Boltzmann Equation

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. The distribution is defined such that
:

f(\mathbf{x},\mathbf{p},t)\,d\mathbf{x}\,d\mathbf{p}

is proportional to the number of particles in the phase-space volume ''d''x ''d'''''p''' at time ''t''.
Consider ''f'' experiencing an external Force F. Then ''f'' must satisfy
:

f(\mathbf{x}+rac{\mathbf{p}}{m}dt,\mathbf{p}+\mathbf{F}dt,t+dt)\,d\mathbf{x}\,d\mathbf{p}-
f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p}=0

if there is no collision at all. However, since collisions do occur, the particle density in the phase-space volume ''d''x ''d'''''p''' changes.
:

f(\mathbf{x}+rac{\mathbf{p}}{m}dt,\mathbf{p}+\mathbf{F}dt,t+dt)\,d\mathbf{x}\,d\mathbf{p}-
f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p}=

+ rac{\partial f}{\partial \mathbf{p}} \cdot \mathbf{F} where L is the Liouville operator describing the evolution of a phase space volume and '''C''' is the collision operator. The non-relativistic form of L is :

\hat{\mathbf{L}}_\mathrm{NR}=rac{\partial}{\partial t}+rac{\mathbf{p}}{m}\cdot
abla_\mathbf{x}+\mathbf{F}\cdot
abla_\mathbf{p},

and the generalization to (general) relativity is
:

\hat{\mathbf{L}}_\mathrm{GR}=\sum_\alpha p^\alpharac{\partial}{\partial x^\alpha}-\sum_{\alpha\beta\gamma}\Gamma^{\alpha}{}_{\beta\gamma}p^\alpha p^\gammarac{\partial}{\partial p^\alpha},

where Γ is the Christoffel Symbol .

	CATEGORIES ABOUT BOLTZMANN EQUATION

	fundamental physics concepts

	statistical mechanics

	equations

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Information About

Boltzmann Equation