Boltzmann relation

Last updated November 28, 2023

In a plasma, the Boltzmann relation describes the number density of an isothermal charged particle fluid when the thermal and the electrostatic forces acting on the fluid have reached equilibrium.

Equation

If the local electrostatic potentials at two nearby locations are φ₁ and φ₂, the Boltzmann relation for the electrons takes the form:^[2]

n_{\text{e}}(\phi _{2})=n_{\text{e}}(\phi _{1})e^{e(\phi _{2}-\phi _{1})/k_{\text{B}}T_{\text{e}}}

where n_e is the electron number density, T_e is the temperature of the plasma, and k_B is the Boltzmann constant.

Derivation

A simple derivation of the Boltzmann relation for the electrons can be obtained using the momentum fluid equation of the two-fluid model of plasma physics in absence of a magnetic field. When the electrons reach dynamic equilibrium, the inertial and the collisional terms of the momentum equations are zero, and the only terms left in the equation are the pressure and electric terms. For an isothermal fluid, the pressure force takes the form

F_{\rm {fluid}}=-k_{\text{B}}T_{\text{e}}\nabla n_{\text{e}},

while the electric term is

F_{\rm {electric}}=en_{\text{e}}\nabla \phi

.

Integration leads to the expression given above.

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the Poisson equation because the electron and ion densities are not known a priori, and if they were, because of quasineutrality the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the electron density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation.

Inaccurate situations

Discrepancies with the Boltzmann relation can occur, for example, when oscillations occur so fast that the electrons cannot find a new equilibrium (see e.g. plasma oscillations) or when the electrons are prevented from moving by a magnetic field (see e.g. lower hybrid oscillations).

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References

Wesson, John; et al. (2004). Tokamaks. Oxford University Press. ISBN 978-0-19-850922-6.

↑ Chen, Francis F. (2006). Introduction to Plasma Physics and Controlled Fusion (2nd ed.). Springer. p. 75. ISBN 978-0-306-41332-2.
↑ Inan, Umran S. (2011). Principles of plasma physics for engineers and scientists. Marek Gołkowski. Cambridge: Cambridge University Press. ISBN 978-0-511-91683-0. OCLC 700691127.

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[Chen-1] Chen, Francis F. (2006). Introduction to Plasma Physics and Controlled Fusion (2nd ed.). Springer. p. 75. ISBN 978-0-306-41332-2.

[2] Inan, Umran S. (2011). Principles of plasma physics for engineers and scientists. Marek Gołkowski. Cambridge: Cambridge University Press. ISBN 978-0-511-91683-0. OCLC 700691127.

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