The Boltzmann Equation and Its Applications

Cercignani, Carlo

doi:10.1007/978-1-4612-1039-9

Cited by 2,421 publications

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“…It depends on a quantum scale, dictated by the small Planck's constant ∼ 10 −34 . As another example of a semilinear PDEs with a pivotal role in mathematical physics, we mention the Boltzmann equation [67,35], which provides a microscopic description of the dynamics of many particles in dilute gases.…”

Section: Examples Of Time-dependent Pdes Atomic Physics Is Dominatedmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Section: Examples Of Time-dependent Pdes Atomic Physics Is Dominatedmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

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“…Some integral parameters of the shock wave front can be obtained by gas dynamics equations with additional thermodynamic relations, for weak shocks the hydrodynamic approach can give the shock front structure too [50]. For strong shocks it is necessary to use the kinetic representation, for rarefied gases the Boltzmann kinetic equation gives the framework for studying the structure of strong shocks [51]. This equation describes the dynamics of the one-particle distribution function f (v, x), where v is the vector of particle velocity, and x is the particle position in space.…”

Section: Tamm-mott-smith Approximation For Kinetics Of Shock Wavesmentioning

confidence: 99%

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban

Karlin

2004

Physica A: Statistical Mechanics and its Applications

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Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given ansatz manifold which is not tangent to the Lyapunov function levels.Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.)In example, we apply the thermodynamic projector to derivation the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally, as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2 m , and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of "molecular individualism". This is the third result.

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“…This permits to obtain a 1D reduced order model: (1) Mq(t) + Bq(t) + Kq(t) = F (t), where M is the generalised mass, B the damping coefficient and K the generalised stiffness. In terms of these parameters, the quality factor can be simply computed as…”

Section: Introductionmentioning

confidence: 99%

“…Anyway, among the different sources, gas damping often provides a meaningful contribution. The length scale and the working pressure are such that the collisions between molecules can be neglected; this regime is known as free-molecule flow [1][2][3][4][5][6]. The deterministic model implemented, proposed in [7,8], is an application of the collisionless Boltzmann equation which rests on the following assumptions: i) the mean free-molecule path is much larger than the typical dimension d of the flow (i.e.…”

Section: Introductionmentioning

confidence: 99%

Integral equations for free-molecule ow in MEMS: recent advancements

Fedeli

Frangi

2017

Communications in Applied and Industrial Mathematics

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We address a Boundary Integral Equation (BIE) approach for the analysis of gas dissipation in near-vacuum for Micro Electro Mechanical Systems (MEMS). Inspired by an analogy with the radiosity equation in computer graphics, we discuss an efficient way to compute the visible domain of integration. Moreover, we tackle the issue of near singular integrals by developing a set of analytical formulas for planar polyhedral domains. Finally a validation with experimental results taken from the literature is presented.

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The Boltzmann Equation and Its Applications

Cited by 2,421 publications

References 0 publications

A review of numerical methods for nonlinear partial differential equations

A review of numerical methods for nonlinear partial differential equations

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Integral equations for free-molecule ow in MEMS: recent advancements

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