequalities rest, and (hence) which might be rejected in the face of empirical data violating the inequalities. We thus suggest that the phrase ‘local realism’ should be banned from future discussions of these issues, and urge physicists to revisit the foundational questions behind Bell’s Theorem.     

On Visibility in the Afshar Two-Slit Experiment

A modified version of Young’s experiment by Shahriar Afshar indirectly reveals the presence of a fully articulated interference pattern prior to the post-selection of a particle in a “which-slit” basis. While this experiment does not constitute a violation of Bohr’s Complementarity Principle as claimed by Afshar, both he and many of his critics incorrectly assume that a commonly used relationship between visibility parameter V and “which-way” parameter K has crucial relevance to his experiment. It is argued here that this relationship does not apply to this experimental situation and that it is wrong to make any use of it in support of claims for or against the bearing of this experiment on Complementarity.     

A global version of the quantum duality principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domainR and for each primepεR we establish an “inner” Galois’ correspondence on the categoryHA of torsionless Hopf algebras overR, using two functors (fromHA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulop, respectively (i.e., they are“quantum function algebras” (=QFA) and“quantum universal enveloping algebras” (=QUEA), atp, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebraH over a fieldk: apply the functors tok[ν] ⊗k H forp=ν. A relevant example occurring in quantum electro-dynamics is studied in some detail. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001     

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one-and two-dimensional substrates focusing our analysis on the probability distribution function P(M,L) of the number M of maximal points (i.e., local “peaks”) of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one-dimensional ballistic growth process in the steady state can be mapped onto “rise-and-descent” sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. [G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004)], that different characteristics of “rise-and-descent” patterns in random permutations can be interpreted in terms of a certain continuous-space Hammersley-type process. For one-dimensional system we compute P(M,L) exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long-ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as L → ∞.     

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.     

From two mesons and (V-A) weak currents to the standard model of quark and lepton interactions

This retrospective paper traces the conceptual evolution of two theories in which the author was involved—the two-meson theory (with H A Bethe) in 1947 and the universal (V-A) theory of weak interactions (with E C G Sudarshan) in 1957—into the present-day standard model of particle interactions. Part 1 is entitled ‘From the pion to QCD and pseudo-Goldstone bosons” and Part 2 “From the muon and neutrino to QFD and chiral anomalies’. Adapted from two lectures delivered at the University of Rochester in October, 1987.     

Cellular Biology in Terms of Stochastic Nonlinear Biochemical Dynamics: Emergent Properties,Isogenetic Variations and Chemical System Inheritability

Based on a stochastic, nonlinear, open biochemical reaction system perspective, we present an analytical theory for cellular biochemical processes. The chemical master equation (CME) approach provides a unifying mathematical framework for cellular modeling. We apply this theory to both self-regulating gene networks and phosphorylation-dephosphorylation signaling modules with feedbacks. Two types of bistability are illustrated in mesoscopic biochemical systems: one that has a macroscopic, deterministic counterpart and another that does not. In certain cases, the latter stochastic bistability is shown to be a “ghost” of the extinction phenomenon. We argue the thermal fluctuations inherent in molecular processes do not disappear in mesoscopic cell-sized nonlinear systems; rather they manifest themselves as isogenetic variations on a different time scale. Isogenetic biochemical variations in terms of the stochastic attractors can have extremely long lifetime. Transitions among discrete stochastic attractors spend most of the time in “waiting”, exhibit punctuated equilibria. It can be naturally passed to “daughter cells” via a simple growth and division process. The CME system follows a set of nonequilibrium thermodynamic laws that include non-increasing free energy F(t) with external energy drive Q _hk ≥0, and total entropy production rate e _p =−dF/dt+Q _hk ≥0. In the thermodynamic limit, with a system’s size being infinitely large, the nonlinear bistability in the CME exhibits many of the characteristics of macroscopic equilibrium phase transition.     

Effect of free electrons on growth of “cancers” in mixed electronic-ionic conductors

The reasons for the anomalously rapid (for solids) growth of “cancers” (filamentary crystals, hills, pores, and other formations) on the surface of samples of superionic conductors with mixed electronic-ionic conductivity (of the type Cu_2−x Se and Ag₂Te) are discussed. The effects are attributed to the acceleration of mobile ions and activated ions in the “core” of the superionic by free electrons in the joint “chemical” diffusion of ions and electrons in the samples. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 97–100 (25 July 1996)     

Extended theories of gravity and their cosmological and astrophysical applications

Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large scale structure and cosmic dynamics. The emerging picture is a spatially flat, homogeneous Universe undergoing the today observed accelerated phase. Despite of the good quality of astrophysical surveys, commonly addressed as Precision Cosmology, the nature and the nurture of dark energy and dark matter, which should constitute the bulk of cosmological matter-energy, are still unknown. Furthermore, up to now, no experimental evidence has been found, at fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of General Relativity in its simplest formulation (a linear theory in the Ricci scalar R, minimally coupled to the standard perfect fluid matter) and claiming for the “correct” theory of gravity as that derived by matching the largest number of observational data, without imposing any theory a priori. As a working hypothesis, accelerating behavior of cosmic fluid, large scale structure, potential of galaxy clusters, rotation curves of spiral galaxies could be reproduced by means of extending the standard theory of General Relativity. In other words, gravity could acts in different ways at different scales and the above “shortcomings” could be due to incorrect extrapolations of the Einstein gravity, actually tested at short scales and low energy regimes. After a survey of what is intended for Extended Theories of Gravity in the so called “metric” and “Palatini” approaches, we discuss some cosmological and astrophysical applications where the issues related to the dark components are addressed by enlarging the Einstein theory to more general f (R) Lagrangians, where f (R) is a generic function of Ricci scalar R, not assumed simply linear. Obviously, this is not the final answer to the problem of “dark-components” but it can be considered as an operative scheme whose aim is to avoid the addition of unknown exotic ingredients to the cosmic pie.     

Experimental study of the energy gap in the spin-wave spectrum in Co/Pd multilayer films

The energy gap appearing in the spin-wave spectrum as a result of Bragg scattering by the modulation period q=2π/(d ₁+d ₂) of a one-dimensional superlattice is observed by the method of spin-wave resonance in Co/Pd multilayer films. It is shown that this gap is asymmetric: The “positive” deviation is from two to three times greater than the “negative” deviation. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 938–941 (25 June 1996)     

Noninertial Observers in Special Relativity and Clock Synchronization Debates

Selleri's arguments that a consideration of noninertial reference frames in the framework of special relativity identify “absolute simultaneity” as being “Nature's choice of synchronization” are considered. In the case of rectilinearly accelerating rockets, it is argued by considering two rockets which maintain a fixed proper separation rather than a fixed separation relative to the inertial frame in which they start from rest, that what seems the most “natural” choice for a simultaneity convention is problem-dependent and that Einstein's definition is the most “natural” (though still conventional) choice in this case. In addition, the supposed problems special relativity has with treating a rotating disk, namely how a pulse of light traveling around the circumference of the disk can have a local speed of light equal to c everywhere but a global speed not equal to c, and how coordinate transformations to the disk can give the Lorentz transformations in the limit of large disk radius but small angular velocity, are addressed. It is shown that the theory of Fermi frames solves both of these problems. It is also argued that the question of defining simultaneity relative to a uniformly rotating disk does not need to be resolved in order to resolve Ehrenfest's paradox.     

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.     

The Migdal jump in the nucleon momentum distribution in nuclear matter is determined by the spin-isospin response function

The renormalization factor Z of the single-particle Green’s function in nuclear matter, equal to the jump in the nucleon momentum distribution (“Migdal jump”), is expressed in terms of the nuclear response function, using approximations that are realistic for nuclear physics. It is shown that the spin-isospin channel makes the dominant contribution to Z. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 7, 529–534 (10 October 1998)     

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.