Thermal Physics · Problem 5

Kinetic Theory — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · boltzmann-equation-thermal-conductivity

Problem

Boltzmann equation and thermal conductivity. Consider a medium with temperature gradient

d\tau/dx

. The particle concentration is constant. (a) Employ the Boltzmann transport equation in the relaxation time approximation to find the first order nonequilibrium classical distribution:

f \simeq f_0 - v_x \tau_c \left(-\frac{3}{2\tau} + \frac{\varepsilon}{\tau^2}\right) f_0 \frac{d\tau}{dx}. \tag{125}

(b) Show that the energy flux in the

x

direction is

J_u = -\left(\frac{d\tau}{dx}\right) \tau_c \int v_x^2 \left(-\frac{3\varepsilon}{2\tau} + \frac{\varepsilon^2}{\tau^2}\right) f_0 \mathfrak{D}(\varepsilon) d\varepsilon\,, \tag{126}

where

v_x^2 = 2\varepsilon/3m

. (c) Evaluate the integral to obtain for the thermal conductivity

K = 5n\tau\tau_c/m

Related theory

Thermal Conductivity In Solids

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