First Order In Gradients Research Articles

Second order hydrodynamics based on effective kinetic theory and electromagnetic signals from QGP

We study the thermal dilepton and photon production from relativistic heavy ion collisions in presence of viscosities by employing the recently developed second order dissipative hydrodynamic formulation estimated within a quasiparticle description of thermal quantum chromo-dynamics (QCD) medium. The sensitivity of shear and bulk viscous pressures to the temperature dependence of relaxation time is analyzed within one- dimensional boost invariant expansion of quark gluon plasma (QGP). The dissipative corrections to the phase-space distribution functions up to first order in gradients are obtained from the Chapman–Enskog like iterative solution of effective Boltzmann equation in the relaxation time approximation. Thermal dilepton and photon production rates for QGP are calculated by employing this viscous modified distribution function. Yields of these particles are quantified for the longitudinal expansion of QGP with different temperature dependent relaxation times. Our analysis employing this second order hydrodynamic model indicates that the spectra of dileptons and photons get enhanced by both bulk and shear viscosities and are well behaved. Also, these particle yields are found to be sensitive to relaxation time. Further, we do a comparison of these particle spectra with a standard hydrodynamic formulation.

On hydrodynamics in the presense of strong external potential field

On the base of the Boltzmann kinetic equation, hydrodynamics of a dilute gas in the presence of the strong external potential field is investigated. First of all, a gravitational field is meant, because the consistent development of hydrodynamics in this environment is of great practical importance. In the present paper it is assumed that it is possible to neglect the influence of the field on the particle collisions. The study is based on the Chapman–Enskog method in a Bogolyubov’s formulation, which uses the idea of the functional hypothesis. Consideration is limited to steady gas states, which are subjected to a simpler experimental study. Chemical potential μ0 of the gas at the point where the external field has zero value and its temperature T are selected as the reduced description parameters of the system. In equilibrium, in the presence of the field, these values do not depend on the coordinates. It is assumed that in thehydrodynamic states T and μ0 are weakly dependent on the coordinates and therefore their gradients, considered on the scale of the free path length of the gas, are small. The kinetic equation, accounting for the functional hypothesis, gives an integro-differential equation for a gas distribution function at the hydrodynamic stage of evolution. This equation is solved in perturbation theory in gradients of T and μ0. The main approximation is analyzed for possibility of the system to be in a local equilibrium by means of comparing it with an equilibrium distribution function. Next, the distribution function is calculated in the first approximation in gradients and it is expressed in terms of solutions Ap , Bp of some first kind integral Fredholm equations. An approach to the approximate solution of these equations is discussed. The found distribution function is used to calculate the fluxes of the number of gas particles and their energy in the first order in gradients T and μ0 . Kinetic coefficients, which describe the structure of these fluxes, are introduced. Matrix elements of the operator of the linearized collision integral (integral brackets) are used for their research. It is a question of validity of the principle of symmetry of kinetic coefficients and definition of their signs.

Relaxation phenomena in electron plasma of semiconductors

The hydrodynamics of the electron subsystems of semiconductors is studied in the approximations of the ideal and real liquid, taking into account processes of relaxation of temperatures and macroscopic velocities of electrons and phonons without assuming the local equilibrium of the system. A set of integral equations for the electron distribution function of the first order in gradients is obtained, which determines the sources in the hydrodynamic equations of the ideal liquid approximation and the dissipative flows of energy and momentum of electrons. The steady states of the system in the ideal liquid approximation are investigated. The exact formulas for the electron mobility of the semiconductor and its conductivity are derived and kinetic coefficients that determine current in a spatially inhomogeneous state are calculated. In the presence of an electric field, the phenomenon of difference of temperatures of the electron and phonon subsystems is predicted. The obtained expressions are specified for the case of temperatures much higher the Debye temperature.

The relativistic fluid dual to vacuum Einstein gravity

Abstract We present a construction of a (d + 2)-dimensional Ricci-flat metric corresponding to a (d + 1)-dimensional relativistic fluid, representing holographically the hydrodynamic regime of a (putative) dual theory. We show how to obtain the metric to arbitrarily high order using a relativistic gradient expansion, and explicitly carry out the computation to second order. The fluid has zero energy density in equilibrium, which implies incompressibility at first order in gradients, and its stress tensor (both at and away from equilibrium) satisfies a quadratic constraint, which determines its energy density away from equilibrium. The entire dynamics to second order is encoded in one first order and six second order transport coefficients, which we compute. We classify entropy currents with non-negative divergence at second order in relativistic gradients. We then verify that the entropy current obtained by pulling back to the fluid surface the area form at the null horizon indeed has a non-negative divergence. We show that there are distinct near-horizon scaling limits that are equivalent either to the relativistic gradient expansion we discuss here, or to the non-relativistic expansion associated with the Navier-Stokes equations discussed in previous works. The latter expansion may be recovered from the present relativistic expansion upon taking a specific non-relativistic limit.

Mach cones in the quark-gluon plasma: Viscosity, speed of sound, and effects of finite source structure

I use the space-time distribution of energy and momentum deposited by a fast parton traversing a perturbative quark-gluon plasma as a source term for the linearized hydrodynamical equations of the medium. A method of solution for the medium response is presented in detail. Numerical results are given for different values of the shear viscosity to entropy density ratio, $\ensuremath{\eta}/s$, and speed of sound, ${c}_{s}$. Furthermore, I investigate the relevance of finite source structure by expanding the source term up to first order in gradients of a \ensuremath{\delta} function centered at the fast parton and comparing the resulting dynamics to that obtained with the full source. It is found that, for the source term used here, the medium response is sensitive to the finite source structure up to distances of several femtometers from the source parton.

Axial currents from CKM matrix CP violation and electroweak baryogenesis

The first principle derivation of kinetic transport equations suggests that a CP-violating mass term during the electroweak phase transition can induce axial vector currents. Since the important terms are of first order in gradients there is a possibility to construct new rephasing invariants that are proportional to the CP phase in the Cabibbo–Kobayashi–Maskawa matrix and to circumvent the upper bound of CP-violating contributions in the Standard Model, the Jarlskog invariant. Qualitative arguments are given that these new contributions still fail to explain electroweak baryogenesis in extensions of the Standard Model with a strong first order phase transition.

Large amplitude spin waves in ultra-cold gases

We discuss the theory of spin waves in non-degenerate ultra-cold gases, and compare various methods which can be used to obtain appropriate kinetic equations. We then study non-hydrodynamic situations, where the amplitude of spin waves is sufficiently large to bring the system far from local equilibrium. The full position and momentum dependence of the distribution function must then be retained. In the first part of the article, we compare two general methods which can be used to derive a kinetic equation for a dilute gas of atoms (bosons or fermions) with two internal states (treated as a pseudo-spin 1/2). The collisional methods are in the spirit of Boltzmann's original derivation of his kinetic equation where, at each point of space, the effects of all sorts of possible binary collisions are added. We discuss two different versions of collisional methods, the Yvon-Snider approach and the S matrix approach. The second method uses the notion of mean field, which modifies the drift term of the kinetic equation, in the line of the Landau theory of transport in quantum liquids. For a dilute cold gas, it turns out that all these derivations lead to the same drift terms in the transport equation, but differ in the precise expression of the collision integral and in higher order gradient terms. In the second part of the article, the kinetic equation is applied to spin waves (or internal conversion) in trapped ultra-cold gases. Numerical simulations are used to illustrate the strongly non-hydrodynamic character of the spin waves recently observed with trapped 87Rb atoms. The decay of the phenomenon, which takes place when the system relaxes back towards equilibrium, is also discussed, with a short comment on decoherence. In two appendices we calculate the Wigner transform of the interaction term in the S matrix method, to first order in gradients; Appendix A.1 treats the case of spin-independent interactions, Appendix A.2 that of spin-dependent interactions.

Dense fluid transport for inelastic hard spheres.

The revised Enskog theory for inelastic hard spheres is considered as a model for rapid flow granular media at finite densities. A normal solution is obtained via the Chapman-Enskog method for states near the local homogeneous cooling state. The analysis is performed to first order in the spatial gradients, allowing identification of the Navier-Stokes order transport coefficients associated with the heat and momentum fluxes. In addition, the cooling rate is calculated to first order in the gradients and expressed in terms of the transport coefficients. The transport coefficients are determined from linear integral equations analogous to those for elastic collisions. The solubility conditions for these equations are confirmed and the transport coefficients are calculated as explicit functions of the density and restitution coefficient using a Sonine polynomial expansion. The results are not limited to small dissipation. Finally, the analysis is repeated using a simpler kinetic model. Excellent agreement is obtained with the results from the revised Enskog equation.

Higher corrections to the mass current in weakly inhomogeneous superfluid3He−A

Two new general representations (the series and the integral) for the mass current $\vj$ in weakly inhomogeneous superfluid A-phase of Helium--3 are obtained near zero of temperature by solving the Dyson-Gorkov equation. These representations result in additional correcting contribution to the standard leading expression for $\vj$ which is of first order in gradients of the orbital angular momentum vector $\hl$. The total supplementary term is found as integral, and, provided the London limit holds, the procedure is advanced to expand it at T=0 asymptotically by the Laplace method in powers of gradients of $\hl$. Three special static orientations of $\hl$ with respect to its curl are considered to calculate the higher correcting terms up to third order. Coefficients at the quadratic terms are estimated numerically, new cubic contributions are found which contain the logarithm of the London parameter.

Quasihydrodynamics of nanofluid mixtures

The functional perturbation theory method developed earlier [L. A. Pozhar and K. E. Gubbins, J. Chem. Phys. 94, 1367 (1991)] and used for derivation of the transport theory of pure dense, strongly inhomogeneous fluids [L. A. Pozhar and K. E. Gubbins, J. Chem. Phys. 99, 8970 (1993)] is exploited to develop the transport theory for mixtures of dense, strongly inhomogeneous fluids. The generalized Enskog-like kinetic equations have been solved using the 13-moment approximation method to obtain linearized quasihydrodynamic equations of first order in gradients of continuum variables and to derive explicit, tractable expressions for the transport coefficients of such mixtures. The derived transport coefficients are expressed in terms of equilibrium structure factors (the number density and the pair and direct correlation functions) of the corresponding inhomogeneous fluid mixtures. Diffusion in such mixtures is considered in detail for several particular cases.

Bose-Einstein correlations in thermal field theory

Two-particle correlation functions are calculated for bosons emitted from a localized thermal source (the “glow” of a “hot spot”). In contrast to existing work, non-equilibrium effects up to first order in gradients of the particle distribution function are taken into account. The spectral width of the bosons is shown to be an important quantity: if it is too small, they do not equilibrate locally and therefore strongly increase the measured correlation radius.

Transport coefficients in a weakly ionized nonequilibrium plasma.

The Boltzmann equation for an electron gas in a weakly ionized plasma is solved for an inhomogeneous plasma in a steady state but outside equilibrium (no Maxwell distribution). The inhomogeneity concerns both the electron density and the imposed dc electric field. With the emphasis on the case of a molecular gas discharge, the perturbed part of the electron distribution is evaluated only to the first order in gradients. The result is incorporated in the electron transport equations, where it leads to a renormalization of transport coefficients. Based on this procedure, a corrected version of the so-called quasihomogeneous equations is given.

The normal current in a thin film of superfluid3He-A

A thin film of superfluid 3He-A film containing a polar domain wall has a normal current flowing in it even at zero temperature. The authors calculate this current from the Gorkov-Nambu equations in a way similar to that of Volovik (1985) for bulk 3He-A. An earlier calculation by the authors working to first order in gradients yielded a singular normal current (proportional to a delta function). In the present work the authors show that the singular nature is an artefact of the gradient expansion. An additional length scale in the problem smooths out the delta singularity. They show that the current obtained has its origins in a zeroth Landau level.