u(x) 4 Figure 6.10. A one-dimensional po- tential well. The higher the temper- ature, the farther the particle will stray from the equilibrium point.
Consider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure 6.10 (attached). The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics.
If we keep the cubic term in the Taylor series as well, the integrals in the formula for x become difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from X0 by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series for u(x).
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