# The equipartition theorem

We now want to cover an important topic in statistical mechanics, the *equipartition theorem*. In very general words it can be formulated as follows: *the mean energy of a particle is equal to times the number of microscopic degrees of freedom*.

We now see some examples and then we shall prove the theorem in general.

## Real gas[edit | edit source]

Let us begin with a gas with the following Hamiltonian:

## Harmonic oscillator in a heat bath[edit | edit source]

Let us now consider a single one-dimensional harmonic oscillator in a heat bath (the presence of the heat bath justifies the use of the canonical partition function); in other words we are considering a single particle with Hamiltonian:

We can now move on and prove the equipartition theorem in general.

**Theorem**(Equipartition)

Every term in the Hamiltonian of a system that appears *only* quadratically contributes to the total energy with . In other words, if we can write the Hamiltonian of the system in the form:

*any*of the variables and is an Hamiltonian that does

*not*depend on , then:

*Proof*

The partition function of the system is:

Therefore, if we have an ideal gas of particles its energy will be:

## An application of the equipartition theorem: the specific heat of crystals[edit | edit source]

Now, we could ask if the model of a gas of harmonic oscillators is actually realistic; in other words, are there cases where a particle can be actually considered a harmonic oscillator?
Let us suppose that the particles of our system are subjected to a potential of the following kind^{[1]}:

Clearly, the equilibrium configurations for the particles will be ; furthermore, if the temperature of the system is very small the fluctuations of the particles around this equilibrium will be very small and therefore we can expand the potential around :

Now, these facts can be used in order to describe the properties of crystals.
Let us in fact suppose to have a system of particles with Hamiltonian:

*normal modes of vibration*) we have rewritten the Hamiltonian as a sum of

*decoupled*harmonic oscillators. Note in fact that in the positions are coupled by the matrix , which in general is not diagonal

^{[2]}, while now the new variables are not. Note however that these new variables are not related to the positions or the momenta of any of the atoms: they are just some generalised coordinates, which allow us to rewrite the Hamiltonian in a simpler way

^{[3]}. Thus, since our system is equivalent to a set of harmonic oscillators, its energy will be:

## One last application of the equipartition theorem: specific heat of biatomic gases[edit | edit source]

If we apply what we have seen for the equipartition theorem to a biatomic gas, what do we expect? In this case the atoms in a molecule can oscillate around their equilibrium positions and the molecules can rotate. Choosing a reference frame where the vector that connects the two atoms is directed along the axis, the Hamiltonian of such a system will be:

This is due to the fact that in reality the molecules are quantum systems and the observed behaviour of comes from their properties on a quantum level.
To be more explicit: it is a known fact that in quantum mechanics the energy levels of a system are discrete (for example, for a harmonic oscillator ); in particular any system will have a non-null *zero-point energy*, i.e. the lowest possible energy level, and then all the possible excited states. If the temperature of the system is low enough it will occupy its lowest possible energy state; if we then increase the thermal energy of the system () at a certain point will be large enough to allow the system to pass to the first excited states, and then to the others.
For example the harmonic oscillator has equally spaced energy levels, with the spacing equal to ; if the temperature of the system is such that the system cannot acquire the necessary energy to pass to the first excited state (this will be possible as soon as ).
In the case of the biatomic gas the observed behaviour is due exactly to this mechanism: the first excited state of the vibrational spectrum has an energy higher to the first excited state of the rotational spectrum, so increasing the temperature at a certain point we will be able to give enough energy to the molecules to pass from their ground state to the first rotational excited state, and similarly when we further increase the temperature we will be able to make them pass to the first vibrational excited state.

- ↑ This is the
*Lennard-Jones potential*, which we will cover in some more detail in Mean field theories for weakly interacting systems. It is a very realistic potential for interatomic interactions. - ↑ The positions and the oscillations of the atoms in a crystal are strongly correlated: when an atom is excited and oscillates faster it "gives" some of its energy to the nearby atoms increasing their energy of oscillation. If this was not the case, the crystal would disgregate.
- ↑ Note, however, that if our fundamental assumption is not valid, i.e. if the displacements of the atoms from their equilibrium positions are not small, then also these new variables would be coupled: in this case in fact we couldn't have neglected the third derivatives of , which using the expression of the Hamiltonian in terms of the normal modes of vibration would have been a coupling term between the .