(a) Consider a plasma in which the magnetic field is so weak that it presents little impediment to the flow of heat and electric current. Suppose that the plasma has a gradient ∇T_e of its electron temperature and also has an electric field E. It is a familiar fact that the temperature gradient will cause heat to flow and the electric field will create an electric current. Not so familiar, but somewhat obvious if one stops to think about it, is that the temperature gradient also creates an electric current and the electric field also causes heat flow. Explain in physical terms why this is so.

(b) So long as the mean free path of an electron between substantial deflections by electrons and protons, ℓ_{D, e} = (3k_BT_e/m_e)^1/2t_{D, e}, is short compared to the lengthscale for substantial temperature change, T_e/|∇T_e|, and short compared to the lengthscale for the electrons to be accelerated to near the speed of light by the electric field, m_ec²/(eE), the fluxes of heat q and of electric charge j will be governed by nonrelativistic electron diffusion and will be linear in ∇T and E:

The coefficients κ (heat conductivity), κ_e (electrical conductivity), β, and α are called thermoelectric transport coefficients. Use kinetic theory (Chap. 3), in a situation where ∇T = 0, to derive the conductivity equations j = κ_eE and q = −βE, and the following approximate formulas for the transport coefficients:

Show that, aside from a coefficient of order unity, this κe, when expressed in terms of the plasma’s temperature and density, reduces to the Fokker-Planck result Eq. (20.26a).

(c) Use kinetic theory, in a situation where E = 0 and the plasma is near thermal equilibrium at temperature T, to derive the conductivity equations q = −κ∇T and j = α∇T, and the approximate formulas

Show that, aside from a coefficient of order unity, this κ reduces to the Fokker-Planck result Eq. (20.26b).

(d) It can be shown (Spitzer and Harm, 1953) that for a hydrogen plasma

By studying the entropy-governed probability distributions for fluctuations away from statistical equilibrium, one can derive another relation among the thermoelectric transport coefficients, the Onsager relation.

Eqs. (20.28c) and (20.28d) determine α and β in terms of κ_e and κ. Show that your approximate values of the transport coefficients, Eqs. (20.28a) and (20.28b), are in rough accord with Eqs. (20.28c) and (20.28d).

(e) If a temperature gradient persists for sufficiently long, it will give rise to sufficient charge separation in the plasma to build up an electric field (called a “secondary field”) that prevents further charge flow. Show that this cessation of charge flow suppresses the heat flow. The total heat flux is then q = −κ_{T effective} ∇T , where

Eq. (20.26)

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q=-KVT BE, j=K₂E + aVT. (20.27)