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2004
Abstract The evolution of the velocity of the distribution function is governed by the Boltzmann equation. This chapter derives the Boltzmann equation for the homogeneous cooling granular gas and discusses the properties of the collision in general.
Nikolai V. Brilliantov +1 more
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Abstract The evolution of the velocity of the distribution function is governed by the Boltzmann equation. This chapter derives the Boltzmann equation for the homogeneous cooling granular gas and discusses the properties of the collision in general.
Nikolai V. Brilliantov +1 more
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2002
In the Langevin equation (Chap. 8), irreversibility was introduced phenomenologically through a damping term. Kinetic theories have the goal of explaining and quantitatively calculating transport processes and dissipative effects due to scattering of the atoms (or in a solid, of the quasiparticles) .
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In the Langevin equation (Chap. 8), irreversibility was introduced phenomenologically through a damping term. Kinetic theories have the goal of explaining and quantitatively calculating transport processes and dissipative effects due to scattering of the atoms (or in a solid, of the quasiparticles) .
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2016
The phase coordinate of a particle is its position and momentum: \((\vec{r},\vec{p}\:)\). It is a six-dimensional variable, \(\left (x,y,z,p_{x},p_{y},p_{z}\right )\), a point in a six-dimensional phase space. In this book, instead of momentum \(\vec{p}\) we will mostly use two variables: particle energy E and a unit vector: \(\vec{\Omega } =\vec{ p ...
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The phase coordinate of a particle is its position and momentum: \((\vec{r},\vec{p}\:)\). It is a six-dimensional variable, \(\left (x,y,z,p_{x},p_{y},p_{z}\right )\), a point in a six-dimensional phase space. In this book, instead of momentum \(\vec{p}\) we will mostly use two variables: particle energy E and a unit vector: \(\vec{\Omega } =\vec{ p ...
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2017
In 1879, Ludwig Boltzmann established the equation governing the behavior of particles in a gas of molecules, without interaction except for collisions between the said molecules. A parallel can in fact be drawn between the behavior of neutrons in matter and that of a ideal gas whose molecules would “disappear” as a result of some collisions, similar ...
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In 1879, Ludwig Boltzmann established the equation governing the behavior of particles in a gas of molecules, without interaction except for collisions between the said molecules. A parallel can in fact be drawn between the behavior of neutrons in matter and that of a ideal gas whose molecules would “disappear” as a result of some collisions, similar ...
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2015
The Einstein equations are insufficient to describe the Universe unless we complement them with a model of its content. In this chapter we shall introduce the Boltzmann formalism, a statistical treatment of the Universe matter content that naturally coexists with general relativity. Each matter component (photons, baryons, cold dark matter, dark energy
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The Einstein equations are insufficient to describe the Universe unless we complement them with a model of its content. In this chapter we shall introduce the Boltzmann formalism, a statistical treatment of the Universe matter content that naturally coexists with general relativity. Each matter component (photons, baryons, cold dark matter, dark energy
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2009
Abstract The statistical evolution of a classical system of N particles with pair interactions can in principle be studied by means of the BBGKY hierarchy for the reduced distribution functions. If no approximation is made, the evolution equation of the one-particle distribution function involves the two-particle distribution function ...
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Abstract The statistical evolution of a classical system of N particles with pair interactions can in principle be studied by means of the BBGKY hierarchy for the reduced distribution functions. If no approximation is made, the evolution equation of the one-particle distribution function involves the two-particle distribution function ...
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2011
Ludwig Eduard Boltzmann (1844–1906), the Austrian physicist whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms and molecules (microscopic properties) determine the phenomenological (macroscopic) properties of matter such as the viscosity, thermal conductivity, and ...
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Ludwig Eduard Boltzmann (1844–1906), the Austrian physicist whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms and molecules (microscopic properties) determine the phenomenological (macroscopic) properties of matter such as the viscosity, thermal conductivity, and ...
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2016
Boltzmann equation is the fundamental equation on the distribution function and is reduced to (8.12) in Sect. 8.2. When the collisional term is negligible, (8.12) becomes Vlavov equation. Collisional term under the assumption of Markoff process is reduced to Fokker–Planck collision term (8.22) in Sect. 8.3.
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Boltzmann equation is the fundamental equation on the distribution function and is reduced to (8.12) in Sect. 8.2. When the collisional term is negligible, (8.12) becomes Vlavov equation. Collisional term under the assumption of Markoff process is reduced to Fokker–Planck collision term (8.22) in Sect. 8.3.
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2010
Consider a monatomic gas with N molecules enclosed in a recipient of volume V. One molecule of this gas can be specified at a given time by its position x = (x1,x2,x3) and velocity c = (c1,c2,c3). Hence, a molecule can be specified as a point in a six-dimensional space spanned by its coordinates and velocity components, the so-called μ-phase space.
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Consider a monatomic gas with N molecules enclosed in a recipient of volume V. One molecule of this gas can be specified at a given time by its position x = (x1,x2,x3) and velocity c = (c1,c2,c3). Hence, a molecule can be specified as a point in a six-dimensional space spanned by its coordinates and velocity components, the so-called μ-phase space.
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