The equipartition law of classical thermodynamics

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3.3 The equipartition law of classical thermodynamics

We now will investigate systems for which the one particle energy is written as

$E (\vec{ξ}) = \sum_{i, j = 1}^{s} a_{i j} ξ_{i} ξ_{j} (bilinear form) .$

(3.10)

This function is homogeneous of second order, i.e.:

$\sum_{j = 1}^{s} ξ_{j} \frac{\partial E}{\partial ξ_{j}} = 2 E .$

(3.11)

For classical particles the Boltzmann approximation holds:

$f (E, T) = \frac{exp (- \frac{E}{k T})}{Z} .$

(3.12)

For the inner energy we find:

$U = ⟨ E ⟩ = ∭ E (\vec{ξ}) \frac{exp (- \frac{E (\vec{ξ})}{k T})}{Z} d ξ_{1} . . . d ξ_{s} = \frac{1}{2} \sum_{j = 1}^{s} ∭ ξ_{j} \frac{\partial E (\vec{ξ})}{\partial ξ_{j}} \frac{exp (- \frac{E (\vec{ξ})}{k T})}{Z} d ξ_{1} . . . d ξ_{s} .$

(3.13)

For the norm we find:

$1 = ∭ \frac{exp (- \frac{E (\vec{ξ})}{k T})}{Z} d ξ_{1} . . . d ξ_{s} .$

(3.14)

Finally we get

$U = \frac{- k T}{2 Z} \sum_{j = 1}^{s} ∭ ξ_{j} \frac{\partial}{\partial ξ_{j}} [exp (- \frac{E}{k T})] d ξ_{1} . . . d ξ_{s},$

(3.15)

and after partial integration:

$U = \frac{- k T}{2 Z} \sum_{j = 1}^{s} [ξ_{j} exp {(- \frac{E (\vec{ξ})}{k T})}_{b o u n d a r i e s} - ∭ exp (- \frac{E (\vec{ξ})}{k T}) d ξ_{1} . . . d ξ_{s}] .$

(3.16)

The first term vanishes at the boundaries, the second one is the partition function; thus we find

$U = \frac{k T}{2} s .$

(3.17)

Independent of the special form of the energy function each degree of freedom adds $0.5 k T$ to the inner energy of the system.
The specific heat capacity is

$C = \frac{k}{2} s,$

(3.18)

independent of the temperature.

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