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. If the temperature is reasonably low (but still high enough for classical me- chanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x) in a Taylor series about the equilibrium point xo: d²u du u(x) = u(xo) + (x – xo) da 2 dr2 du (x - xo)3 dr3 + +....

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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.

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.
If the temperature is reasonably low (but still high enough for classical me-
chanics to apply), the particle will spend most of its time near the bottom
of the potential well. In that case we can expand u(x) in a Taylor series
about the equilibrium point xo:
d²u
du
u(x) = u(xo) + (x – xo)
da
2
dr2
du
(x - xo)3
dr3
+
+....
Transcribed Image Text: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. If the temperature is reasonably low (but still high enough for classical me- chanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x) in a Taylor series about the equilibrium point xo: d²u du u(x) = u(xo) + (x – xo) da 2 dr2 du (x - xo)3 dr3 + +....
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