Thermal Physics · Problem 13

Thermal Radiation and Planck Distribution — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · energy-fluctuations-in-a-solid-at-low-temperatures

Problem

Energy fluctuations in a solid at low temperatures. Consider a solid of NN atoms in the temperature region in which the Debye T3T^3 law is valid. The solid is in thermal contact with a heat reservoir. Use the results on energy fluctuations from Chapter 3 to show that the root mean square fractional energy fluctuation F\mathscr{F} is given by F2=⟨(ε−⟨ε⟩)2⟩/⟨ε⟩2≈0.07N(θT)3.(57)\mathscr{F}^2 = \langle(\varepsilon - \langle\varepsilon\rangle)^2\rangle/\langle\varepsilon\rangle^2 \approx \frac{0.07}{N}\left(\frac{\theta}{T}\right)^3. \tag{57} Suppose that T=10−2 KT = 10^{-2}\,\text{K}; θ=200 K\theta = 200\,\text{K}; and N≈1015N \approx 10^{15} for a particle 0.01 cm on a side; then F≈0.02\mathscr{F} \approx 0.02. At 10−5 K10^{-5}\,\text{K} the fractional fluctuation in energy is of the order of unity for a dielectric particle of volume 1 cm3^3.

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