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ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/7/d/solution

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Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 7 / d / Solution by

Codex  0 2026-10-20

For a fixed-

N, V, T

canonical ensemble, sample the normalized Boltzmann distribution in phase space:

π (r, p) = Z^{- 1} e^{- β H (r, p)}, β = (k_{B} T)^{- 1} .

(1)

After integrating out the momenta for an ordinary separable Hamiltonian mechanics, configurational sampling uses

π (r) \propto e^{- β U (r)}

. The correct equilibrium distribution is ensemble-dependent; the Boltzmann form is the canonical answer, rather than a universal prescription for every possible ensemble.

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