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	History and outlook of statistical physics :: The Boltzmann Equation (Site not responding. Last check: 2007-10-14)
		In 1872 Ludwig Boltzmann in Graz generalized Maxwell's approach for the kinetic theory of dilute gases to nonequilibrium processes, so that he could investigate the transition from nonequilibrium to equilibrium.
		A further assumption in Boltzmann's expression for the collision term is that the velocities of the colliding molecules must be uncorrelated, which was later called the assumption of "molecular chaos" by Jeans.
		The H-theorem and the Boltzmann equation met with violent objections from physicists and from mathematicians.
	statisticfunction.net /boltzmann/index.html (330 words)

	Boltzmann's Work in Statistical Physics (Stanford Encyclopedia of Philosophy)
		However, Boltzmann's ideas on the precise relationship between the thermodynamical properties of macroscopic bodies and their microscopic constitution, and the role of probability in this relationship are involved and differed quite remarkably in different periods of his life.
		Boltzmann is often portrayed as a staunch defender of the atomic view of matter, at a time when the dominant opinion in the German-speaking physics community, led by influential authors like Mach and Ostwald, disapproved of this view.
		Actually, Boltzmann's subsequent work in gas theory in the next decade and a half was predominantly concerned with technical applications of his 1872 Boltzmann equation, in particular to gas diffusion and gas friction.
	plato.stanford.edu /entries/statphys-Boltzmann (12986 words)

	Ludwig Boltzmann - Wikipedia, the free encyclopedia
		Boltzmann was appointed to the Chair of Theoretical Physics at the University of Munich in Bavaria, Germany in 1890.
		Thus in 1900 Boltzmann went to the University of Leipzig, on the invitation of Wilhelm Ostwald.
		This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space.
	en.wikipedia.org /wiki/Ludwig_Boltzmann (1914 words)

	Boltzmann Transport Equation
		The equation is applied to analysis of the general currents within a system, the transport coefficients and the relationships between them.
		Boltzmann transport equation relates the properties of a non-equilibrium system, expressed by a non-equilibrium distribution, in terms of local equilibrium distributions.
		Boltzmann transport equation is applied to calculation of the general currents in a medium with particle and temperature gradients.
	urila.tripod.com /Boltzmann.htm (952 words)

	Boltzmann equation - Wikipedia, the free encyclopedia
		The Boltzmann equation, devised by Ludwig Boltzmann, describes the statistical distribution of particles in a fluid.
		The Boltzmann equation is used to study how a fluid transports physical quantities such as heat and charge, and thus to derive transport properties such as electrical conductivity, Hall conductivity, viscosity, and thermal conductivity.
		The Boltzmann equation is an equation for the time t evolution of the distribution (properly a density) function f(x, p, t) in one-particle phase space, where x and p are position and momentum, respectively.
	en.wikipedia.org /wiki/Boltzmann_equation (407 words)

	Physics Today September 2000 (Site not responding. Last check: 2007-10-14)
		Minimization of equation 1 with respect to the ion concentrations leads to the condition that they obey the Boltzmann distribution.
		The electrostatic self-energy of a macro-ion is computed by inserting the solution of the PB equation into equation 1 for an isolated macro-ion, and then subtracting the free energy with all charges set equal to zero.
		When this calculation is carried out for a charged rod, the self-energy is found to be positive; the increase in entropic free energy induced by the confinement of the ions near the rod exceeds the lowering of their electrostatic energy.
	www.aip.org /pt/vol-53/iss-9/captions/p38box1.html (478 words)

	Boltzmann Equation -- from Eric Weisstein's World of Physics
		Boltzmann first wrote down his equation in 1872, although Planck was actually the first to write it in this form.
		The second Boltzmann equation is a diffusion equation used in neutron transport theory,
		A third Boltzmann equation concerns particles in a gravitational field, and is given by
	scienceworld.wolfram.com /physics/BoltzmannEquation.html (150 words)

	Boltzmann, Ludwig (1844-1906) -- from Eric Weisstein's World of Scientific Biography
		Boltzmann's statistical interpretation led him to conclude that entropy
		Because of Boltzmann's dense and difficult style, his work was disseminated only after its exigesis by Ehrenfest in 1911.
		Boltzmann is buried in the Central Cemetery in Vienna.
	scienceworld.wolfram.com /biography/Boltzmann.html (151 words)

	Numeric Solution of the Non-Linear Poisson-Boltzmann Equation (Site not responding. Last check: 2007-10-14)
		An iterative sparse linear equation solver is used to solve for the electric field.
		Our fundamental method is to represent each field in the equation by a uniform grid of function values and use a non-linear partial differential equation solver to solve the discretized Poisson-Boltzmann equation (or equivalent) and thus obtain the potential field u.
		The resulting computed free energy transfer was therefore (-82.1583 + 1.0377) = -81.1087 as compared to the Born equation result of -81.977 (a relative error of 1%).
	www.chemcomp.com /feature/pboltz.htm (2338 words)

	Numerical Solution of the Boltzmann Transport Equation
		Numerical solution of the Boltzmann transport equation finds application in different fields such as nuclear reactor design, radiation shielding calculations, radiative transfer in stellar atmospheres, semiconductor device design, radiation oncology, and high energy physics, to name a few.
		In the first class, deterministic methods, the transport equation is discretized using a variety of methods and then solved directly or iteratively.
		We conclude these introductory remarks on the numerical solution of the Boltzmann transport equation, by noting that the rest of this chapter is divided into following sections: in section 1.2 we explain the motivation and objective for this research work.
	www.sdsc.edu /~majumdar/thesis/node2.html (468 words)

	Explanation of Misleading Nernst Slope by Boltzmann Equation (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-14)
		The factors affecting the slope, the electrode potential and it’s measurements, the importance of potential mechanism, and the modified Boltzmann equation are presented.
		The original Boltzmann equation is for an electrode with a single type of ion.
		Contrary to what's shown in the Nernst equation, that the slope is infinitely linear, E = q/C is linear only in a limited range depending on the nature of ions, the electrode surface and thickness, etc. Here, the usefulness of a mechanism or theory depends on how good it can explain the electrode potential phenomena.
	www.weissresearch.com.cob-web.org:8888 /Nernst.htm (1451 words)

	A New Hierarchy System on the Basis of a Master Boltzmann Equation for Microscopic Density (Site not responding. Last check: 2007-10-14)
		It is shown that Boltzmann's equation written in terms of microscopic density (namely the unaveraged Boltzmann Function) has a wider range of validity as well as finer resolvability for fluctuations than the conventional Boltzmann equation governing Boltzmann's function.
		In fact the new Boltzmann equation for ideal gases has implications as a microscopically exact continuity equation like Klimontovich's equation for plasmas, and can be derived without invoking any statistical concepts, e.g., distribution functions, or molecular chaos.
		The Boltzmann equation in the older formalism is obtained by averaging this equation only under a restricted condition of the molecular chaos.
	www.ca-homes.com /science/tsuge-5.htm (293 words)

	Springer Online Reference Works
		defines the hydrodynamic equations of an ideal fluid (the Euler equations).
		, to which correspond the Navier–Stokes equations with explicit expressions for the diffusion coefficients, for the heat conduction and for the two viscosity coefficients.
		The above methods of solution for a single-particle distribution function may be obtained immediately from the Bogolyubov chain of equations with a suitably small value for the parameter of inhomogeneity for the hydrodynamic approximation (that is, omitting the kinetic equation).
	eom.springer.de /c/c021510.htm (638 words)

	The adaptive multilevel finite element solution of the Poisson–Boltzmann equation on massively parallel computers
		The Poisson–Boltzmann equation is a second-order elliptic partial differential equation which describes the electrostatic potential around a fixed charge distribution in an ionic solution.
		In brief, this method is used to solve the linear matrix equations in an iterative fashion to determine improvements w to the solution.
		By using new methods for the parallel solution of elliptic partial differential equations, it is possible to leverage the teraflops computing power of massively parallel computers to perform electrostatic calculations on biological systems at scales approaching the cellular level.
	www.research.ibm.com /journal/rd/453/baker.html (4835 words)

	Solution of Boltzmann equation
		To estimate the evolution, I present a solution of the time-dependent collisional Boltzmann equation for orbits in a fixed potential.
		The net change in the phase-space distribution function, then, due to the resonant heating takes the form of a collisional Boltzmann equation where the right-hand-side collision term depends on the gradient of the phase-space distribution function (see Appendix for additional detail).
		The now linear partial differential equation may be solved by finite-difference on a three-dimensional grid (e.g.
	donald.phast.umass.edu /~weinberg/papers/lmc2/node4.html (570 words)

	Simulation Techniques for the Boltzmann Equation
		Ludwig Eduard Boltzmann was born on February 20, 1844 in Vienna (Austria) and died on September 05, 1906 in Duino near Trieste (Italy).
		Pictures of the young Boltzmann, the old Boltzmann, his tombstone, and a detail of his tombstone.
		The Boltzmann transport equation is a integro partial differential equation; the left-hand side describes the convective transport of the molecules, and the right-hand side models collisions between the molecules.
	www.math.umbc.edu /~gobbert/boltzmann.html (1593 words)

	1.1 The relativistic Boltzmann equation (Site not responding. Last check: 2007-10-14)
		At present there is no analogous result for the relativistic Boltzmann equation and this must be regarded as an interesting open problem.
		There is however a recent result [74] for the homogeneous relativistic Boltzmann equation where global existence is shown for small initial data and bounded scattering kernel.
		In [5] it is shown that the gain-term-only classical and relativistic Boltzmann equations blow up for initial data not restricted to a small neighbourhood of trivial data.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2005-2/articlesu1.html (781 words)

	The Boltzmann equation (Site not responding. Last check: 2007-10-14)
		He is reputed to have smuggled wine into the Faculty Club during his 1904 visit to Berkeley--at that time Berkeley was a dry town.
		This equation is incomplete because we have not calculated the collision term on the right hand side.
		We shall continue with an examination of the consequences of the Boltzmann equation which do not depend on the exact form of the collision term.
	grus.berkeley.edu /~jrg/ay202/node32.html (205 words)

	The Boltzmann Equation (Site not responding. Last check: 2007-10-14)
		The Boltzmann equation is the basis for the standard models of electron transport in semiconductors in a semi-classical approximation.
		It consists of the classical Liouville equation (79) augmented by a master operator of precisely the form (84) to describe collisions between electrons and other particles.
		The Boltzmann equation is commonly written in the form [35]:
	www.utdallas.edu /~frensley/technical/qtrans/node13.html (191 words)

	The Boltzmann Equation (Site not responding. Last check: 2007-10-14)
		The classical Boltzmann equation is derived and the macroscopic quantities of mass, velocity and energy are defined in terms of the distribution function which describes the fluid.
		It is shown that the Boltzmann description of the fluid satisfies the fluid conservation equations.
		An outline of the derivation of the Navier-Stokes equation and a discussion of the equilibrium distribution are given for the binary collision model.
	www.ph.ed.ac.uk /~jmb/thesis/node18.html (124 words)

	The Poisson-Boltzmann Equation.
		The difficulty of doing this, even via numerical simulation carried out on modern computers, has led to the development of approximate treatments of the equilibrium properties of ``ionic atmospheres" surrounding static charge distributions.
		Note that there are three contributions to the electric charge distribution that appears on the r.h.s.
		The introduction of this equation raises several questions.
	bessie.che.uc.edu /tlb/rctb6/node19.html (534 words)

	Congen V2.1.2178 - Poisson-Boltzmann Electrostatics
		The equation can be solved over a volume enclosing a molecule of interest using finite difference methods(5) (6) (7) appropriate for the solution of boundary value problems.
		The implementation of the finite difference solution of the Poisson-Boltzmann equation in CONGEN is written in C and uses dynamic storage allocation throughout so any size grid can be accommodated.
		This is a critical parameter in the calculation of electrostatics using the Poisson-Boltzmann equation.
	www.congenomics.com /congen/congen_9.html (4410 words)

	Lattice Boltzmann Methods
		The lattice Boltzmann method is a powerful technique for the computational modeling of a wide variety of complex fluid flow problems including single and multiphase flow in complex geometries.
		It is an approach that bridges microscopic phenomena with the continuum macroscopic equations.
		Nicos Martys, John Hagedorn and Judith Devaney, Lattice Boltzmann Simulations of Single and Multi-Component Flow in Porous Media in Mesoscopic Modeling: Techniques and Applications, Nicolaides and Bick (Ed.), Marcel Dekker, Inc., (to be published).
	math.nist.gov /mcsd/savg/parallel/lb (871 words)

	APBS: Adaptive Poisson-Boltzmann Solver
		PMG is developed and maintained by the Holst Research Group at UC San Diego, and is designed to solve the nonlinear Poisson-Boltzmann equation and similar problems with linear space and time complexity through the use of box methods, inexact Newton methods, and algebraic multilevel methods.
		FEtk is developed and maintained by the Holst Research Group at UC San Diego, and is designed to solve general coupled systems of nonlinear partial differential equations accurately and efficiently using adaptive multilevel finite element methods, inexact Newton methods, algebraic multilevel methods.
		FETK is a portable collection of finite element modeling class libraries written in an object-oriented version of C. It is designed to solve general coupled systems of nonlinear partial differential equations using adaptive finite element methods, inexact Newton methods, and algebraic multilevel methods.
	apbs.sourceforge.net (519 words)

	Giulio M. Occhionero: The Boltzmann Equation
		In the most general definition, a Boltzmann Equation (BE) is an equation for the evolution of a probability density during time furnished with an initial condition.
		Then I have exactly solved the equation for BGK scattering in the presence of an external force.
		The use of Boltzmann equations is widening fastly in pure and applied sciences and solutions are needed in several fields of applications ranging from civil engineering (traffic control) to electronics and medicine; from military simulations (nuclear explosions, battle simulations and behavior modeling etc.) to high-end financial-market research.
	www.occhionero.info /Boltz.htm (518 words)

	Derivation of the C- and F-Processes Model From the Boltzmann Transport Equation
		For a steady-state one-dimensional case, equation eq:Boltzmann describes the particle diffusion and is reduced to
		Instead of dealing directly with equation eq:joseph3 which describes the total heat conduction process, we next independently derive from the BTE the following two equations which are associated with the C-processes and the F-processes,
		Note that equations bte2p1 and bte2p2 can indeed explain and lead to equation eq:joseph3; however, the opposite is not true.
	www-users.cs.umn.edu /~xiangmin/te/node8.html (672 words)

	LECTURE NOTES ON THE DISCRETIZATION OF THE BOLTZMANN EQUATION (Site not responding. Last check: 2007-10-14)
		Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community.
		Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni)
		Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schürrer)
	www.worldscibooks.com /mathematics/5155.html (290 words)

	Lattice Boltzmann Model Theory (Site not responding. Last check: 2007-10-14)
		Lattice Boltzmann methods (LBMs) are a class of mesoscopic particle based approaches to simulate fluid flows.
		The lattice Boltzmann method solves this problem by pre-averaging the lattice gas.
		He, X. and Luo, L. Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation.
	www.science.uva.nl /research/scs/projects/lbm_web/lbm.html (553 words)

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