abstract

The first half of this review describes the development in mathematical models of Brownian motion after Einstein’s and Smoluchowski’s seminal papers and current applications to optical tweezers. This instrument of choice among single-molecule biophysicists is also an instrument of such precision that it requires an understanding of Brownian motion beyond Einstein’s and Smoluchowski’s for its calibration, and can measure effects not present in their theories. This is illustrated with some applications, current and potential. It is also shown how addition of a controlled forced motion on the nano-scale of the thermal motion of the tweezed object can improve the calibration of the instrument in general, and make calibration possible also in complex surroundings. The second half of the present review, starting with Sect. 9, describes the co-evolution of biological motility models with models of Brownian motion, including recent results for how to derive cell-type-specific motility models from experimental cell trajectories.

abstract

The asymptotic behavior of the solutions of some nonlinear variational inequalities with highly oscillating coefficients modeling chemical reactive flows through the exterior of a domain containing periodically distributed reactive solid obstacles, with period \(\varepsilon \), is analyzed. In this kind of boundary-value problems there are involved two distinct sources of oscillations, one coming from the geometrical structure of the domain and the other from the fact that the medium is heterogeneous. We focus on the only case in which a real interaction between both these sources appears, i.e. the case in which the obstacles are of the so-called critical size and we prove that the solution of such a boundary-value problem converges to the solution of a new problem, associated to an operator which is the sum of a standard homogenized one and extra zero order terms coming from the geometry and the nonlinearity of the problem.

abstract

As a continuation of our previously published work, the dynamic phase transitions are studied, within a mean-field approach, in the kinetic spin-3/2 Blume–Emery–Griffiths (BEG) model in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The dynamic phase transitions (DPTs) are obtained and the phase diagrams are constructed in two different planes, namely reduced temperature (\(T\)) and biquadratic interaction (\(k\)), (\(T,\,k\)) plane where found seven fundamental types of phase diagrams for both positive and negative values of crystal-field interaction (\(d\)) and magnetic field amplitude (\(h\)), and also (\(T,\ d\)) plane in which obtained ten distinct topologies for different values of \(k\) and \(h\). Phase diagrams exhibit one or two dynamic tricritical points, a dynamic double critical end point, and besides a disordered and two ordered phases, seven coexistence phase regions exist in which occurring of all these strongly depend on the values of \(k,\ d\) and \(h\).

abstract

We present a mechanism for controlling directed transport of particles in inertia ratchets. We study a parameter regime where two attractors — each transporting particles in different directions co-exist in phase space; and show that a proper control of direction of transport can be achieved by using adaptive backstepping based synchronisation technique.

abstract

Using the Mathematica program we calculate numerically the difference of the diagonal matrix elements of the time dependent effective Hamiltonian for the neutral \(K\) meson complex. We consider the exactly solvable neutral \(K\) meson model based on the one-pole approximation for the mass density. The so-called Khalfin’s Theorem is numerically examined. Some characteristic parameters for this system are also calculated. The results of all calculations are presented in the graphical form. The calculations are made assuming the total system is CPT-invariant and CP-noninvariant.

abstract

We study chaotic inflation with a quadratic potential in all dimensions. The slow-roll parameters, the spectral indices of scalar and tensor perturbations and also their running have been calculated in all dimensions.

abstract

This study is purposed to elaborate the problem of energy and momentum distribution of string and domain wall in the context of two different approaches of gravity such as general relativity and tele-parallel gravity. In this connection, we have calculated energy-momentum of domain wall and string by using the energy-momentum definitions of Einstein, Bergmann–Thomson and Landau–Lifshitz in general relativity and tele-parallel gravity. In our analysis we obtained that (i) general relativity and tele-parallel gravity are equivalent theories for string and domain wall (ii) different energy-momentum complexes do not provide the same energy and momentum densities neither in general relativity nor in tele-parallel gravity for domain wall.

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We find a class of exact solutions of differentially rotating dust in the framework of General Relativity. There exist asymptotically flat spacetimes of the flow with positive mass function that for radii sufficiently large is monotone and tends to zero at infinity. Some of the spacetimes may have non-vanishing total angular momentum.

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A concise description of the curvature structures borne by the Infeld-van der Waerden \(\gamma \varepsilon \)-formalisms is provided. The derivation of the wave equations that control the propagation of gravitons and geometric photons in generally relativistic space-times is then carried out explicitly.

abstract

Three new models with V-shaped field potentials \(U\) are considered: a complex scalar field \(X\) in \(1+1\) dimensions with \(U(X)=|\,X\,|\), a real scalar field \({\mit \Phi }\) in 2+1 dimensions with \(U({\mit \Phi }) = |{\mit \Phi }|\), and a real scalar field \( \varphi \) in \(1+1\) dimensions with \(U(\varphi ) = \varphi {\mit \Theta }(\varphi )\) where \({\mit \Theta }\) is the step function. Several explicit, self-similar solutions are found. They describe interesting dynamical processes, for example, “freezing” a string in a static configuration.

abstract

We calculate the mass and width of the \(\sigma \)-meson in nuclear medium by considering that it couples to two virtual pions and to a pair of nucleon–antinucleon states and to particle–hole states. The mass is calculated by using the spectral function in the Walecka model, finding that it is about 520 MeV. In addition, we have obtained the value of 700 MeV for the width of its spectral function, showing that it has increased respect to that in vacuum. We find that there is a reduction in the mass value compared with that in vacuum. This result is consistent with those reported by other authors who have used different models predicting a decreasing of the mass as a function of the density.

abstract

We discuss effects related to the fact that the final state particles of the reaction \(\mathrm {e} ^+e^- \to t \bar t H\) are actually produced and they decay off mass shell. For the intermediate mass Higgs boson, which decays preferably into a \(b\bar b\)-quark pair, the reaction will be observed through reactions with 8 fermions in the final state. Such reactions, already in the lowest order of the standard model, receive contributions typically from a few dozen thousands of the Feynman diagrams, the vast majority of which constitute background to the signal of associated production of the top quark pair and Higgs boson. In order to illustrate pure off mass shell effects we neglect the background contributions and compare the “signal” cross section with the cross section in the narrow width approximation for \(e^+ e^- \to b u \bar {d} \;\bar b \mu ^- \bar {\nu }_{\mu } \; b \bar {b}\), which is one of possible detection channels of the associated production of the top quark pair and Higgs boson at a linear collider.

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We present a framework for fitting the parton density functions obtained from the Monte Carlo solutions of the QCD evolution equations of various types to the \(F_2\) data. To speed up the fitting it is enough to perform the Monte Carlo simulation only once. The actual parton density functions as functions of the fitting parameters are then reconstructed by fast one-dimensional numerical integration. Such a tool is necessary in order to determine initial values for parton density functions in case of non-DGLAP Monte Carlo evolutions.

abstract

The four reggeized gluon states for non-vanishing Lorentz conformal spin \(n_h\) are considered. To calculate their spectrum the \(Q\)-Baxter method is used. As a result we describe normalizable trajectory-like states, which form continuous spectrum, as well as discrete point-like solutions, which turn out to be non-normalizable. The point-like solutions exist due to symmetry of the Casimir operator where conformal weights \((h,\overline {h})\!\rightarrow \!(h,1\!-\!\overline {h})\).

abstract

Dirac-like linearisation of \({\mathbf x}^2+{\mathbf p}^2\) with noncommuting position and momentum variables leads to the representation of the standard U\((1)\otimes \) SU(3) symmetry of the three-dimensional harmonic oscillator in the relevant Clifford algebra and the emergence of a formula which we previously proposed to identify with the Gell-Mann–Nishijima–Glashow relation between charge, third component of weak isospin and weak hypercharge. This matrix representation exhibits features not present in the standard treatment of harmonic oscillator. We show that these features, strictly corresponding to the structure of a single quark–lepton generation in the Standard Model, may be understood from the point of view of specific O(6) phase-space transformations, which go beyond U\((1)\otimes \) SU(3), and modify standard canonical commutation relations. It is demonstrated that the whole structure of a single quark–lepton generation corresponds to assuming that the imaginary unit appearing in the canonical commutation relations may acquire an additional “\(+\)” or “\(-\)” sign separately for each of the three directions.

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Presence of intermittency and Levy stability in 14.5A GeV/\(c\) \(^{28}\)Si nucleus collisions is investigated using Monte Carlo approach. Our experimental data reveals the presence of intermittency. Levy stability analysis is carried out; values of Levy index for the experimental and simulated data are found to be \(1.511 \pm 0.061\) and \(1.491 \pm 0.041\), respectively. An attempt is also made to obtain Renyi Dimensions also called self-similar dimensions, \(D_q\), and multifractal spectrum, \(f(\alpha )\). The Renyi dimensions, \(D_q\), are observed to decrease with increasing order of the moment, \(q\). The self-similar multifractal spectrum is found to be a convex curve with a maximum around \(q=0\). Simulation technique is used and an analytical continuation is applied to find the multifractal spectrum for the fractional values of \(q\).

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Small momentum transfer elastic pion–proton cross-section at high energies is calculated assuming the proton is composed of two constituents, a quark and a diquark. We find that it is possible to fit very precisely the data when (i) the pion acts as a single entity (no constituent quark structure) and (ii) the diquark is rather large, comparable to the size of the proton.

abstract

The equation of motion of massive spherical shell expanding in the field of its own gravitational potential has been solved within the special relativity mechanics, assuming fixed total energy of such system. The initial velocity of such “shell-universe” is always finite and equal to the velocity of light. When the total energy is less than the rest mass energy of the shell, the expansion terminates in time and the shell collapses, while otherwise it expands indefinitely; at long times it resembles the Friedman model of universe. For zero total energy the shell radius goes in time \(t\) as \(\sin ({\mit \Omega }\,t)\), where the “frequency” \({\mit \Omega }\) is proportional to the rest mass of the shell. A given “lifetime” of the expansion-terminated shell universe can be achieved in two ways: “grand” or “small” expansion scenarios. Another version of the model, relying explicitly on the energy–gravitational-mass equivalence, leads to similar (but not identical) predictions. The predictions of the model are compared with the predictions of the GRT “dust shell” model. Possible impact of this special relativity model of expanding universe on its general relativity counterpart is suggested.