Hydrodynamic Limit Research Articles

Probing double-distribution-function models in discrete-velocity Boltzmann methods for highly compressible flows: Particles-on-demand realization

The double distribution function approach is an efficient route toward an extension of kinetic solvers to compressible flows. With a number of realizations available, an overview and comparative study in the context of high-speed compressible flows is presented. We discuss the different variants of the energy partition, analyses of hydrodynamic limits, and a numerical study of accuracy and performance with the particles on demand realization. Out of three considered energy partition strategies, it is shown that the nontranslational energy split requires a higher-order quadrature for proper recovery of the Navier-Stokes-Fourier equations. The internal energy split, on the other hand, while recovering the correct hydrodynamic limit with fourth-order quadrature, comes with a nonlocal—both in space and time—source term that contributes to higher computational cost and memory overhead. Based on our analysis, the total energy split demonstrates the optimal overall performance. Published by the American Physical Society 2024

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

Hydrodynamic limit for the non-cutoff Boltzmann equation

This work deals with the non-cutoff Boltzmann equation for all types of potentials, in both the torus \mathbf{T}^{3} and in the whole space \mathbf{R}^{3} , under the incompressible Navier–Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is, for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmann equation towards the incompressible Navier–Stokes–Fourier system.

On Hydrodynamic Limits of the Vlasov–Navier–Stokes System

We introduce a framework to justify hydrodynamic limits of the Vlasov–Navier–Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport–Navier–Stokes (resp. Inhomogeneous Navier–Stokes) systems.

Kinetic description and macroscopic limit of swarming dynamics with continuous leader–follower transitions

In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.

Environmental averaging

Many classical examples of models of self-organized dynamics, including the Cucker–Smale, Motsch–Tadmor, multi-species, and several others, include an alignment force that is based upon density-weighted averaging protocol. Those protocols can be viewed as special cases of “environmental averaging”. In this paper we formalize this concept and introduce a unified framework for systematic analysis of alignment models.A series of studies are presented including the mean-field limit in deterministic and stochastic settings, hydrodynamic limits in the monokinetic and Maxwellian regimes, hypocoercivity and global relaxation for dissipative kinetic models, several general alignment results based on chain connectivity and spectral gap analysis. These studies cover many of the known results and reveal new ones, which include asymptotic alignment criteria based on connectivity conditions, new estimates on the spectral gap of the alignment force that do not rely on the upper bound of the macroscopic density, uniform gain of positivity for solutions of the Fokker–Planck-alignment model based on smooth environmental averaging. As a consequence, we establish unconditional relaxation result for global solutions to the Fokker–Planck-alignment model, which presents a substantial improvement over previously known perturbative results.

Radiation pressure-driven Rayleigh–Taylor instability in compressible strongly magnetized ultra-relativistic degenerate plasmas

The radiation pressure and strong magnetic fields are prominent in the structures of Rayleigh–Taylor (R–T) instability in the interior of white dwarfs. This paper investigates the radiation pressure-driven R–T instability in a compressible and magnetized ultra-relativistic degenerate strongly coupled plasma. The equation of state has been derived for such systems incorporating ultra-relativistic degenerate electrons with their radiation pressure and ion gas compressibility. The dispersion relation of the density gradients driven R–T instability is analyzed using the generalized hydrodynamic fluid model in the strongly coupled and weakly coupled limits. It is observed that the R–T instability criterion has been modified significantly due to radiation pressure, ion gas compressibility and degeneracy parameters. In the kinetic limit, the instability region is shorter than the hydrodynamic limit due to the dominance of plasma frequency over the viscoelastic relaxation frequency. The outcomes are explored in analyzing the development of R–T instability in the strongly magnetized carbon–oxygen white dwarfs. The radiation pressure, electron temperature and ion density strongly suppress the growth rate of the R–T instability in the interior of white dwarfs. The strong magnetic fields introduce asymmetry to the system by destabilizing the R–T unstable modes. The present results are also useful for understanding the R–T instability in the star formation and dense plasmas in inertial confinement fusion in some limiting cases.

Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries

Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries

How general is the Jensen–Varadhan large deviation functional for 1D conservation laws?

Starting from a microscopic particle model whose hydrodynamic limit under hyperbolic space-time scaling is a 1D conservation law, we derive the large deviation rate function encoding the probability to observe a density profile which is a non entropic shock, and compare this large deviation rate function with the classical Jensen-Varadhan functional, valid for the totally asymmetric exclusion process and the weakly asymmetric exclusion process in the strong asymmetry limit. We find that these two functionals have no reason to coincide, and in this sense Jensen-Varadhan functional is not universal. However, they do coincide in a small Mach number limit, suggesting that universality is restored in this regime. We then compute the leading order correction to the Jensen-Varadhan functional.

Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails

Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails

Discrete Boltzmann model with split collision for nonequilibrium reactive flows* *This work is supported by the National Natural Science Foundation of China (under Grant Nos. U2242214, 51806116 and 91441120), the Guangdong Basic and Applied Basic Research Foundation (under Grant Nos. 2022A1515012116 and 2024A1515010927), the Natural Science Foundation of Fujian Province (under Grant Nos. 2021J01652, 2021J01655), and the China Scholarship Council

A multi-relaxation-time discrete Boltzmann model (DBM) with split collision is proposed for both subsonic and supersonic compressible reacting flows, where chemical reactions take place among various components. The physical model is based on a unified set of discrete Boltzmann equations that describes the evolution of each chemical species with adjustable acceleration, specific heat ratio, and Prandtl number. On the right-hand side of discrete Boltzmann equations, the collision, force, and reaction terms denote the change rates of distribution functions due to self- and cross-collisions, external forces, and chemical reactions, respectively. The source terms can be calculated in three ways, among which the matrix inversion method possesses the highest physical accuracy and computational efficiency. Through Chapman–Enskog analysis, it is proved that the DBM is consistent with the reactive Navier–Stokes equations, Fick's law and the Stefan–Maxwell diffusion equation in the hydrodynamic limit. Compared with the one-step-relaxation model, the split collision model offers a detailed and precise description of hydrodynamic, thermodynamic, and chemical nonequilibrium effects. Finally, the model is validated by six benchmarks, including multicomponent diffusion, mixture in the force field, Kelvin–Helmholtz instability, flame at constant pressure, opposing chemical reaction, and steady detonation.

Bifractality in the one-dimensional Wolf-Villain model.

We introduce a multifractal optimal detrended fluctuation analysis to study the scaling properties of the one-dimensional Wolf-Villain (WV) model for surface growth. This model produces coarsened surface morphologies for long timescales (up to 10^{9} monolayers) and its universality class remains an open problem. Our results for the multifractal exponent τ(q) reveal an effective local roughness exponent consistent with a transient given by the molecular beam epitaxy (MBE) growth regime and Edwards-Wilkinson (EW) universality class for negative and positive q values, respectively. Therefore, although the results corroborate that long-wavelength fluctuations belong to the EW class in the hydrodynamic limit, as conjectured in the recent literature, a bifractal signature of the WV model with an MBE regime at short wavelengths was observed.

Gravitational instability in magnetized viscoelastic fluid with radiation pressure effect

This study examines the gravitational instability of a finitely conducting magnetized viscoelastic fluid, under the influence of radiation pressure within the framework of a generalized hydrodynamic fluid model. The general dispersion relation, valid for strongly and weakly coupled fluids under kinetic and hydrodynamic limits respectively, is derived using normal mode analysis for a transverse wave propagation mode. The obtained Jeans instability criteria, in terms of critical Jeans wavenumbers under both limits, are modified by the presence of radiative effects, viscoelastic compressional speed, and Alfvén wave velocity. The findings highlight that viscoelastic compressional speed and radiative pressure slow the Jeans instability's growth rate, exerting a stabilizing effect on the initiation of gravitational collapse. The present investigation provides valuable insights into the role of radiative mechanisms in the gravitational collapse within the strongly coupled region of molecular cloud clumps.

Hybrid model of condensate and particle dark matter: Linear perturbations in the hydrodynamic limit

We analyze perturbations of self-interacting, scalar field dark matter that contains modes in both a coherent condensate state and an incoherent particlelike state. Starting from the coupled equations for the condensate, the particles’ phase-space distribution, and their mutual gravitational potential, first derived from first principles in earlier work by the authors, we derive a hydrodynamic limit of two coupled fluids and study their linearized density perturbations in an expanding universe, also including particle pressure under an assumption for an equation of state consistent with the dynamical equations. We find that away from the condensate-only or particle-only limits, and for certain ranges of the parameters, such self-interacting mixtures can significantly enhance the density power spectrum above the standard linear ΛCDM value at localized wave numbers, even pushing structure formation into the nonlinear regime earlier than expected in ΛCDM for these scales. We also note that such mixtures can lead to degeneracies between models with different boson masses and self-coupling strengths, in particular, between self-coupled models and noncoupled fuzzy dark matter made up of heavier bosons. These findings open up the possibility of a richer phenomenology in scalar field dark matter models and could further inform efforts to place observational limits on their parameters. Published by the American Physical Society 2024

Hydrodynamics for Asymmetric Simple Exclusion on a Finite Segment with Glauber-Type Source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Partial Differ Equ 4(9):1017–1034, https://doi.org/10.1080/03605307908820117, 1979) and Otto (C R Acad Sci Paris 322(1):729–734, 1996), discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

Richtmyer-Meshkov instability when a shock wave encounters with a premixed flame from the burned gas

We present a linear stability analysis of the Richtmyer-Meshkov instability that develops when a shock wave reaches a sinusoidally perturbed premixed flame from behind. In the hydrodynamic regime, when acoustic contributions dominate the flame growth rate, the problem is analytically addressed by the direct integration of the sound wave equations at both sides of the flame, which are bounded by the reflected and transmitted shock waves and the flame front that acts as a contact surface in the hydrodynamic limit. The resolution involves: a hyperbolic change of variables to modify the triangular spatio-temporal domain, a transformation in the Laplace variable, the resolution of the functional equations in the frequency domain, and the final inverse Laplace transform. The latter involves a novel resolution method that is proven beneficial for long-time dynamics. Asymptotic analysis is also carried out to describe the early time and late time hydrodynamic response. The nonuniform flow field resulting from distorted oscillating shocks is characterized by acoustic, rotational, and entropic disturbances, each of which exerts a substantial influence on flame dynamics. These disturbances contribute to the intricate interplay of factors shaping the behavior of the flame in response to the nonuniform flow. In particular, the sensitivity of local flame propagation to temperature disturbances is investigated. This work contributes to a deeper understanding of Richtmyer-Meshkov instability dynamics and offers insights into instability reduction through the modulation of temperature disturbances generated by the transmitted shock.

The Multiscale Hybrid Method with a Localized Constraint. II. Hybrid Equations of Motion Based on Variational Principles

A multiscale modelling framework that employs molecular dynamics and hydrodynamics principles has been developed to describe the dynamics of hybrid particles. Based on the principle of least action, the equations of motion for hybrid particles were derived and verified by using the Gauss principle of least constraints testifying to their accuracy and applicability under various system constraints. The proposed scheme has been implemented in a popular open-source molecular dynamics code GROMACS. The simulation for liquid argon under equilibrium conditions in the hydrodynamic limit (s = 1) has demonstrated that the standard deviation of the density exhibits a remarkable agreement with predictions from a pure hydrodynamics model, validating the robustness of the proposed framework.

On the Hydrodynamics of Active Matter Models on a Lattice

Active matter has been widely studied in recent years because of its rich phenomenology, whose mathematical understanding is still partial. We present some results, based on [8, 17] linking microscopic lattice gases to their macroscopic limit, and explore how the mathematical state of the art allows to derive from various types of microscopic dynamics their hydrodynamic limit. We present some of the crucial aspects of this theory when applied to weakly asymmetric active models. We comment on the speci fic challenges one should consider when designing an active lattice gas, and in particular underline math- ematical and phenomenological differences between gradient and non-gradient models. Our purpose is to provide the physics community, as well as member of the mathematical community not specialized in the mathematical derivation of scaling limits of lattice gases, some key elements in de ning microscopic models and deriving their hydrodynamic limit.

Mechanical stability of homogeneous holographic solids under finite shear strain

We study the linear stability of holographic homogeneous solids (HHS) at finite temperature and in presence of a background shear strain by means of a large scale quasi-normal mode analysis which extends beyond the hydrodynamic limit. We find that mechanical instability can arise either as a result of a complex speed of sound — gradient instability — or of a negative diffusion constant. Surprisingly, the simplest HHS models are linearly stable for arbitrarily large values of the background strain. For more complex HHS, the onset of the diffusive instability always precedes that of the gradient instability, which becomes the dominant destabilizing process only above a critical value of the background shear strain. Finally, we observe that the critical strains for the two instabilities approach each other at low temperatures. We conclude by presenting a phase diagram for HHS as a function of temperature and background shear strain which shows interesting similarities with the physics of superfluids in presence of background superfluid velocity.

Radiation pressure and galactic cosmic rays-driven gravitational instability in rotating and magnetized viscoelastic fluids

This paper studies the combined effects of radiation and galactic cosmic ray pressures on the gravitational instability of magnetized and rotating viscoelastic fluids. The dispersion relations are derived using the normal mode analysis and discussed in the hydrodynamic (weakly coupled fluid) and kinetic (strongly coupled fluid) limits. These dispersion relations are analyzed separately for transverse and longitudinal wave propagation modes. Jeans instability criteria are obtained for kinetic and hydrodynamic limits for both modes of wave propagation, and it is found that the critical Jeans wavenumbers in each case are modified due to the presence of viscoelastic effects, radiation and cosmic rays pressures, and Alfvên wave velocity. It is also observed that the radiation pressure, cosmic ray pressure and viscoelastic parameters suppress the growth rate and thus have stabilizing effects on the Jeans instability. However, cosmic ray diffusion has a destabilizing effect on the onset of gravitational instability. The effects of various parameters on the growth rate of instability are calculated numerically and the outcomes are depicted graphically. The results of the present analysis shall be helpful in understanding the impact of cosmic rays and radiative mechanisms on the gravitational collapse in the viscoelastic region of molecular cloud clumps.