Nonlocal Collision Research Articles

Damping of spin waves

We show that, in ideal-spin hydrodynamics, the components of the spin tensor follow damped wave equations. The damping rate is related to nonlocal collisions of the particles in the fluid, which enter at first order in ℏ in a semiclassical expansion. This rate provides an estimate for the timescale of spin equilibration and is computed by considering a system of spin-1/2 fermions interacting via a quartic self-interaction as well as via (screened) one-gluon exchange. It is found that the relaxation times of the components of the spin tensor can become very large compared to the usual dissipative timescales of the system. Our results suggest that the spin degrees of freedom in a heavy-ion collision may not be in equilibrium by the time of freeze-out, and thus should be treated dynamically. Published by the American Physical Society 2024

Exponential ergodicity for damping Hamiltonian dynamics with state-dependent and non-local collisions

In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes that is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in (Stochastic Process. Appl. (2022) 146 114–142) on exponential ergodicity of stochastic Hamiltonian systems with Lévy noises and a complement of (Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022a) 916–944) upon exponential ergodicity for Andersen dynamics with constant jump rate functions.

Lattice Boltzmann model for diffusion equation with reduced truncation errors: Applications to Gaussian filtering and image processing

Several image processing approaches require blurring an image using the Gaussian filter. In the present study the filtering method based on a novel four-velocity lattice Boltzmann model with non-local collision kernel is proposed. By applying the Chapman–Enskog expansion up to super-Burnett terms it is shown that the present scheme is equivalent to the diffusion equation with relatively small hyper-viscous defects (truncation errors). Compared to the conventional lattice Boltzmann model the proposed scheme has a small computational overhead but significantly improves accuracy due to the reduced hyper-viscosity. To validate the model the following test problems are considered: diffusion in time of a Gaussian bell, sinusoidal wave, image filtering, feature detection with the application of the modified SIFT method. The numerical experiments indicate that the novel model gives more precise results than the standard lattice Boltzmann model for non-small values of diffusion coefficient and in several cases its computational cost is competitive with OpenCV blurring techniques. In addition, the model can be efficiently used in feature detection algorithms like SIFT.

Relativistic spin transport theory for spin-1/2 fermions

Global polarization effect is an important physical phenomenon reflecting spin-orbit couplings in heavy ion collisions. Since STAR’s observation of the global polarization of <inline-formula><tex-math id="M2">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222470_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222470_M2.png"/></alternatives></inline-formula> hyperons in Au+Au collisions in 2017, this effect has attracted a lot of interests in the field. In the hot and dense matter produced in heavy ion collisions, the spin-orbit couplings come from non-local collisions between particles, in which the orbital angular momentum involves the space and momentum information of the colliding particles, so it is necessary to describe the particle collisions with spin-orbit couplings in phase space. In addition, the spin-orbit coupling is a quantum effect, which requires quantum theory. In combination of two aspects, the quantum kinetic theory based on covariant Wigner functions has become a powerful tool to describe the global polarization effect. In this paper, we introduce the quantum kinetic theory for spin-1/2 Fermion system based on Wigner functions as well as the spin transport theory developed on this basis. The recent research progress of spin transport theory provides a solid theoretical foundation for simulating the space-time evolution of spin polarization effects in heavy ion collisions.

Quadrature-based lattice Boltzmann model for non-equilibrium dense gas flows

The Boltzmann equation becomes invalid as the size of gas molecules is comparable with the average intermolecular distance. A better description is provided by the Enskog collision operator, which takes into account the finite size of gas molecules. This extension implies nonlocal collisions as well as an increase in collision frequency, making it computationally expensive to solve. An approximation of the Enskog collision operator, denoted the simplified Enskog collision operator, is used in this work to develop a quadrature-based lattice Boltzmann model for non-ideal monatomic dense gases. The Shakhov collision term is implemented in order to fine-tune the Prandtl number. This kinetic model is shown to be able to tackle non-equilibrium flow problems of dense gases, namely, the sound wave and the shock wave propagation. The results are compared systematically with the results of the more accurate but computationally intensive particle method of solving the Enskog equation. The model introduced in this paper is shown to have good accuracy for small to moderate denseness of the fluid (defined as the ratio of the molecular diameter to the mean free path), and due to the efficiency in terms of computational time, it is suitable for practical applications.

Lorentz-covariant nonlocal collision term for spin- 1/2 particles

We revisit the derivation of the nonlocal collision term in the Boltzmann equation for spin-1/2 particles, using both the Wigner-function approach by de Groot, van Leeuwen, and van Weert, and the Kadanoff-Baym equation in $T$-matrix approximation. Contrary to previous calculations, our results maintain full Lorentz covariance of the nonlocal collision term.

Lorentz-covariant kinetic theory for massive spin- 1/2 particles

We construct a matrix-valued spin-dependent distribution function (MVSD) for massive spin-$1/2$ fermions and study its properties under Lorentz transformations. Such transformations result in a Wigner rotation in spin space and in a nontrivial matrix-valued shift in space-time, which corresponds to the side jump in the massless case. We express the vector and axial-vector components of the Wigner function in terms of the MVSD and show that they transform in a Lorentz-covariant manner. We then construct a manifestly Lorentz-covariant Boltzmann equation which contains a nonlocal collision term encoding spin-orbit coupling. Finally, we obtain the spin-dependent distribution function in local equilibrium by demanding detailed balance.

Relativistic dissipative spin hydrodynamics from kinetic theory with a nonlocal collision term

We derive relativistic dissipative spin hydrodynamics from kinetic theory featuring a nonlocal collision term using the method of moments. In this framework, the components of the spin tensor are dynamical variables which obey relaxation-type equations. We find that the corresponding relaxation times are determined by the local part of the collision term, while the nonlocal part contributes to the Navier-Stokes terms in these equations of motion. The spin relaxation time scales are comparable to those of the usual dissipative currents. Finally, the Navier-Stokes limit of the Pauli-Lubanski vector receives contributions proportional to the shear tensor of the fluid, which implies that the polarization of hadrons observed in heavy-ion collisions is influenced by dissipative effects.

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.

A Fast Petrov--Galerkin Spectral Method for the Multidimensional Boltzmann Equation Using Mapped Chebyshev Functions

Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this method requires a domain truncation which is unphysical since the collision operator is defined in $\mathbb{R}^d$. In this paper, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we are able to construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT). We demonstrate the superior accuracy of the proposed method in comparison to the Fourier spectral method through a series of 2D and 3D examples.

Relativistic first-order spin hydrodynamics via the Chapman-Enskog expansion

In this paper, we present a detailed derivation of relativistic first-order spin hydrodynamics using the Chapman-Enskog method to linearize the nonlocal collision term for massive fermions proposed in \cite{Weickgenannt:2021cuo}, which well describes spin-orbit coupling in the collision process and is relevant for the research on local spin polarization. Based on this collisional term, we provide a formal discussion about first-order spin hydrodynamics and determine the motion equations of fluid variables and nonequilibrium corrections to the energy-momentum and spin tensors. The results indicate that the motion equations show no differences compared to spinless first-order hydrodynamics and the energy-momentum tensor receives no corrections from spin as far as first-order theory is concerned, which calls for the construction of second-order theory of fluids naturally incorporating the effect of spin-orbit coupling.

From Kadanoff-Baym to Boltzmann equations for massive spin- 1/2 fermions

We derive Boltzmann equations for massive spin-1/2 fermions with local and nonlocal collision terms from the Kadanoff--Baym equation in the Schwinger--Keldysh formalism, properly accounting for the spin degrees of freedom. The Boltzmann equations are expressed in terms of matrix-valued spin distribution functions, which are the building blocks for the quasi-classical parts of the Wigner functions. Nonlocal collision terms appear at next-to-leading order in $\hbar$ and are sources for the polarization part of the matrix-valued spin distribution functions. The Boltzmann equations for the matrix-valued spin distribution functions pave the way for simulating spin-transport processes involving spin-vorticity couplings from first principles.

Generating Spin Polarization from Vorticity through Nonlocal Collisions

We derive the collision term in the Boltzmann equation using the equation of motion for the Wigner function of massive spin-1/2 particles. To next-to-lowest order in ℏ, it contains a nonlocal contribution, which is responsible for the conversion of orbital into spin angular momentum. In a proper choice of pseudogauge, the antisymmetric part of the energy-momentum tensor arises solely from this nonlocal contribution. We show that the collision term vanishes in global equilibrium and that the spin potential is, then, equal to the thermal vorticity. In the nonrelativistic limit, the equations of motion for the energy-momentum and spin tensors reduce to the well-known form for hydrodynamics for micropolar fluids.

Non-dissipative hydrodynamic equations based on a nonlocal collision integral

We consider a slightly non-uniform one-component gas with weak potential interaction. The basis of the investigation is the known kinetic equation in the case of small interaction which is truncated up to the second order of smallness. This equation contains a nonlocal collision integral of the second order in small interaction. In this paper we consider the hydrodynamic stage of the system evolution, and, in contrast to the standard hydrodynamics, we take into account the non-locality of the collision integral. We propose the following set of the reduced description parameters which are the densities of the conserved quantities: the particle number density, the momentum density, and the total energy density. It should be stressed that in contrast to the standard hydrodynamics, the kinetic energy is not conserved, and only the total system energy is conserved if the nonlocal collision integral is used. Definitions of the system velocity and temperature are proposed; it should be stressed that the proposed temperature definition is based on the total system energy rather than on the kinetic one. The hydrodynamics in the leading order in small gradients is investigated, and it is shown that the system one-particle distribution function in the leading-in-gradients order coincides with the Maxwellian one. Particle number density, velocity and temperature time evolution equations (hydrodynamic equations) are derived in the non-dissipative case. The leading-in-interaction orders of the obtained equations coincide with the corresponding equations in the framework of the standard hydrodynamics. The corrections of the first and second order in small interaction are also obtained.

Nonequilibrium thermodynamics with binary quantum correlations.

The balance equations for thermodynamic quantities are derived from the nonlocal quantum kinetic equation. The nonlocal collisions lead to molecular contributions to the observables and currents. The corresponding correlated parts of the observables are found to be given by the rate to form a molecule multiplied with its lifetime which can be considered as collision duration. Explicit expressions of these molecular contributions are given in terms of the scattering phase shifts. The two-particle form of the entropy is derived extending the Landau quasiparticle picture by two-particle molecular contributions. There is a continuous exchange of correlation and kinetic energies condensing into the rate of correlated variables for energy and momentum. For the entropy, an explicit gain remains and Boltzmann's H theorem is proved including the molecular parts of the entropy.

Boltzmann equation with a nonlocal collision term and the resultant dissipative fluid dynamics

Starting with the relativistic Boltzmann equation where the collision term was generalized to include gradients of the phase-space distribution function, we recently presented a new derivation of the equations for the relativistic dissipative fluid dynamics. We compared them with the corresponding equations obtained in the standard Israel-Stewart and related approaches. Our method generates all the second-order terms that are allowed by symmetry, some of which have been missed by the traditional approaches, and the coefficients of other terms are altered. The first-order or Navier-Stokes equation too receives a small correction. Here we outline this work for the general audience.

Relaxed micro–macro schemes for kinetic equations

In Bennoune et al. (2008) [1], Lemou and Mieussens (2008) [6] we recently developed a general approach to design asymptotic preserving (AP) schemes for kinetic equations which are able to solve both macroscopic (small Knudsen number ε) and kinetic scales using the same model and the same numerical parameters. The strategy is based on micro/macro decompositions of the distribution function and can be applied to a large class of kinetic models (Boltzmann, Landau, etc.). However, the so-obtained schemes are shown to be AP for only close-to-equilibrium initial data in the non-linear case and, in general, they require costly inversions of non-local collision operators. In the present work, we introduce a new formulation of this strategy with the following properties: i) Initial data does not need to be well prepared, and may be independent of ε. ii) No inversion (even linear) of collision operators is needed and time-implicit schemes are obtained for the asymptotic models in the limit ε → 0 , making the schemes free from the usual diffusive CFL constraint. iii) The numerical schemes are consistent with the models for all values of ε > 0 , and degenerate into consistent discretizations of the asymptotic model (Euler, Navier–Stokes, diffusion) when ε goes to 0, the numerical parameters (time–space–velocity steps) being fixed. Preliminary numerical validations of this approach are done on the non-local linear transport equation and its diffusion limit.

Quantum lattice-gas model for computational fluid dynamics.

Quantum-computing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specifically, the computational simulation of fluid dynamics). The quantum algorithm is called a quantum lattice gas. An analytical treatment of the microscopic quantum lattice-gas system is carried out to predict its behavior at the mesoscopic scale. At the mesoscopic scale, a lattice Boltzmann equation with a nonlocal collision term that depends on the entire system wave function, governs the dynamical system. Numerical results obtained from an exact simulation of a one-dimensional quantum lattice model are included to illustrate the formalism. A symbolic mathematical method is used to implement the quantum mechanical model on a conventional workstation. The numerical simulation indicates that classical viscous damping is not present in the one-dimensional quantum lattice-gas system.

Viscosity in Keplerian Disks: Steady-State Velocity Distribution and Non-local Collision Effects

A kinetic model based on a numerical algorithm, rather than Boltzman's equation, yields the viscosity and velocity distribution for colliding, finite-size particles in a planetary ring. Results are similar to those of many-particle simulations, and show that non-local effects due to the finite size are dominant in many cases of interest. Only for small particles does the viscosity decrease with increasing optical thickness sufficiently for the standard ringlet instability model to apply. This numerical kinetic theory will allow study of multi-size particle distributions, as well as scattering due to gravitational interactions or alternative collision models.

Moderately dense gas quantum kinetic theory: Transport coefficient expressions

Expressions for the transport coefficients of a moderately dense gas are obtained, based on a recently derived density corrected quantum Boltzmann equation. Linearization of the equations determining the pair correlation and the ‘‘free’’ singlet density operators about local equilibrium is discussed first. The rate of change of the pair correlations is treated as dynamic effects for pairs of particles relaxing to local equilibrium via a relaxation time model arising from interactions with ‘‘third particles.’’ In contrast, the singlet density operator satisfies a Boltzmann equation with binary collisions. Spatially inhomogeneous corrections to the collision superoperator are included. Contributions to the transport coefficients arise from the perturbation from local equilibrium through fluxes associated with kinetic, collisional and, for the thermal conductivity, potential energy mechanisms. A comparison is made between the classical limit of the transport coefficient expressions obtained here and the classical expressions previously derived from the Boltzmann equation with the nonlocal collision corrections of Green and Bogoliubov.