Boltzmann Operator Research Articles

Mathematical Models for the Large Spread of a Contact-Based Infection: A Statistical Mechanics Approach

In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.

Finite-Temperature Correlation Functions Obtained from Combined Real- and Imaginary-Time Propagation of Variational Thawed Gaussian Wavepackets.

We apply the so-called variational Gaussian wavepacket approximation (VGA) for conducting both real- and imaginary-time dynamics to calculate thermal correlation functions. By considering strongly anharmonic systems, such as a quartic potential and a double-well potential at high and low temperatures, it is shown that this method is partially able to account for tunneling. This is contrary to other popular many-body methods, such as ring polymer molecular dynamics and the classical Wigner method, which fail in this respect. It is a historical peculiarity that no one has considered the VGA method for representing both the Boltzmann operator and the real-time propagation. This method should be well suited for molecular systems containing many atoms.

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model.

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

Fredholm property of the linearized Boltzmann operator for a polyatomic single gas model

Fredholm property of the linearized Boltzmann operator for a polyatomic single gas model

An explicit coercivity estimate of the linearized quantum Boltzmann operator without angular cutoff

The quantum Boltzmann-Bose equation describes a large system of Bose-Einstein particles in the weak-coupling regime. If the particle interaction is governed by the inverse power law, the corresponding collision kernel has angular singularity. In this paper, we give a constructive proof of the coercivity estimate for the linearized quantum Boltzmann-Bose operator to capture the effects of the singularity and the fugacity. Precisely, the estimate explicitly reveals the dependence on the fugacity parameter before the Bose-Einstein condensation. With the coercivity estimate, the global in time well-posedness of the inhomogeneous quantum Boltzmann-Bose equation in the perturbative framework and stability of the Bose-Einstein equilibrium can be established.

From the Vlasov-Poisson-Boltzmann System to the Vlasov-Poisson-Landau System with Coulomb Potential

In the celebrated work [Y. Guo., J. Amer. Math. Soc., 25 (2012), pp. 759–812], Guo proved global well-posedness of the Vlasov–Poisson–Landau (VPL) system with Coulomb potential in algebraically weighted Sobolev spaces. Thanks to this breakthrough, there are rich developments on the VPL system and also on the Vlasov–Poisson–Boltzmann (VPB) system with cutoff or noncutoff Boltzmann kernels. In this work, we are interested in the connection between the two important physical systems where particles interact through the Coulomb force and are influenced by their self-consistent electrostatic field. The Coulomb potential yields the famous Rutherford scattering cross section which is too singular in the angular variable for the Boltzmann equation to be meaningful. With an angular cutoff and a proper scaling as in [R. Alexandre and C. Villani, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 61–95], we prove global well-posedness of VPB in algebraically weighted Sobolev spaces. To this end, we develop a new two-stage energy method and merely rely on the -norm (from the dissipation of the linearized Boltzmann operator) to control the nonlinear terms containing the electrostatic field. Consequently, the same well-posedness result also holds for the VPB system with all the other soft potentials, which removes the technical constraint in [R. Duan and S. Liu, Comm. Math. Phys., 324 (2013), pp. 1–45]. Moreover, when the cutoff parameter tends to zero, convergence of VPB to VPL is established with an explicit rate.

Solutions to the non-cutoff Boltzmann equation in the grazing limit

It is known that in the parameters range $-2 \leq \gamma <-2s$ spectral gap does not exist for the linearized Boltzmann operator without cutoff but it does for the linearized Landau operator. This paper is devoted to the understanding of the formation of spectral gap in this range through the grazing limit. Precisely, we study the Cauchy problems of these two classical collisional kinetic equations around global Maxwellians in torus and establish the following results that are uniform in the vanishing grazing parameter $\epsilon$: (i) spectral gap type estimates for the collision operators; (ii) global existence of small-amplitude solutions for initial data with low regularity; (iii) propagation of regularity in both space and velocity variables as well as velocity moments without smallness; (iv) global-in-time asymptotics of the Boltzmann solution toward the Landau solution at the rate $O(\epsilon)$; (v) continuous transition of decay structure of the Boltzmann operator to the Landau operator. In particular, the result in part (v) captures the uniform-in-$\epsilon$ transition of intrinsic optimal time decay structures of solutions that reveals how the spectrum of the linearized non-cutoff Boltzmann equation in the mentioned parameter range changes continuously under the grazing limit.

The grazing collisions limit from the linearized Boltzmann equation to the Landau equation for short-range potentials

The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $ \Phi(x) $ in three dimensions. We then consider the limit of weak interaction given by $ \Phi_ \epsilon(x) = \epsilon \Phi(x) $. Here $ \epsilon\rightarrow 0 $ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $ s\geq 0 $ at the origin, i.e. $ \phi(x) \sim |x|^{-s} $ as $ |x|\rightarrow 0 $. For $ s\in [0,1] $, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $ s&gt;1 $ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $ s = 1 $ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.

The Boltzmann equation with an external force on the torus : incompressible Navier–Stokes–Fourier hydrodynamical limit

We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajec-tories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier-Stokes-Fourier system with an external force from the Boltzmann equation. Neither smallness, nor time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cutoff Boltzmann kernels. As a by-product the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.

De Giorgi Argument for Weighted L^2cap L^infty Solutions to the Non-cutoff Boltzmann Equation

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the L^infty -setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in L^2 cap L^infty with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted L^2cap L^infty -space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable L^p-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in Alonso et al. (Rev Mat Iberoam, 2020), we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both L^2 and L^infty -spaces.

Neural-network based collision operators for the Boltzmann equation

Kinetic gas dynamics in rarefied and moderate-density regimes have complex behavior associated with collisional processes. These processes are generally defined by convolution integrals over a high-dimensional space (as in the Boltzmann operator), or require evaluating complex auxiliary variables (as in Rosenbluth potentials in Fokker-Planck operators) that are challenging to implement and computationally expensive to evaluate. In this work, we develop a data-driven neural network model that augments a simple and inexpensive BGK collision operator with a machine-learned correction term, which improves the fidelity of the simple operator with a small overhead to overall runtime. The composite collision operator has a tunable fidelity and, in this work, is trained using and tested against a direct-simulation Monte-Carlo (DSMC) collision operator.

Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions

In this paper, we investigate a compactness property of the linearized Boltzmann operator in the context of a polyatomic gas whose molecules undergo resonant collisions. The peculiar structure of resonant collision rules allows to tensorize the problem into a velocity-related one, neighbouring the monatomic case, and an internal energy-related one. Our analysis is based on a specific treatment of the internal energy contributions. We also propose a geometric variant of Grad's proof of the same compactness property in the monatomic case.

Efficient world-line-based quantum Monte Carlo method without Hubbard–Stratonovich transformation

By precisely writing down the matrix element of the local Boltzmann operator ({mathrm{e}}^{-tau h}, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard–Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.

On the Cauchy problem of Boltzmann equation with a very soft potential

This work proves the global existence to Boltzmann equation in the whole space with very soft potential \(\gamma \in [0,d)\) and angular cutoff, in the framework of small perturbation of equilibrium state. In this article, we generalize the estimate on linearized collision operator L to the case of very soft potential and obtain the spectrum structure of the linearized Boltzmann operator correspondingly. The global classic solution can be derived by the method of strongly continuous semigroup. For soft potential, the linearized Boltzmann operator could not give spectral gap; hence, we have to consider a weighted velocity space in order to obtain algebraic decay in time.

A Hybrid Smart Quantum Particle Swarm Optimization for Multimodal Electromagnetic Design Problems

Quantum behaved particle swarm optimization (QPSO) has been one of the most widely used algorithm in engineering world. Since its debut in 2004, QPSO has been used for resolving numerous complicated multimodal problems. Moreover, considering the adaptability and versatility, it has resolved a variety of real-world and test problems. To tackle numerical and engineering optimization problems, we introduce novel hybrid algorithm QPSODE. The novel hybrid algorithm integrates Quantum behaved particle swarm optimization (QPSO) with differential evolution (DE) strategy. A crossover and selection (influenced by DE) is used in the QPSODE&#x2019;s position updating mechanism. During the selection process, the Boltzmann operator is applied to the position vectors of two randomly chosen particles, not to their individual optimum placements. Therefore, unlike the QPSO, a particle is only relocated to a new position if it has a higher fitness value, implying the application of a selection strategy across the whole search space. Additionally, the hybrid algorithm is improved by introducing proper parameters tuning, control parameter, path disparity. The hybrid algorithm enhances the algorithm&#x2019;s performance by speeding up the convergence and avoiding the premature convergence, the main flaw in the earlier algorithms. The proposed algorithm is put to test, by using 19 well-known benchmark test functions and the engineering optimization problem for superconducting magnetic energy storage (SMES). In terms of the quality of the resulting outputs, QPSODE outperforms various state-of-the-art approaches.

Compactness property of the linearized Boltzmann operator for a diatomic single gas model

&lt;p style='text-indent:20px;'&gt;In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal{B} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, we prove that the linearized Boltzmann operator &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal{L} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of this model is a Fredholm operator. For this, we write &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathcal{L} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathcal{K} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is compact. The result is established after inspecting the kernel form of &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathcal{K} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and proving it to be &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ L^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; integrable over its domain using elementary arguments.This implies that &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \mathcal{K} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a Hilbert-Schmidt operator.&lt;/p&gt;

Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation

In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the Gevrey index depending only on the angular singularity. This together with [24] provides a self-contained well-posedness theory for both existence and regularity of global solutions for initial data of low regularity in the framework of perturbations. For the proof we treat in a subtle way the commutator between the regularization operators and the Boltzmann collision operator involving rough coefficients, and this enables us to combine the classical Hörmander's hypoelliptic techniques together with the global symbolic calculus established for the linearized Boltzmann operator so as to improve the regularity of solutions at positive time.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

A Divergence-Free Wigner Transform of the Boltzmann Operator Based on an Effective Frequency Theory.

The centroid effective frequency representation of path integrals as developed by Feynman and Kleinert was originally aimed at calculating partition functions and related quantities in the canonical ensemble. In its path integral formulation, only closed paths were relevant. This formulation has been used by the present authors in order to calculate the many-body Wigner function of the Boltzmann operator, which includes also open paths. This usage of the theory outside of the original intention can lead to mathematical divergence issues for potentials with barriers, particularly at low temperature. In the present paper, we modify the effective frequency theory of Feynman and Kleinert by also including open paths in its variational equations. In this way, a divergence-free approximation to the Boltzmann operator matrix elements is derived. This generalized version of Feynman and Kleinert’s formulation is thus more robust and can be applied to all types of barriers at all temperatures. This new version is used to calculate the Wigner functions of the Boltzmann operator for a quartic oscillator and for a double well potential and both static and dynamic properties are studied at several temperatures. The new theory is found to be essentially as precise as the original one. Its advantage is that it will always deliver a well-defined, even if approximate, Wigner function, which can, for instance, be used for sampling initial conditions for molecular dynamics simulations. As will be discussed, the theory can be systematically improved by including higher-order Fourier modes into the nonquadratic part of the trial action.

Asymptotic behavior of linearized Boltzmann equations for soft potentials with cut-off

We consider the asymptotic behavior of the linearized Boltzmann equation for soft potentials with cut-off. By introducing a new decomposition of the linearized Boltzmann operator, we analyze the spectrum of the linearized Boltzmann operator and obtain the asymptotic behaviors of the linearized Boltzmann equation for \(\gamma\in (-3,0)\), extending the result in [12] for \(\gamma\in (-1,0)\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/46/abstr.html