Canonical energy fluctuation

Canonical energy variance equals

\partial_{β}^{2} lo g Z = k_{B} T^{2} C_{V}

Relative energy fluctuation of a monatomic ideal gas ( $σ_{U} / U = 2/ (3 N)$ )

At fixed particle number and volume, a classical monatomic ideal gas has

U = 3 N k_{B} T /2

and

C_{V} = 3 N k_{B} /2

Var (E) = k_{B} T^{2} C_{V}

therefore gives

σ_{U} / U = 2/ (3 N)

. Absolute fluctuations grow as

N

, while relative fluctuations decrease as

N^{- 1/2}

. A particle reservoir changes the fluctuation problem because energy and particle-number fluctuations are then correlated.

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