Energy of a system with quadratic degrees of freedom

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.6.

The definition of temperature in terms of entropy is

\displaystyle \frac{1}{T}\equiv\frac{\partial S}{\partial U} \ \ \ \ \ (1)

 

Schroeder quotes a theorem that states that for any system with only quadratic degrees of freedom in the high temperature limit, the multiplicity {\Omega} is proportional to {U^{Nf/2}} where {U} is the energy, {N} is the number of molecules and {f} is the number of degrees of freedom per molecule. For an ideal gas, this is given by Schroeder’s equation 2.40:

\displaystyle \Omega\approx\frac{V^{N}\left(2\pi mU\right)^{3N/2}}{h^{3N}N!\left(3N/2\right)!} \ \ \ \ \ (2)

Since {f=3} for a monatomic ideal gas (3 translational degrees of freedom only), the theorem is valid here. Similarly, for an Einstein solid in the high temperature limit

\displaystyle \Omega\approx\left(\frac{qe}{N}\right)^{N} \ \ \ \ \ (3)

and since {q}, the number of energy quanta, is proportional to {U}. Each oscillator in an Einstein solid is equivalent to a one-dimensional harmonic oscillator, which has two degrees of freedom. [Rather, it has two quadratic terms in its energy: {\frac{1}{2}kx^{2}} for potential energy and {\frac{1}{2}mv^{2}} for its kinetic energy. Each of these is interpreted as a ‘degree of freedom’.] Thus again, {\Omega\propto U^{Nf/2}=U^{N}}.

In the general case, we have

\displaystyle \Omega=AU^{Nf/2} \ \ \ \ \ (4)

 

for some constant {A} (that is, {A} doesn’t depend on {U}). The entropy is therefore

\displaystyle S=k\ln\Omega=k\ln A+\frac{kNf}{2}\ln U \ \ \ \ \ (5)

 

and the temperature is given by

\displaystyle \frac{1}{T} \displaystyle = \displaystyle \frac{kNf}{2U}\ \ \ \ \ (6)
\displaystyle U \displaystyle = \displaystyle \frac{1}{2}NfkT \ \ \ \ \ (7)

which is just what the equipartition theorem predicts ({\frac{1}{2}kT} energy per degree of freedom).

The formula is valid only for large energies, since if {U} is small enough, {\Omega} eventually becomes less than 1 and the entropy 5 can become negative, which isn’t physically possible.

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