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A system is composed of N identical classical oscillators, each of mass m, defined on a one-dimensional lattice. The potential for the oscillators has the form
(1)(Thus, the oscillators are harmonic for n = 2 and anharmonic otherwise).
Find the average thermal energy at temperature T.
Hint: An integral that appears in the course of evaluating the partition function cannot be computed in terms of elementary functions. Fortunately,it amounts only to an unimportant overall coefficient.
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Answer
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Classical partition function for a single oscillator is
(2)By change of variables, we bring this to the form
(3)where
(4)A learned reader would recognize this as the Euler Gamma-function. However, knowing this is not necessary. The product $\Gamma(1/2)\Gamma(1/n)$ is just a numerical coefficient, which will disappear from the final result.
The average energy is given by
(5)This result resembles the equipartition theorem in the sense that material constants do not enter.
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