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AbstractAbstract
[en] Transport of particles in media whose cross sections are random functions of space and time is considered. The linear transport equation, in the presence of this spacetime noise, is viewed as a Boltzmann-Langevin equation, the solution of which generates a stochastic process, the angular flux, for different realizations of the cross section. A Gaussian model of fluctuations is adopted with a prescribed mean, variance and correlation function. For white noise in time, but with otherwise arbitrary spatial correlation, an exactly closed equation for the ensemble averaged angular flux is obtained and seen to be identical to the transport equation but with renormalised cross sections. Similar exact closures are demonstrated for the second moment and the two-point space-angle correlation of the angular flux. Standard methods may be adapted to solve these averaged transport equations which are valid in the small correlation time limit. (Author)
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[en] Boltzmann's equation is solved for the diffusion of nonabsorbing particles through a thick slab. The dimensionless thickness D=dμs, where d is the thickness and μs is the rate of scattering. Our solution neglects terms of order O(e-D) and is exact in the limit that D much-gt 1. Simple expressions are obtained for the reflection and transmission coefficients, as well as for the angular distributions on the front and back sides of the slab. copyright 1995 American Institute of Physics
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[en] The Uehling endash Uhlenbeck evolution equations for gases of identical quantum particles either fermions or bosons, in the case in which the collision kernel does not depend on the distribution function, are considered. The existence of solutions and their asymptotic relations with solutions of the hydrodynamic equations both at the level of the Euler system and at the level of the Navier endash Stokes system are proved. copyright 1997 American Institute of Physics
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[en] We discuss steady boundary value problems for the Boltzmann equation with inflow and diffusive boundary conditions in one, two, and three dimensions, with suitable truncations of the collision kernel. General existence and uniqueness results are obtained if the domain is sufficiently small. In one dimension, the existence of solutions on general intervals is obtained by abstract fixed-point theory
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Chen Nan-Xian; Sun Bo-Hua, E-mail: nanxian@mail.tsinghua.edu.cn2017
AbstractAbstract
[en] Within about a year (1916–1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time . Since then, this divergence problem has puzzled researchers for a century. Using a modified Möbius series inversion formula, we propose a modified Chapman–Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also the best-so-far explanation on some unbelievable scenarios occurring in previous practice. (paper)
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Available from http://dx.doi.org/10.1088/0256-307X/34/2/020502; Country of input: International Atomic Energy Agency (IAEA)
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[en] The macroscopic equations derived from the lattice Boltzmann equation are not exactly the Navier-Stokes equations. Here the cubic deviation terms and the methods proposed to eliminate them are studied. The most popular two- and three-dimensional models (D2Q9, D3Q15, D3Q19, D3Q27) are considered in the paper. It is demonstrated that the compensation methods provide only partial elimination of the deviations for these models. It is also shown that the compensation of Qian and Zhou (1998 Europhys. Lett. 42 359) using the compensation parameter K = 1 in a D2Q9 or D3Q27 model can eliminate all the cross terms perfectly, but the deviation terms ∂xρu3x, ∂yρu3y and ∂zρu3z still survive the compensation
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S0305-4470(06)13396-X; Available online at http://stacks.iop.org/0305-4470/39/3127/a6_12_019.pdf or at the Web site for the Journal of Physics. A, Mathematical and General (ISSN 1361-6447) http://www.iop.org/; Country of input: International Atomic Energy Agency (IAEA)
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Journal of Physics. A, Mathematical and General; ISSN 0305-4470;
; CODEN JPHAC5; v. 39(12); p. 3127-3136

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AbstractAbstract
[en] A dissipative integro-differential operator L arising in the linearization of Boltzmann's equation in one-speed particle transport theory is considered. Under assumptions ensuring that the point spectrum of L is finite a scalar multiple of the characteristic functions of L is found and a condition for the absence of spectral singularities is indicated. Using the techniques of non-stationary scattering theory and the Sz.-Nagy-Foias functional model direct and inverse wave operators with the completeness property are constructed. The structure of the operator L in the invariant subspace corresponding to its continuous spectrum is studied
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Available from http://dx.doi.org/10.1070/SM2001v192n01ABEH000539; Country of input: International Atomic Energy Agency (IAEA)
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Sbornik. Mathematics; ISSN 1064-5616;
; v. 192(1); p. 141-162

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[en] We present a variational formulation of the steady Boltzmann equation for semiconductors. In this formulation, the distribution function is replaced by a weighted distribution function, and the symmetry of the drift operator is obtained by using the parity operator. We show that the solutions of the Boltzmann equation for the weighted distribution function are stationary functions of a suitable functional, which takes into account realistic boundary conditions. After introducing a general numerical framework, the approach proposed is tested in the bulk case, by computing an approximate expression for carrier mobility in silicon.
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Nuovo Cimento della Societa Italiana di Fisica. B, General Physics, Relativity, Astronomy and Mathematical Physics and Methods; ISSN 1594-9982;
; v. 123(1); p. 95-120

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[en] The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to analyse transport properties in spatially inhomogeneous states close to the simple shear flow. A normal solution is obtained via a Chapman-Enskog-like expansion around a local shear flow distribution f(0) that retains all the hydrodynamic orders in the shear rate. The constitutive equations for the heat and momentum fluxes are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the reference state and the corresponding generalized transport coefficients are exactly determined in terms of the coefficient of restitution α and the shear rate a. Since f(0) applies for arbitrary values of the shear rate and is not restricted to weak dissipation, the transport coefficients turn out to be nonlinear functions of both parameters a and α. A comparison with previous results obtained for inelastic hard spheres from a kinetic model of the Boltzmann equation is also carried out
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S1751-8113(07)49461-6; Country of input: International Atomic Energy Agency (IAEA)
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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121;
; v. 40(35); p. 10729-10757

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Krusiial, M.D.; Oberman, N.R.
Proceedings of the second United Nations international conference on the peaceful uses of atomic energy. V. 31. Theoretical and experimental aspects of controlled nuclear fusion1958
Proceedings of the second United Nations international conference on the peaceful uses of atomic energy. V. 31. Theoretical and experimental aspects of controlled nuclear fusion1958
AbstractAbstract
[en] Our purpose is to derive from the Boltzmann equation in the small m/e limit, criteria useful in the discussion of stability of plasmas in static equilibrium. At first we ignore collisions but later show their effects may be taken into account. Our approach yields a generalization of the usual energy principles for investigating the stability of hydromagnetic systems to situations where the effect of heat flow along magnetic lines is not negligible, and hence to situations where the strictly hydrodynamic approach is inapplicable. In the first two sections we characterize our general method of approach and delineate the properties of the small m/e limit which we use to determine the constants of the motion and the condition for static equilibrium. In the next two sections we calculate the first and second variations of the energy and conclude with a statement of the general stability criterion. In the final three sections we state several theorems which relate our stability criterion to those of ordinary hydromagnetic theory, we show how to take into account the effect of collisions, and briefly discuss the remaining problem of incorporating the charge neutrality condition into the present stability theory. (author)
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United Nations, Geneva (Switzerland); 400 p; 1958; p. 137-143; 2. United Nations international conference on the peaceful uses of atomic energy; Geneva (Switzerland); 1-13 Sep 1958; P--365-USA; 8 refs
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