PX284 - D1 - equipartition theorem

suppose, $E (x) = α x^{2}$ , where $α$ is a constant, $E$ is in the classical limit, and $x \in [- \infty, + \infty]$
in this case, the boltzmann distribution can be written as:

P (x) = \frac{e^{- β α x^{2}}}{\int_{- \infty}^{+ \infty} e^{- β α x^{2}} d x}

the average energy is:

⟨ E ⟩ = \int_{- \infty}^{+ \infty} E (x) P (x) d x = \frac{\int_{- \infty}^{+ \infty} α x^{2} e^{- β α x^{2}} d x}{\int_{- \infty}^{+ \infty} e^{- β α x^{2}} d x} = \frac{1}{2 β} = \frac{1}{2} k_{B} T

standard gaussian integrals:

\begin{aligned} \int_{- \infty}^{+ \infty} e^{- β α x^{2}} d x & = \sqrt{\frac{π}{2 a}} \\ \int_{- \infty}^{+ \infty} x^{2} e^{- β α x^{2}} d x & = \frac{1}{2} \sqrt{\frac{π}{2 a^{3}}} \end{aligned}

in general, for a system with $n$ independent quadratic modes or degrees of freedom, ie: $E = α_{1} x_{1}^{2} + α_{2} x_{2}^{2} \dots + α_{n} x_{n}^{2}$ , the expected energy is given by:

⟨ E ⟩ = \frac{n}{2} k_{B} T

this called is the equipartition theorem

equipartition theorem

for a system in contact with a reservoir at temperature $T$ , each independent quadratic energy term in the energy (mode) contributes $\frac{1}{2} k_{B} T$ to the average energy

eg:
- for the energy of a 1D harmonic oscillator, there are two quadratic terms:

E = k x^{2} + \frac{1}{2} m v^{2}

- therefore there are two modes of freedom, and the average energy:

⟨ E ⟩ = k_{B} T

the internal energy $(U)$ is associated with the average energy $(⟨ E ⟩)$ , so:

C_{V} = {(\frac{\partial U}{\partial T})}_{V} = \frac{n}{2} k_{B}