The Boltzmann equation with a soft potential

The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ?f/?t+Lf=0 withf(ξ,t=0) given. The linear operatorL operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay forf, but in this paper it is shown thatf decays likee ^?λ t ^β with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.     

Analysis of a time fractional wave-like equation with the homotopy analysis method

The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when ?f=?g=−1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.     

The onset of convection in a couple stress fluid saturated porous layer using a thermal non-equilibrium model

The stability of a couple stress fluid saturated horizontal porous layer heated from below and cooled from above when the fluid and solid phases are not in local thermal equilibrium is investigated. The Darcy model is used for the momentum equation and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is employed to obtain the condition for the onset of convection. The effect of thermal non-equilibrium on the onset of convection is discussed. It is shown that the results of the thermal non-equilibrium Darcy model for the Newtonian fluid case can be recovered in the limit as couple stress parameter C→0. We also present asymptotic analysis for both small and large values of the inter phase heat transfer coefficient H. We found an excellent agreement between the exact solutions and asymptotic solutions when H is very small.     

Boltzmann'sH theorem and the Loschmidt and the Zermelo paradoxes

The Umkehreinwand of Loschmidt and the Wiederkehreinwand of Zermelo have been reexamined. The former paradox depends on the augument that for a dynamical system, upon the reversal of the velocities of all the molecules, theH function retraces its sequence of values so thatdH/dt will change its sign. The latter paradox depends on the argument that theH function returns infinitely close to its value after a Poincare' quasi-period and therefore cannot be decreasing all the time. While the main contention of the two paradoxes is correct, that theH theorem is inconsistent with classical dynamical laws, the arguments there can be considerably simplified and the “paradoxes” answered more directly. If the distribution functionf(q _K ,p _K ,t) is governed by an equation which is time-reversal invariant (such as the Liouville equation for a closed dynamical system), then it can be shown immediately thatdH/dt=0,H=cons. In this case, both paradoxes disappear, but together with them, thedH/dt<0 part of theH theorem also has disappeared, i.e., there is no second law of thermodynamics. Iff(q _K ,p _K ,t) is governed by an equation which is not time-reversal invariant (such as the Boltzmann equation, or the Master Equation for Markovian processes), then (1) there is no argument forf andH(t) to retrace their sequence of values upon the reversal of all the velocities of the system, (2) there is no quasiperiod in whichf andH(t) return to their earlier values. In this case, both paradoxes disappear also, but then one must go beyond classical dynamics in order to maintain theH theorem.     

The general exact bijective continuous solution of Feigenbaum's functional equation

Solutionsf:?→? of Feigenbaum's functional equationf(f(x))=α^?1 f(αx), where α≠0 is a fixed real number, account for many of the fascinating properties of the behaviour of successive iterates of (one parameter families of) nonlinear maps. In connection with the phenomenon of intermittency, interesting families of exact solutions have recently been found (for α>0). These solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations. In this paper, thegeneral exact continuous bijective solution is found for any α≠0, positive or negative. In particular, it is shown that, forany α≠0, there are solutions which areinequivalent to translations. And it is shown that bijective solutions equivalent to translations exist only when 0<α<1. These results considerably enlarge the stock of available exact solutions of Feigenbaum's equation.     

Kantowski-Sachs Universe Filled with Perfect Fluid in f(R,T) Theory of Gravity

The exact solutions of the field equations in respect of Kantowski-Sachs universe filled with perfect fluid in the framework of f(R,T) theory of gravity (Harko et al. in Phys. Rev. D 84:024020, 2011) is derived. A cosmological model with an appropriate choice of the function f(T) is constructed. The physical behavior of the cosmological model is studied. Some important features of astrophysical phenomena, like Hubble’s parameter H(z), luminosity distance (d _L ) and distance modulus μ(z) with red-shift are also discussed.     

Conformal anisotropic relativistic charged fluid spheres with a linear equation of state

We obtain two new families of compact solutions for a spherically symmetric distribution of matter consisting of an electrically charged anisotropic fluid sphere joined to the Reissner–Nordstrom static solution through a zero pressure surface. The static inner region also admits a one parameter group of conformal motions. First, to study the effect of the anisotropy in the sense of the pressures of the charged fluid, besides assuming a linear equation of state to hold for the fluid, we consider the tangential pressure p _⊥ to be proportional to the radial pressure p _r , the proportionality factor C measuring the grade of anisotropy. We analyze the resulting charge distribution and the features of the obtained family of solutions. These families of solutions reproduce for the value C=1, the conformal isotropic solution for quark stars, previously obtained by Mak and Harko. The second family of solutions is obtained assuming the electrical charge inside the sphere to be a known function of the radial coordinate. The allowed values of the parameters pertained to these solutions are constrained by the physical conditions imposed. We study the effect of anisotropy in the allowed compactness ratios and in the values of the charge. The Glazer’s pulsation equation for isotropic charged spheres is extended to the case of anisotropic and charged fluid spheres in order to study the behavior of the solutions under linear adiabatic radial oscillations. These solutions could model some stage of the evolution of strange quark matter fluid stars.     

Universe Filled with Dark Energy (DE) from a Wet Dark Fluid (WDF) in f(R,T) Gravity

We studied the Bianchi type-V universe filled with dark energy (DE) from a wet dark fluid (WDF) in the framework of f(R,T) gravity (Harko in Phys. Rev. D 84:024020, 2011). A new equation of state for the dark energy (DE) component of the universe has been used. It is modeled on the equation of state p=w(ρ?ρ ^?) which can be describing a liquid, for example water. The exact solutions to the corresponding field equations are obtained for exponential and power-law volumetric expansion. It is observed that the universe can approach to isotropy monotonically even in the presence of wet dark fluid. Also we have discussed the well-known astrophysical phenomena, namely the look-back time, proper distance, the luminosity distance and angular diameter distance with redshift.     

Quasi-periodic solutions of the sine-Gordon equation

Two-phase quasi-periodic solutions of the sine-Gordon equation are discussed. It is shown that the solutions expressed by the multidimensional θ-function form a broader class than the well-known factorised solutions 4 arctan [f(u)g(v)]. Expressions describing the spatial and temporal period are also given.     

Separation of Spin 0, 1/2, 1 Field Equations in Lemaître-Tolman-Bondi Cosmological Model with Cosmological Constant

The problem of variable separation of the scalar field equation is approached within the Lemaître-Tolman-Bondi (LTB) cosmological model with cosmological constant Λ. Parametric solutions of the cosmological Newton-like equation of the model are preliminary determined that result factorized in the parameter and in the radial dependence. The result holds on a sufficient condition that relates the two arbitrary integration functions of the model. The condition is of the same type of the one that ensures, in absence of cosmological term, the separability of the spin field equations for spin 0, 1/2, 1. It is then shown that the scalar field equation results automatically separable in the class of LTB models determined. The separated radial equation results independent of Λ, while the separated time equation strictly depends on Λ. The separability of the field equations is then checked to hold, in the same context, for spinor field equation of spin 1/2 and spin 1.     

Spatially Homogeneous Rotating Solution in f(R) Gravity and Its Energy Contents

In this paper, the metric approach of f(R) theory of gravity is used to investigate the exact vacuum solutions of spatially homogeneous rotating spacetimes. For this purpose, R is replaced by f(R) in the standard Einstein-Hilbert action and the set of modified Einstein field equations reduce to a single equation. We adopt the assumption of constant Ricci scalar which maybe zero or non-zero. Moreover, the energy density of the non-trivial solution has been evaluated by using the generalized Landau-Lifshitz energy-momentum complex in the perspective of f(R) gravity for some appropriate f(R) model, which turns out to be a constant quantity.     

Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer by hydromagnetic three-dimensional free convection over a permeable stretching surface with radiation

The thermal-diffusion and diffusion-thermo effects on the heat and mass transfer characteristics of free convection past a continuously stretching permeable surface in the presence of magnetic field, blowing/suction and radiation are studied. The fluid viscosity is assumed to vary with temperature. The resulting, governing three-dimensional equations are transformed using a similarity transformation and then solved numerically by the shooting method. Comparison with previously published work is performed and full agreement is obtained. A parametric study showing the effects of variable viscosity parameter β, magnetic parameter M, Dufour number Df, Soret number Sr, radiation parameter R and blowing/suction parameter f₀ on the velocity, temperature, and concentration field of a hydrogen-air mixture as a non-chemical reacting fluid pair, as well as the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number is carried out. These are illustrated graphically and in tabular form to depict special features of the solutions.     

On the cosmological viability of the Hu-Sawicki type modified induced gravity

It has been shown recently that the normal branch of a DGP braneworld scenario self-accelerates if the induced gravity on the brane is modified in the spirit of f(R) modified gravity. Within this viewpoint, we investigate cosmological viability of the Hu-Sawicki type modified induced gravity. Firstly, we present a dynamical system analysis of a general f(R)-DGP model. We show that in the phase space of the model, there exist three standard critical points; one of which is a de Sitter point corresponding to accelerating phase of the universe expansion. The stability of this point depends on the effective equation of state parameter of the curvature fluid. If we consider the curvature fluid to be a canonical scalar field in the equivalent scalar-tensor theory, the mentioned de Sitter phase is unstable, otherwise it is an attractor, stable phase. We show that the effective equation of state parameter of the model realizes an effective phantom-like behavior. A cosmographic analysis shows that this model, which admits a stable de Sitter phase in its expansion history, is a cosmologically viable scenario.     

On exact spatially nonuniform solutions of the Boltzmann equation in the generalized Kac model

It is shown that the Boltzmann equation in the generalized Kac model can be cast into a nonlinear differential equation of the second order for a certain class of functions f(t, x, v). Some explicit solutions of this equation are pointed out.     

Flows of a generalized second grade fluid in a cylinder due to a velocity shock

Unsteady axial flows of second grade fluids with generalized fractional constitutive equation in a circular cylinder are studied. Flows are generated by a time-dependent pressure gradient in the axial direction, an external magnetic field perpendicular on the flow direction and by the cylinder motion. Two different problems are analyzed; one in which the cylinder velocity supports a shock at the instant t = 0 and another in which the cylinder motion is a translation with time-dependent velocity along the axis of cylinder. The generalized fractional constitutive equation of second grade fluid is described by the Caputo time-fractional derivative. Analytical solutions for the velocity field are obtained by using the Laplace transform with respect to time variable and the finite Hankel transform of order zero with respect to the radial coordinate. The influence of the fractional parameter of Caputo derivative on the fluid velocity has been studied by numerical simulations and graphical illustrations. It is found that the fractional fluid flows are faster than the ordinary second grade fluid.     

Strong energy condition and the repulsive character of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

The Raychaudhuri equation enables to examine the whole spacetime structure without specific solutions of Einstein’s equations, playing a central role for the understanding of the gravitational interaction in cosmology. In General Relativity, without considering a cosmological constant, a non-positive contribution in the Raychaudhuri equation is usually interpreted as the manifestation of the attractive character of gravity. In this case, particular energy conditions—indeed the strong energy condition—must be assumed in order to guarantee the attractive character. In the context of f(R) gravity, however, even assuming the standard energy conditions one may have a positive contribution to the Raychaudhuri equation. Besides providing a simple way to explain the observed cosmic acceleration, this fact opens the possibility of a repulsive character of this kind of gravity. In order to discuss physical bounds on f(R) models, we address the attractive/non-attractive character of f(R) gravity considering the Raychaudhuri equation and assuming the strong energy condition along with recent estimates of the cosmographic parameters.     

Fluid–fluid phase equilibria in disordered porous media. Nonadditive hard sphere mixture

We report a computer simulation and integral equation study of fluid–fluid phase equilibria of nonadditive hard sphere binary mixture adsorbed in disordered hard sphere matrix. The mixture exhibits phase separation with critical density ρ_c^f lower than its bulk counterpart. It is found that ρ_c^f decreases with increasing both porosity and nonadditivity parameter.     

The spectrum of multi-flavor QCD2 and the non-abelian Schwinger equation

Massless QCD₂ is dominated by classical configurations in the larre-N_f limit. We use this observation to study the theory by finding solutions to equations of motion, which are the non-abelian generalization of the Schwinger equation. We find that the spectrum consists of massive mesons with M² = e²N_f/2π, which correspond to abelian solutions. We generalize previously discovered non-abelian solutions and discuss their interpretation. We prove a no-go theorem ruling out the existence of soliton solutions. Thus the semi-classical approximation shows no baryons in the case of massless quarks, a result derived before in the strong-coupling limit only.     

Phase space analysis of interacting dark energy in f(T) cosmology

In this paper, we examine the interacting dark energy model in f(T) cosmology. We assume dark energy as a perfect fluid and choose a specific cosmologically viable form f(T) = ????T. We show that there is one attractor solution to the dynamical equation of f(T) Friedmann equations. Further we investigate the stability in phase space for a general f(T) model with two interacting fluids. By studying the local stability near the critical points, we show that the critical points lie on the sheet u* = (c ? 1)v* in the phase space, spanned by coordinates (u, v, ??, T). From this critical sheet, we conclude that the coupling between the dark energy and matter c ?? (?2, 0).     

Effects of CDTT model on the stability of spherical collapse in Palatini f(R) gravity

The objective of this paper is to study the stability of an adiabatic anisotropic collapsing sphere in the context of Palatini f(R) gravity. In this framework, we construct the collapse equation with the help of the contracted Bianchi identities of the effective as well as the usual energy-momentum tensor. The perturbation scheme is applied on the fluid variables which accordingly cause a perturbation on the Ricci scalar. We explore the instability ranges in the Newtonian and post-Newtonian regimes. It is concluded that the stability of the star is governed by adiabatic index Γ ₁, which depends on the energy density profile, anisotropic pressure and dark source terms of the chosen f(R) model. We also explore our results when f(R)→R.