Single-particle Distribution Function Research Articles

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Investigating the role of electric fields on flow harmonics in heavy-ion collisions

Using the blast-wave model, we explore the effect of electric fields on spectra and flow harmonics (especially the elliptic flow) for charged pions and protons. We incorporate the first-order correction to the single-particle distribution function due to the electric fields and the dissipative effect while calculating the invariant yields of hadrons in the Cooper–Frye prescription at the freezeout hypersurface. We find a noticeable correction to the directed and elliptic flow of pions and protons for unidirectional and azimuthal asymmetric electric fields of a magnitude in the transverse plane. Further, we observe mass dependency of the directed flow generated due to the electric fields. The splitting of particle and antiparticle’s elliptic flow due to the electric fields is also discussed.

Reduced kinetic model of polyatomic gases

Kinetic models of polyatomic gas typically account for the internal degrees of freedom at the level of the two-particle distribution function. However, close to the hydrodynamic limit, the internal (rotational) degrees of freedom tend to be well represented just by rotational kinetic energy density. We account for the rotational energy by augmenting the ellipsoidal statistical Bhatnagar–Gross–Krook (ES–BGK) model, an extension of the BGK model, at the level of the single-particle distribution function with an advection–diffusion–relaxation equation for the rotational energy. This reduced model respects the$H$theorem and recovers the compressible hydrodynamics for polyatomic gases as its macroscopic limit. As required for a polyatomic gas model, this extension of the ES–BGK model not only has the correct specific heat ratio but also allows for three independent tunable transport coefficients: thermal conductivity, shear viscosity and bulk viscosity. We illustrate the effectiveness of the model via a lattice Boltzmann method implementation.

Particle–particle drag force in inertial bidisperse gas–particle suspensions

Particle-resolved direct numerical simulations are employed to investigate the particle–particle drag force in the bidisperse gas–particle suspensions where the particles are smooth and the single-particle velocity distribution function is Maxwellian. The particle Reynolds number ranges from 6.7 to 123.8, and in this range the particle inertia is high enough that the lubrication force is not essential. It is found that the relation derived by the kinetic theory of granular flow (KTGF) highly overestimates the particle–particle drag force. This is because the pre-collision velocities of colliding particles are not completely uncorrelated with each other. From the time sequence of collision events, it is observed that the particle pair that has just collided will probably collide again after a short time due to the restriction of the particle motion in dense suspensions. Since the post-collision velocities of the first collision cannot relax entirely in such a short time, the relative velocity before the subsequent collision is statistically smaller than the domain-averaged relative velocity. Consequently, the particle–particle drag force is over-predicted when the domain-averaged relative velocity is used. For this reason, this work assumes that the particle–particle drag force is determined by the relative velocity within a local region near large particles. When the local region is set to be the spherical shells centred on the centres of large particles and with an outer radius of a mean free path of small particles, the KTGF-based relation can reasonably predict the particle–particle drag force.

Relativistic second-order dissipative spin hydrodynamics from the method of moments

We derive relativistic second-order dissipative fluid-dynamical equations of motion for massive spin-1/2 particles from kinetic theory using the method of moments. Besides the usual conservation laws for charge, energy, and momentum, such a theory of relativistic dissipative spin hydrodynamics features an equation of motion for the rank-3 spin tensor, which follows from the conservation of total angular momentum. Extending the conventional method of moments for spin-0 particles, we expand the spin-dependent distribution function near local equilibrium in terms of moments of the momentum and spin variables. We work to next-to-leading order in the Planck constant $\hbar$. As shown in previous work, at this order in $\hbar$ the Boltzmann equation for spin-1/2 particles features a nonlocal collision term. From the Boltzmann equation, we then obtain an infinite set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from local equilibrium. In order to close this system of moment equations, a truncation procedure is needed. We employ the "14+24-moment approximation", where "14" corresponds to the components of the charge current and the energy-momentum tensor and "24" to the components of the spin tensor, which completes the derivation of the equations of motion of second-order dissipative spin hydrodynamics. For applications to heavy-ion phenomenology, we also determine dissipative corrections to the Pauli-Lubanski vector.

Momentum Distribution Functions and Pair Correlation Functions of Unpolarized Uniform Electron Gas in Warm Dense Matter Regime

In this paper we continued our research of the uniform electron gas in a warm dense matter regime, focusing on the momentum distribution functions and pair correlation functions. We use the single–momentum path integral Monte Carlo method, based on the Wigner formulation of quantum statistics to calculate both momentum- and coordinate-depending distributions and average values of quantum operators for many-fermion Coulomb systems. We discovered that the single-particle momentum distribution function deviates from the ideal Fermi distribution and forms the so-called “quantum tails” at high momenta, if non-ideality is strong enough in both degenerate and non-degenerate cases. This effect is always followed by the appearance of the short-range order on pair correlation functions and can be explained by the tunneling through the effective potential wells surrounding the electrons. Furthermore, we calculated the average kinetic and potential energies in the wide range of states, expanding our previous results significantly.

Soft modes in Fermi liquids at arbitrary temperatures

We use kinetic-theory methods to analyze Landau Fermi-liquid theory, and in particular to investigate the number and nature of modes in Fermi liquids that are soft in the long-wavelength limit, both in the hydrodynamic and the collisionless regimes. In the hydrodynamic regime we show that Fermi-liquid theory is consistent with Navier-Stokes hydrodynamics at all temperatures, as expected. The soft modes are the ones familiar from classical hydrodynamics that are controlled by the five conservation laws; namely, two first-sound modes, two shear diffusion modes, and one heat diffusion mode. These modes have a particle-like spectrum and are soft, or scale invariant, at all temperatures. In the collisionless regime we show that the entire single-particle distribution function is soft with a continuous part of the spectrum. This continuous soft mode, which is well known but often not emphasized, has important physical consequences, e.g., for certain quantum phase transitions. In addition, there are the well-known soft zero-sound excitations that describe angular fluctuations of the Fermi surface; their spectra are particle-like. They are unrelated to conservation laws, acquire a mass at any nonzero temperature, and their number depends on the strength of the quasiparticle interaction. We also discuss the fates of these two families of soft modes as the temperature changes. With increasing temperature the size of the collisionless regime shrinks, the damping of the modes grows, and eventually all of the collisionless modes become overdamped. In their stead the five hydrodynamic modes appear in the hydrodynamic regime at asymptotically low frequencies. The two families of soft modes are unrelated and have very different physical origins. In charged Fermi liquids the first-sound modes in the hydrodynamic regime and the $\ensuremath{\ell}=0$ zero-sound modes in the collisionless regime get replaced by plasmons, all other modes remain soft.

Long-range oscillations of a single-particle distribution function for a molecular system of hard spheres near a solid surface in the Percus–Yevick approximation

The properties of molecular systems are defined by asymptotic behavior of distribution functions that have descending character with long-range oscillating order. We consider the molecular system of hard spheres near hard wall. The decrement and period of oscillation are calculated for the distribution function which defines the local density profile.

Remarks on the phenomenological Tsallis distributions and their link with the Tsallis statistics

From the Tsallis unnormalized (or Tsallis-2) statistical mechanical formulation, Büyükkiliç et al (1995 Phys. Lett. A 197, 209) derived the expressions for the single-particle distribution functions (known as the phenomenological Tsallis distributions) for particles obeying the Maxwell–Boltzmann, Bose–Einstein and the Fermi–Dirac statistics using the factorization approximation. In spite of the fact that this paper was published long time ago, its results are still extensively used in many fields of physics, and it is considered that it was this paper that established the connection between the phenomenological Tsallis distributions and the Tsallis statistics. Here we show that this result is incorrect: the mistake lies in the fact that the probability distribution function was derived from the maximum entropy principle using the definition of the generalized expectation values (of the Tsallis-2 statistics), but the single-particle distribution functions were calculated on the basis of this probability distribution of the Tsallis-2 statistics using the standard definition of the expectation values of the Tsallis normalized (or Tsallis-1) statistics. Considering the definition of the expectation values for thermodynamic quantities which is consistent with the definition of expectation values of the constraints in the maximum entropy principle of the Tsallis-2 formulation, we have proved that the single-particle (classical and quantum) distribution functions in the factorization approximation differ from the well-known phenomenological Tsallis distributions.

Relativistic non-resistive viscous magnetohydrodynamics from the kinetic theory: a relaxation time approach

We derive the relativistic non-resistive, viscous second-order magnetohydrodynamic equations for the dissipative quantities using the relaxation time approximation. The Boltzmann equation is solved for a system of particles and antiparticles using Chapman-Enskog like gradient expansion of the single-particle distribution function truncated at second order. In the first order, the transport coefficients are independent of the magnetic field. In the second-order, new transport coefficients that couple magnetic field and the dissipative quantities appear which are different from those obtained in the 14-moment approximation [1] in the presence of a magnetic field. However, in the limit of the weak magnetic field, the form of these equations are identical to the 14-moment approximation albeit with different values of these coefficients. We also derive the anisotropic transport coefficients in the Navier-Stokes limit.

Determination of the single particle distribution function in a weakly correlated weakly inhomogeneous plasma

Single particle distribution function of plasma particles has been derived from the first member of the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy utilising the pair correlation function evaluated in [Bose, Phys. Plasmas 26, 064501 (2019)] from the second member of the BBGKY hierarchy. This distribution function may be employed to probe the thermodynamic properties of the weakly inhomogeneous plasma systems.

Hydrodynamic generators in relativistic kinetic theory

We resum the non-equilibrium gradient corrections to a single-particle distribution function evolved by the Boltzmann equation in the relaxation time approximation (RTA). We first study a system undergoing Bjorken expansion and show that, for a constant relaxation time, the exact solution of the RTA Boltzmann equation at late times (i.e. after the decay of non-hydrodynamic modes) generates the Borel resummed Chapman-Enskog series. Extending this correspondence to systems without Bjorken symmetry, we construct a (3+1)-dimensional hydrodynamic generator for RTA kinetic theory, which is an integral representation of the Chapman-Enskog series in the limit of vanishing non-hydrodynamic modes. Relaxing this limit we find at earlier times a set of non-hydrodynamic modes coupled to the Chapman-Enskog expansion. Including the dynamics of these non-hydrodynamic modes is shown to control the emergence of hydrodynamics as an effective field theory description of non-equilibrium fluids, which works well even for far-off-equilibrium situations where the Knudsen number is initially large.

Velocity and Absorption Coefficient of Sound Waves in Classical Gases

The velocity and absorption coefficient of plane sound waves in classical gases are obtained by solving the Boltzmann kinetic equation. This is done within the linear response theory as a reaction of the single-particle distribution function to a periodic external field. The nonperturbative dispersion equation is derived in the relaxation time approximation and solved numerically. The obtained theoretical results demonstrate an universal dependence of the sound velocity and scaled absorption coefficient on the variable wт , where w is the sound frequency, and т−1 is the particle collision frequency. In the region of wт ∼ 1, a transition from the frequent- to rare-collision regime takes place. The sound velocity increases sharply, and the scaled absorption coefficient has a maximum – both theoretical findings are in agreement with the data.

Phase diagram of noisy systems of coupled oscillators with a bimodal frequency distribution

We study the properties of large systems of globally coupled oscillators in the presence of noise. When the distribution of the natural frequencies of the oscillators is bimodal and its analytical continuation in the complex plane has only a few poles in the lower half plane, the dynamics of the system, governed by a Fokker–Planck equation for the single particle distribution function, can be reduced to a system of ordinary differential equations describing the dynamics of suitably defined order parameters, the first ones of which are related to the usual synchronization order parameter. We obtain the full phase diagram of the oscillator system, that shows a very rich behaviour, with regions characterized by synchronized states, regions with periodic states, and others with bi-stability, associated to the presence of hysteresis. The latter phenomenon is confirmed by numerical simulations ot the full system of coupled oscillators. We compare our results with those previously obtained for noiseless systems, and we show that for increasing noise the phase diagram changes qualitatively, tending to the simple diagram that is found for systems with unimodal frequency distributions.

Signatures of the Schwinger mechanism assisted by a fast-oscillating electric field

The spontaneous production of electron-positron pairs from the vacuum--in a field configuration composed of a high-frequency electric mode of weak intensity and a strong constant electric field--is investigated. Asymptotic expressions for the single-particle distribution function ruling this nonperturbative process are established by considering the low-density approximation in the Boltzmann-Vlasov equation. An analytical formula for the density rate of yielded particles is established which is shown to manifest a nonperturbative dependence on both the strong and weak electric fields and to interpolate between the tunneling and multi-photon regimes. It is shown that--under appropriate circumstances--the produced plasma of electrons and positrons might reach densities for which their recombinations into high-energy photons occurs copiously. On the basis of this feature, an experimental setup for observing the dynamically-assisted Schwinger effect is put forward.

Two-dimensional off-lattice Boltzmann model for van der Waals fluids with variable temperature

We develop a two-dimensional Lattice Boltzmann model for liquid–vapour systems with variable temperature. Our model is based on a single particle distribution function expanded with respect to the full-range Hermite polynomials. This model relies on a set of 25 off-lattice momentum vectors, whose Cartesian projections are the roots of the Hermite polynomial of order Q=5. The time evolution, spatial advection and gradient computations are performed using finite difference techniques. Our implementation is extensively benchmarked in the following contexts: the planar interface, the Laplace pressure test, the damping of shear and longitudinal waves, as well as the Galilean invariance test. We then investigate the liquid–vapour phase separation between two parallel walls kept at a constant temperature Tw, which is smaller than the critical temperature Tc, and discuss the main features of this process.

Nonresistive dissipative magnetohydrodynamics from the Boltzmann equation in the 14-moment approximation

We derive the equations of motion of relativistic, non-resistive, second-order dissipative magnetohydrodynamics from the Boltzmann equation using the method of moments. We assume the fluid to be composed of a single type of point-like particles with vanishing dipole moment or spin, so that the fluid has vanishing magnetization and polarization. In a first approximation, we assume the fluid to be non-resistive, which allows to express the electric field in terms of the magnetic field. We derive equations of motion for the irreducible moments of the deviation of the single-particle distribution function from local thermodynamical equilibrium. We analyze the Navier-Stokes limit of these equations, reproducing previous results for the structure of the first-order transport coefficients. Finally, we truncate the system of equations for the irreducible moments using the 14-moment approximation, deriving the equations of motion of relativistic, non-resistive, second-order dissipative magnetohydrodynamics. We also give expressions for the new transport coefficients appearing due to the coupling of the magnetic field to the dissipative quantities.

Fractional exclusion statistics and thermodynamics of the Hubbard chain in the spin-incoherent Luttinger liquid regime

Bethe ansatz and bosonization procedures are used to describe the thermodynamics of the strong-coupled Hubbard chain in the \textit{spin-incoherent} Luttinger liquid (LL) regime: $J(\equiv 4t^2/U)\ll k_B T\ll E_F$, where $t$ is the hopping amplitude, $U(\gg t)$ is the repulsive on-site Coulomb interaction, and $k_B T (E_F\sim t)$ is the thermal (Fermi) energy. We introduce a fractional Landau LL approach, whose $U=\infty$ fixed point is exactly mapped onto an ideal gas with two species obeying the Haldane-Wu \textit{exclusion} fractional statistics. This phenomenological approach sheds light on the behavior of several thermodynamic properties in the spin-incoherent LL regime: specific heat, charge compressibility, magnetic susceptibility, and Drude weight. In fact, besides the hopping (mass) renormalization, the fractional Landau LL parameters, due to quasiparticle interaction, are determined and relationships with velocities of holons and spinons are unveiled. The specific heat thus obtained is in very good agreement with previous density matrix renormalization group (DMRG) simulations of the $t$-$J$ model in the spin-incoherent regime. A phase diagram is provided and two thermodynamic paths to access this regime clarifies both the numerical and analytical procedures. Further, we show that the high-$T$ limit of the fractional Landau LL entropy and chemical potential exhibit the expected results of the $t$-$J$ model, under the condition $U\gg k_B T$. Lastly, finite-temperature Lanczos simulations of the single-particle distribution function confirm the characteristics of the spin-incoherent regime and the high-$T$ limit observed in previous DMRG studies.

Radiative heavy quark energy loss in an expanding viscous QCD plasma

We study viscous effects on heavy quark radiative energy loss in a dynamically screened medium with boost-invariant longitudinal expansion. We calculate, to first order in opacity, the energy loss by incorporating viscous corrections in the single-particle phase-space distribution function within relativistic dissipative hydrodynamics. We consider the Grad's 14-moment and the Chapman-Enskog-like methods for the nonequilibrium distribution functions. Our numerical results for the charm quark radiative energy loss show that, as compared to static fluid, an expanding ideal (nonviscous) fluid causes a much smaller energy loss. Viscosity in the evolution lead to somewhat enhanced energy loss which is insensitive to the underlying viscous hydrodynamic models used. Further inclusion of viscous correction induces larger energy loss, the magnitude and pattern of this enhancement crucially depend on the form of viscous corrections used.

Ultrasonic waves in classical gases.

The velocity and absorption coefficient for the plane sound waves in a classical gas are obtained by solving the Boltzmann kinetic equation, which describes the reaction of the single-particle distribution function to a periodic external field. Within the linear response theory, the nonperturbative dispersion equation valid for all sound frequencies is derived and solved numerically. The results are in agreement with the approximate analytical solutions found for both the frequent- and rare-collision regimes. These results are also in qualitative agreement with the experimental data for ultrasonic waves in dilute gases.