One-particle Distribution Function Research Articles

Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation

The propagation speeds of excitations are a crucial input in the modeling of interacting systems of particles. In this paper, we assume the microscopic physics is described by a kinetic theory for massless particles, which is approximated by a generalized relaxation time approximation (RTA) where the relaxation time depends on the energy of the particles involved. We seek a solution of the kinetic equation by assuming a parameterized one-particle distribution function (1-pdf) which generalizes the Chapman–Enskog (Ch-En) solution to the RTA. If developed to all orders, this would yield an asymptotic solution to the kinetic equation; we restrict ourselves to an approximate solution by truncating the Ch-En series to the second order. Our generalized Ch-En solution contains undetermined space-time-dependent parameters, and we derive a set of dynamical equations for them by applying the moments method. We check that these dynamical equations lead to energy–momentum conservation and positive entropy production. Finally, we compute the propagation speeds for fluctuations away from equilibrium from the linearized form of the dynamical equations. Considering relaxation times of the form τ=τ0(−βμpμ)−a, with −∞<a<2, where βμ=uμ/T is the temperature vector in the Landau frame, we show that the Anderson–Witting prescription a=1 yields the fastest speed in all scalar, vector and tensor sectors. This fact ought to be taken into consideration when choosing the best macroscopic description for a given physical system.

Dynamics and kinetic theory of hard spheres under strong confinement.

The kinetic theory description of a low-density gas of hard spheres or disks, confined between two parallel plates separated at a distance smaller than twice the diameter of the particles, is addressed starting from the Liouville equationof the system. The associated Bogoliubov, Born, Green, Kirkwood, and Yvon hierarchy of equationsfor the reduced distribution functions is expanded in powers of a parameter measuring the density of the system in the appropriate dimensionless units. The Boltzmann level of description is obtained by keeping only the two lowest orders in the parameter. In particular, the one-particle distribution function obeys a couple of equations. Contrary to what happens with a Boltzmann-like kinetic equationthat has been proposed for the same system on a heuristic basis, the kinetic theory formulated here admits stationary solutions that are consistent with equilibrium statistical mechanics, in both the absence and presence of external fields. In the latter case, the density profile is rather complex due to the coupling between the inhomogeneities generated by the confinement and by the external fields. The general theory formulated provides a solid basis for the study of the properties of strongly confined dilute gases.

On the dependence of the thermal module of elasticity of a two-component magnetic fluid on frequency, concentration and magnetic field

Purpose. Study of the dependence of the thermal modulus of elasticity of a two-component magnetic fluid on the magnitude of the magnetic field strength, the frequency of external disturbance and the volumetric concentration of magnetic particles.Method. The research method is based on the kinetic theory of liquid systems. Based on previously constructed kinetic equations for one-particle and two-particle distribution functions and a microscopic expression for the heat flux vector, an explicit dynamic expression for the thermal modulus of elasticity of magnetic fluids is obtained. In fast processes in liquids, heat transfer occurs in waves and their propagation is similar to the propagation of second sound in helium II. The thermal modulus of elasticity in liquids appears at high frequencies and ensures the propagation of second sound. The expression for the thermal modulus of elasticity consists of potential and kinetic parts, taking into account structural and translational relaxation processes, respectively. To study the thermoelastic properties of magnetic fluids, appropriate expressions of potential interaction energies were selected for each subsystem, allowing for numerical calculations.Results. Numerical calculations of the frequency and concentration dependence of the dynamic thermal modulus of elasticity in the presence of an external magnetic field in a kerosene-based magnetic fluid were carried out. The calculation results show that an increase in the influence of external disturbances leads to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid. An increase in the volume concentration of magnetic particles and an increase in the magnetic field strength also led to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid.Conclusion. It has been established that, due to taking into account translational and structural relaxation processes, the region of frequency dispersion of the thermal elastic modulus is wide. Numerical calculations carried out at different values of the external magnetic field and the volume concentration of magnetic particles showed that although an increase in the magnetic field and concentration of magnetic particles leads to an increase in the thermal modulus of elasticity, their increase does not affect the change in the frequency dispersion region.

Analysis of Self-Gravitating Fluid Instabilities from the Post-Newtonian Boltzmann Equation.

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy-momentum tensor calculated from a post-Newtonian Maxwell-Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory.

Kinetic gases in static spherically symmetric modified dispersion relations

We study the dynamics of a collisionless kinetic gas in the most general static, spherically symmetric dispersion relation. For a static, spherically symmetric kinetic gas, we derive the most general solution to these dynamics, and find that any solution is given by a one-particle distribution function which depends on three variables. For two particular solutions, describing a shell of monoenergetic orbiting particles and a purely radial inflow, we calculate the particle density as a function of the radial coordinate. As a particular example, we study a κ-Poincaré modification of the Schwarzschild metric dispersion relation and derive its influence on the particle density. Our results provide a possible route towards quantum gravity phenomenology via the observation of matter dynamics in the vicinity of massive compact objects.

Confined granular gases under the influence of vibrating walls

In this paper we study the dynamics of a system composed of inelastic hard spheres or disks that are confined between two parallel vertically vibrating walls (the vertical direction is defined as the direction perpendicular to the walls). The distance between the two walls is supposed to be larger than twice the diameter of the particles so that the particles can pass over each other, but is still much smaller than the dimensions of the walls. Hence, the system can be considered to be quasi-two-dimensional (quasi-one-dimensional) in the hard spheres (disks) case. For dilute systems, a closed evolution equation for the one-particle distribution function is formulated that takes into account the effects of the confinement. Assuming the system is spatially homogeneous, the kinetic equation is solved approximating the distribution function by a two-temperature (horizontal and vertical) Gaussian distribution. The obtained evolution equations for the partial temperatures are solved, finding a very good agreement with molecular dynamics simulation results for a wide range of parameters (inelasticity, height and density) for states whose projection over a plane parallel to the walls is homogeneous. In the stationary state, where the energy lost in collisions is compensated by the energy injected by the walls, the pressure tensor in the horizontal direction is analyzed and its relation with an instability of the homogeneous state observed in the simulations is discussed.

On the Partition Temperature of Massless Particles in High-Energy Collisions

Although partition temperature derived using the Darwin–Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase-space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed.

Axisymmetric, stationary collisionless gas clouds trapped in a Newtonian potential

The properties of an axisymmetric, stationary gas cloud surrounding a massive central object are discussed. It is assumed that the gravitational field is dominated by the central object which is modeled by a nonrelativistic rotationally-symmetric potential. Further, we assume that the gas consists of collisionless, identical massive particles that follow bound orbits in this potential. Several models for the one-particle distribution function are considered and the essential formulae that describe the relevant macroscopical observables, such as the particle and energy densities, pressure tensor, and the kinetic temperature are derived. The asymptotic decay of the solutions at infinity is discussed and we specify configurations with finite total mass, energy and (zero or non-zero) angular momentum. Finally, our configurations are compared to their hydrodynamic analogs. In an accompanying paper, the equivalent general relativistic problem is discussed, where the central object consists of a Schwarzschild black hole.

Axisymmetric, stationary collisionless gas configurations surrounding Schwarzschild black holes

The properties of a stationary gas cloud surrounding a black hole are discussed, assuming that the gas consists of collisionless, identical massive particles that follow spatially bound geodesic orbits in the Schwarzschild spacetime. Several models for the one-particle distribution function are considered, and the essential formulae that describe the relevant macroscopic observables, like the current density four-vector and the stress–energy–momentum tensor are derived. This is achieved by rewriting these observables as integrals over the constants of motion and by a careful analysis of the range of integration. In particular, we provide configurations with finite total mass and angular momentum. Differences between these configurations and their nonrelativistic counterparts in a Newtonian potential are analyzed. Finally, our configurations are compared to their hydrodynamic analogues, the ‘polish doughnuts’.

Ergodicity and slow relaxation in the one-dimensional self-gravitating system.

We show that homogeneous and nonhomogeneous states of the one-dimensional self-gravitating sheets models have different ergodic properties. The former are nonergodic and the one-particle distribution function has a zero collision term if a proper limit is taken for the periodic boundary conditions. As a consequence, homogeneous states of the sheets model are nonergodic and do not relax to the equilibrium state, while nonhomogeneous states are ergodic in a time window of the order of the relaxation time to equilibrium, as similarly observed in other systems with a long range interaction.

Unification of kinetic and hydrodynamic approaches in the theory of dense gases and liquids far from equilibrium

A system of non-Markovian transport equations is obtained for the non-equilibrium one-particle distribution function of particles and the non-equilibrium average value of the density of the potential energy of the interaction of the system particles far from the equilibrium state. Expressions for entropy, the partition function of the non-equilibrium state of the system, as well as non-equilibrium thermodynamic relations were obtained. The generalized structure of transfer nuclei is revealed in detail with the selection of short-range and long-range contributions of interactions between particles. The connection of transport nuclei with generalized diffusion coefficients, friction in the space of coordinates and momentum and the potential part of the thermal conductivity coefficient is established.

Far-from-equilibrium attractors for massive kinetic theory in the relaxation time approximation

We investigate whether early and late time attractors for non-conformal kinetic theories exist by computing the time-evolution of a large set of moments of the one-particle distribution function. For this purpose we make use of a previously obtained exact solution of the 0+1D boost-invariant massive Boltzmann equation in relaxation time approximation. We extend prior attractor studies of non-conformal systems by using a realistic mass- and temperature-dependent relaxation time and explicitly computing the effect of varying both the initial momentum-space anisotropy and initialization time on the time evolution of a large set of integral moments. Our findings are consistent with prior studies, which found that there is an attractor for the scaled longitudinal pressure, but not for the shear and bulk viscous corrections separately. We further present evidence that both late- and early-time attractors exist for all moments of the one-particle distribution function that contain greater than one power of the longitudinal momentum squared.

Globally time-reversible fluid simulations with smoothed particle hydrodynamics

This paper describes an energy-preserving and globally time-reversible code for weakly compressible smoothed particle hydrodynamics (SPH). We do not add any additional dynamics to the Monaghan's original SPH scheme at the level of ordinary differential equation, but we show how to discretize the equations by using a corrected expression for density and by invoking a symplectic integrator. Moreover, to achieve the global-in-time reversibility, we have to correct the initial state, implement a conservative fluid-wall interaction, and use the fixed-point arithmetic. Although the numerical scheme is reversible globally in time (solvable backwards in time while recovering the initial conditions), we observe thermalization of the particle velocities and growth of the Boltzmann entropy. In other words, when we do not see all the possible details, as in the Boltzmann entropy, which depends only on the one-particle distribution function, we observe the emergence of the second law of thermodynamics (irreversible behavior) from purely reversible dynamics.

Relaxation Dynamics in a Long-Range System with Mixed Hamiltonian and Non-Hamiltonian Interactions

Sometimes the dynamics of a physical system is described by non-Hamiltonian equations of motion, and additionally, the system is characterized by long-range interactions. A concrete example is that of particles interacting with light as encountered in free-electron laser and cold-atom experiments. In this work, we study the relaxation dynamics to non-Hamiltonian systems, more precisely, to systems with interactions of both Hamiltonian and non-Hamiltonian origin. Our model consists of $N$ globally-coupled particles moving on a circle of unit radius; the model is one-dimensional. We show that in the infinite-size limit, the dynamics, similarly to the Hamiltonian case, is described by the Vlasov equation. In the Hamiltonian case, the system eventually reaches an equilibrium state, even though one has to wait for a long time diverging with $N$ for this to happen. By contrast, in the non-Hamiltonian case, there is no equilibrium state that the system is expected to reach eventually. We characterize this state with its average magnetization. We find that the relaxation dynamics depends strongly on the relative weight of the Hamiltonian and non-Hamiltonian contributions to the interaction. When the non-Hamiltonian part is predominant, the magnetization attains a vanishing value, suggesting that the system does not sustain states with constant magnetization, either stationary or rotating. On the other hand, when the Hamiltonian part is predominant, the magnetization presents long-lived strong oscillations, for which we provide a heuristic explanation. Furthermore, we find that the finite-size corrections are much more pronounced than those in the Hamiltonian case; we justify this by showing that the Lenard-Balescu equation, which gives leading-order corrections to the Vlasov equation, does not vanish, contrary to what occurs in one-dimensional Hamiltonian long-range systems.

Kinetic Theory and Memory Effects of Homogeneous Inelastic Granular Gases under Nonlinear Drag.

We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog–Fokker–Planck equation for the one-particle velocity distribution function. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects such as Mpemba and Kovacs-like ones.

Chiral effects in classical spinning gas

We consider a statistical mechanics of rotating ideal gas consisting of classical non-relativistic spinning particles. The microscopic structure elements of the system are massive point particles with a nonzero proper angular momentum. The norm of proper angular momentum is determined by spin. The direction of proper angular momentum changes continuously. Applying the Gibbs canonical formalism for the rotating system, we construct the one-particle distribution function, generalising the usual Maxwell–Boltzmann distribution, and the partition function of the system. The model demonstrates a set of chiral effects caused by interaction of spin and macroscopic rotation, including the change of entropy, heat capacity, chemical potential and angular momentum.

Time-dependent projection operator and nonlinear generalized master equations.

By introducing a special time-dependent projection operator P(t), the nonlinear version of the Nakajima-Zwanzig inhomogeneous generalized master equation(GME) for the relevant part of the N-particle distribution function containing the irrelevant initial correlation term is derived. For this projection operator the relevant part of the N-particle distribution function is a product of the distribution function of a group of s (s≤N) selected particles F_{s}(t) and the distribution function F_{N-s}(t) of N-s particles constituting the "environment" for a group of s selected particles. Thus, the linear projection operator approach results in the exact nonlinear GME for the relevant part of the distribution function, which is equivalent to the exact nonlinear GME for the distribution function of the complex of s particles of interest. This equationis further specified in the first approximation in the particle density, and then considered for one-particle and two-particle distribution functions. Given that there is no satisfactory way of dealing with the initial correlation term, the approach is suggested to rigorously include this term into consideration by converting the obtained inhomogeneous GME into homogeneous form containing initial correlations in the kernel governing the evolution of the relevant part of the distribution function. The homogeneous nonlinear GME for a one-particle distribution function is considered and the conditions for its equivalence to the nonlinear Boltzmann equationare discussed.

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

Kinetic Gas Disks Surrounding Schwarzschild Black Holes

We describe stationary and axisymmetric gas configurations surrounding black holes. They consist of a collisionless relativistic kinetic gas of identical massive particles following bound orbits in a Schwarzschild exterior spacetime and are modeled by a one-particle distribution function which is the product of a function of the energy and a function of the orbital inclination associated with the particle's trajectory. The morphology of the resulting configuration is analyzed.

Steady states of active Brownian particles interacting with boundaries

An active Brownian particle is a minimal model for a self-propelled colloid in a dissipative environment. Experiments and simulations show that, in the presence of boundaries and obstacles, active Brownian particle systems approach nontrivial nonequilibrium steady states with intriguing phenomenology, such as accumulation at boundaries, ratchet effects, and long-range depletion interactions. Nevertheless, theoretical analysis of these phenomena has proven difficult. Here, we address this theoretical challenge in the context of non-interacting particles in two dimensions, basing our analysis on the steady-state Smoluchowski equation for the one-particle distribution function. Our primary result is an approximation strategy that connects asymptotic solutions of the Smoluchowski equation to boundary conditions. We test this approximation against the exact analytic solution in a 2D planar geometry, as well as numerical solutions in circular and elliptic geometries. We find good agreement so long as the boundary conditions do not vary too rapidly with respect to the persistence length of particle trajectories. Our results are relevant for characterizing long-range flows and depletion interactions in such systems. In particular, our framework shows how such behaviors are connected to the breaking of detailed balance at the boundaries.