Mean-field approximation

A variational/product-measure approximation that replaces interactions by an effective field determined self-consistently, yielding tractable equations for order parameters and approximate free energies.

Mean-field approximation

The mean-field approximation replaces an interacting many-body system by an effective single-site (or single-particle) problem coupled to a self-consistent average field. It is widely used to obtain qualitative phase diagrams and critical behavior.

Variational formulation (canonical ensemble)

In the canonical ensemble (see canonical ensemble ), the equilibrium Gibbs measure minimizes a free-energy functional. A convenient formulation uses the Shannon/Gibbs entropy (see Gibbs (Shannon) entropy and Shannon entropy ).

For a Hamiltonian $H(\sigma)$ on a finite configuration space and a trial probability measure $q$ (see probability measure ), define

\mathcal{F}_\beta[q] \;=\; \mathbb{E}_q[H] \;-\; \beta^{-1} S(q), \qquad S(q)=-\sum_\sigma q(\sigma)\log q(\sigma).

Then the equilibrium free energy (see statistical free energy ) satisfies

-\beta^{-1}\log Z(\beta)=\inf_q \mathcal{F}_\beta[q].

Mean-field restricts the variational class to product measures (or low-complexity factorizations), e.g.

q(\sigma)=\prod_{i\in\Lambda} q_i(\sigma_i),

thereby neglecting correlations (compare two-point correlations ).

Mean-field equation for the Ising model

Consider the nearest-neighbor Ising Hamiltonian on a regular lattice with coordination number $z$ :

H(\sigma)=-J\sum_{\langle i,j\rangle}\sigma_i\sigma_j - h\sum_i \sigma_i.

Mean-field replaces neighbors by their average magnetization $m=\mathbb{E}[\sigma_i]$ , giving an effective single-site energy

H_i^{\text{MF}}(\sigma_i) = -\sigma_i\,(J z\, m + h).

The single-site distribution is proportional to $\exp(\beta \sigma_i(J z m+h))$ , so the self-consistency condition is

m=\tanh\!\big(\beta(J z\, m+h)\big).

Nontrivial solutions at $h=0$ correspond to spontaneous magnetization (see spontaneous magnetization ).

Mean-field free-energy density (and stability)

A standard mean-field free-energy density as a function of $m$ is

f_{\beta,h}^{\text{MF}}(m) ={} -\frac{J z}{2}\,m^2 - h m +\beta^{-1}\left[ \frac{1+m}{2}\log\frac{1+m}{2} +\frac{1-m}{2}\log\frac{1-m}{2} \right].

Equilibrium magnetizations are minimizers of $f_{\beta,h}^{\text{MF}}(m)$ ; multiple local minima correspond to metastability (see metastable states ).

This $m$ -dependent free energy is the simplest instance of a Landau theory (see Landau free-energy functional ).

When mean-field works (and when it fails)

Mean-field becomes accurate when fluctuations are suppressed, for example:

in genuine long-range/fully-connected models (it is exact for Curie–Weiss ),
in high dimension, or when correlation lengths are short (see correlation length ).

It typically fails quantitatively near critical points in low dimensions because it neglects long-wavelength fluctuations that dominate critical behavior.