From kinetic theory to dissipative fluid dynamics

We present the results of deriving the Israel-Stewart equations of relativistic dissipative fluid dynamics from kinetic theory via Grad’s 14-moment expansion. Working consistently to second order in the Knudsen number, these equations contain several new terms which are absent in previous treatments.     

The relativistic Burnett equations and sound propagation

A relaxation-time model for the relativistic Boltzmann equation of a single-component gas is solved to second, or “Burnett”, order using the relativistic version of the Chapman-Enskog method developed by Marle. Expressions are obtained from this second order solution for the “Burnett” contributions to the heat flux and pressure tensor of the gas. Using the “Burnett” equations, which incorporate these contributions, expressions are then derived for the dispersion and absorption of sound in the gas which agree, in the classical limit, with the results of Wang Chang and Uhlenbeck.     

Dressed ion acoustic solitary waves in quantum plasmas with two polarity ions and relativistic electron beams

A theory for dressed quantum ion acoustic waves (QIAWs), which includes higher-order corrections when QIAWs are investigated by the reductive perturbation method, is presented for unmagnetized plasmas containing positive and negative ions and weakly relativistic electron beams. The properties of the QIAWs are investigated using a quantum hydrodynamic model, from which a Korteweg–de Vries equation is derived using the reductive perturbation method. An equation including higher-order dispersion and nonlinearity corrections is also derived, and the physical parameter space is discussed for the importance of these corrections.     

Propagation of a TE surface mode in a relativistic electron beam-quantum plasma system

The dispersion properties of a transverse electric (TE) surface waves propagating along the interface between a magneto-quantum plasma-relativistic beam system and vacuum are studied by using the quantum hydrodynamic model. The general dispersion relations are derived and analyzed in some special cases of interest. Moreover, the effects of density gradients for the beam and plasma on the dispersion properties of surface waves are investigated. The kind of dispersion relations depends strongly on the ambient magnetic field Bo via the gyro-frequency ωc, the quantum parameters, and the width of the plasma layer as well as the relativistic factor for the electron beam. It is found that the quantum effects play a crucial role to facilitate the propagation of TE surface waves.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

Exact scaling relations in relativistic hydrodynamic turbulence

We consider the steady state statistics of turbulence in general classes of dissipative hydrodynamic equations, where the fluctuations are sustained by a random source concentrated at large scales. It is well known that in some particular cases, such as non-relativistic incompressible turbulence, a Kolmogorov-type exact scaling relation for a correlation function holds. We show that all such scaling relations follow from a general relation on the current-density correlation function. The derivation does not require an energy cascade picture and suggests that this traditional interpretation of the Kolmogorov relation for incompressible turbulence may be misleading. Using this we derive exact scaling results for compressible turbulence in relativistic hydrodynamics, which reduce in the slow motion limit to the Kolmogorov relation. We discuss the experimental implications of the results.     

Heat conduction in relativistic neutral gases revisited

The kinetic theory of dilute gases to first order in the gradients yields linear relations between forces and fluxes. The heat flux for the relativistic gas has been shown to be related not only to the temperature gradient but also to the density gradient in the representation where number density, temperature and hydrodynamic velocity are the independent state variables. In this work we show the calculation of the corresponding transport coefficients from the full Boltzmann equation and compare the magnitude of the relativistic correction.     

Dispersion relation of longitudinal oscillation in relativistic plasmas with nonextensive distribution

The generalized dispersion equation for longitudinal oscillation in an unmagnetized, collisionless, isotropic and relativistic plasma is derived in the context of nonextensive q-distribution. The analytical expression for the dispersion relation is obtained in an ultra-relativistic regime, which is related to the q-parameter and temperature. In the limit q→1, the result based on the relativistic Maxwellian distribution is recovered. Using the numerical simulation method, we obtain the full dispersion curve which cannot be given by an analytic method. It is shown that the numerical solution is in good agreement with the analytical result in the long-wavelength, short-wavelength and near light wave region for ultra-relativistic plasmas.     

Mapping exchange in relativistic Hartree-Fock

We show that formally for the standard ansatz relativistic point-coupling mean-field (RMF-PC) model a Lagrangian density is not equivalent in Hartree and Hartree-Fock approximations. The equivalency can be achieved only if we use a “complete” ansatz at the cost of introducing new parameters in the model. An approximate treatment of the exchange terms from standard RMF-PC indicates that these effects cannot be easily, if at all, absorbed by a Dirac-Hartree approximation.     

Isospin equilibration in relativistic heavy-ion collisions

We study the mixing and the kinetic equilibration of projectile and target nucleons in relativistic heavy-ion collisions in the energy regime between 150 AMeV and 2 AGeV in a coupled-channel BUU (CBUU) approach. We find that equilibrium in the projectile-target degrees of freedom is in general not reached even for large systems at low energy where elastic nucleon-nucleon collisions dominate. Inelastic nucleon excitations are more favorable for equilibration and their relative abundance increases both with energy and mass. Experimentally, the projectile/target admixture can be determined by measuring the degree of isospin equilibration in isospin asymmetric nuclear collisions. For one of the most promising systems currently under investigation, ⁹⁶ ₄₄Ru +⁹⁶ ₄₀Zr, we investigate the influence of the equation of state and the inelastic in-medium cross section. Received: 23 September 1998 / Revised version: 3 December 1998     

Kinetic model analysis of time-dependent problems in polyatomic gases

In this work we analyze time-dependent problems like sound propagation and light scattering in dilute polyatomic gases by using a kinetic model of the Boltzmann equation that replaces the collision operator by a single relaxation-time term which is compatible with Grad's 6-moment approximation. Comparison of the theoretical results with available experimental data in nitrogen, oxygen, carbon dioxide and methane shows that the model equation can be used to describe the acoustic properties and the light scattering spectrum of polyatomic gases in both hydrodynamic and kinetic regimes as long as the external oscillation frequency is smaller than the frequency required for the translational and the internal degrees of freedom to come to thermal equilibrium.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Derivation of fluid dynamics from kinetic theory with the 14-moment approximation

We review the traditional derivation of the fluid-dynamical equations from kinetic theory according to Israel and Stewart. We show that their procedure to close the fluid-dynamical equations of motion is not unique. Their approach contains two approximations, the first being the so-called 14-moment approximation to truncate the single-particle distribution function. The second consists in the choice of equations of motion for the dissipative currents. Israel and Stewart used the second moment of the Boltzmann equation, but this is not the only possible choice. In fact, there are infinitely many moments of the Boltzmann equation which can serve as equations of motion for the dissipative currents. All resulting equations of motion have the same form, but the transport coefficients are different in each case.     

X波段内同轴膜片加载相对论返波振荡器的特性研究

 把同轴膜片作为相对论返波振荡器的慢波结构,从Maxwell方程组和Floquet定理出发,导出了电磁波在同轴膜片加载相对论返波振荡器中的色散关系。通过编程计算得出TM_0n模式的色散曲线,数值分析了周期长度和导体半径对器件工作频率的影响,运用PIC程序模拟了返波振荡器中注波互作用过程。讨论了纵向聚焦磁场、电子注到慢波结构的距离、槽深和槽宽对输出功率和效率的影响。研究表明：增加器件的周期长度,可以减小器件的截止频率和同一模式下的工作频率; 调节内导体半径的大小,几乎不影响器件的工作频率。周期长度取11 mm,槽深取2 mm,槽宽取3.6 mm,漂移长度取7.5 mm的结构,在520 kV电压、8 kA电流和0.7 T的轴向聚焦磁场条件下,器件能输出2.8 GW的峰值功率,功率效率约33.2%。该慢波结构加工方便,具有广阔的应用前景。     

Non-linear relativistic diffusions

We obtain a non-linear generalization of the relativistic diffusion. We discuss diffusion equations whose non-linearity is a consequence of quantum statistics. We show that the assumptions of the relativistic invariance and an interpretation of the solution as a probability distribution substantially restrict the class of admissible non-linear diffusion equations. We consider relativistic invariant as well as covariant frame-dependent diffusion equations with a drift. In the latter case we show that there can exist stationary solutions of the diffusion equation besides the equilibrium solution corresponding to the quantum or Tsallis distributions. We define the relative entropy as a function of the diffusion probability and prove that it is monotonically decreasing in time when the diffusion tends to equilibrium. We discuss its relation to the thermodynamic behavior of diffusing particles.     

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Jüttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.     

Newtonian versus relativistic nonlinear cosmology

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.     

相对论返波管注-波作用不稳定区的研究

 以环形电子注驱动的、正弦型周期慢波结构的相对论返波管（RBWO）为模型,理论推导出RBWO慢波结构的色散方程,画出了不包括和包括电子注的色散曲线,然后采用在复平面上绘制等高线图进行逼近的方法对注-波作用不稳定区工作频率的实部和虚部进行了求解,绘出了完整的包括电子注的慢波结构色散关系曲线,并估算了微波的时间增长率和能量转换效率。研究得出：当a（电子注电流参数）较大时,电磁波与慢电荷波的相互作用是不稳定的;而在a较小时,注-波相互作用具有明显的三波作用的特点;微波增长是通过注-波不稳定作用引起的;随着a的增大,微波时间增长率先增大后减小,在a为0.02时有最大值1.37,此时的能量转换效率为19%;增大电子注半径或慢波结构的波纹深度都可以增强注-波不稳定作用。     

Surface waves on the relativistic quantum plasma half-space

We present a theoretical investigation on the propagation of surface waves on the relativistic quantum plasma half-space. The dispersion relations of surface plasmon polaritons (SPPs) and electrostatic surface waves containing relativistic quantum corrected terms are derived. Results show that the frequency of SPPs has a blue-shift, and surface Langmuir oscillations can propagate on the cold plasma half-space due to quantum effects. Numerical evaluation indicates that quantum effects to SPPs and electrostatic surface waves are significant and observable.     

Fluid models from kinetic models using a geometric averaging procedure

We interpret the Lorentz force equation as a geodesic equation associated with a non-linear connection. Using a geometric averaging procedure, we prove that for narrow and smooth one-particle distribution functions whose supports are invariant under the flow of the Lorentz equation, a bunch of charged point particles can be described by a charged cold fluid model in the ultra-relativistic regime. The method used to prove this result does not require additional hypotheses on the higher moments of the distribution. This is accomplished by estimating the expressions that include the differential operators appearing in the charged cold fluid model equation. Under the specified conditions of narrowness and ultra-relativistic dynamics, it turns out that these differential expressions are close to zero, justifying the use of the charged cold fluid model. The method presented in this work can also be applied to justify the use of warm plasmas and other models. Finally, a possible relation with chromohydrodynamics is discussed.