# Variational methods

Variational methods in statistical mechanics are very important since the provide a tool to formulate mean field theories which are valid for any temperature range and with order parameters of essentially arbitrary complexity. Their central idea is what one would expect: if is the Hamiltonian of a physical system and is a set of arbitrary trial states, then we can obtain the energy of the ground state of the system by minimizing with respect to ; since every will be in general a function or an even more complex object, in general is a functional so its minimization must be intended in the sense of functional analysis. We will see that however the only mean value of the Hamiltonian won't be sufficient since we know that the equilibrium configurations of the system are given by the minima of the free energy. In other words, we will compute the free energy using some trial states and then minimize it in the ways we will explain.

Such variational methods are also used in quantum mechanics when a system is too complex and its Schro"dinger equation can't be solved exactly: in this case one introduces a set of trial wave functions and minimizes the functional with respect to , so that both the wave functions and the energy of the ground state of the system can be found.
In statistical mechanics variational methods are performed using the phase space equilibrium probability density of the system.
In particular, the approach of variational methods in statistical mechanics is based upon two inequalities which we now show.

**Theorem**

Let be a random variable (it can be either discrete or continuous), and call its probability density ; clearly, for any function of the mean value of is defined as:

- If is the exponential function then this inequality holds:

- If is the Hamiltonian of a system and its free energy, then:

*Proof*

- Supposing a real number, from the Taylor expansion of the exponential we have and so (we omit the subscript on mean values for simplicity):

- The canonical partition function of the system can be written as:

Remember also that since is a probability distribution it must satisfy:

Now, the free energy of the system is a functional of the probability density and from what we have just seen we can set an upper bound to :

*real*free energy will be given by the minimization of . In particular, the ground state configuration of the system will be given by the form of that minimizes , which can be easily determined in general:

*sole*-th degree of freedom. In other words we are approximating the probability distribution so that the degrees of freedom are

*statistically independent*

^{[1]}, namely:

- The most used one consists in parametrizing with an appropriately defined order parameter that can describe an eventual phase transition; in this way becomes a (real) function of , and the minimization becomes simpler since it reduces to minimizing a simple function.

The parametrization must of course satisfy the constraints:

- Another possible approach consists in considering itself as a variational parameter, and minimizing with respect to it. This is a more general approach, but this time it's harder to establish a connection between as a functional of and as a function of the order parameter that describes a phase transition.

- ↑ This is physically equivalent to what we have done in the Weiss mean field theory for the Ising model (see Weiss mean field theory for the Ising model), where we neglected the correlations between spins.