Rate function for magnetization

The large-deviation rate function governing probabilities of atypical magnetization values under a Gibbs measure; its minimizers describe typical phases and its shape encodes coexistence and metastability.

Rate function for magnetization

Let $\sigma_i \in \{-1,+1\}$ be spins in a volume $\Lambda$ with $|\Lambda|=N$ . The (intensive) magnetization is

m_N \;=\; \frac{1}{N}\sum_{i\in\Lambda}\sigma_i.

Under an equilibrium Gibbs measure (see finite-volume Gibbs measure ), $m_N$ often satisfies a large deviation principle (LDP) at speed $N$ :

\mathbb{P}(m_N \approx m)\;\asymp\;\exp\big(-N\, I(m)\big),

where $I$ is the rate function. See large deviation principle and rate function .

Connection to the pressure (log-partition density)

In the canonical ensemble with external field $h$ (see canonical ensemble and canonical partition function ), exponential tilting of magnetization corresponds to shifting the field.

Indeed,

\mathbb{E}_{\beta,h}\!\left[e^{t\,N m_N}\right] ={} \frac{Z_N(\beta,\,h+t/\beta)}{Z_N(\beta,\,h)}.

If the pressure

p(\beta,h) \;=\; \lim_{N\to\infty}\frac{1}{N}\log Z_N(\beta,h)

exists (see pressure (log-partition density) ), then the scaled cumulant generating function is

\Lambda_{\beta,h}(t)=p(\beta,h+t/\beta)-p(\beta,h).

When $\Lambda_{\beta,h}$ is differentiable, the Gartner–Ellis mechanism yields a convex rate function given by the Legendre–Fenchel transform

I_{\beta,h}(m)=\sup_{t\in\mathbb{R}}\Big\{t m-\Lambda_{\beta,h}(t)\Big\}.

Non-differentiability of $p(\beta,h)$ in $h$ is a signature of phase coexistence and can produce flat pieces in $I_{\beta,h}$ .

Interpreting the shape of $I(m)$

Single phase (no coexistence): $I(m)$ has a unique minimizer at the typical magnetization.
Symmetry breaking / coexistence: at $h=0$ below critical temperature in Ising-type systems, $I(m)$ may have multiple minimizers (or a flat interval of minimizers at speed $N$ ), reflecting competing phases; see phase transitions and spontaneous magnetization .
Metastability: local (non-global) minima of $I(m)$ correspond to metastable magnetizations in a static large-deviation sense; see metastable states .

A key caution at coexistence: intermediate magnetizations may be realized by phase separation with probability costs scaling like surface area (not volume). In that case, the speed- $N$ rate function can develop a plateau (zero cost on an interval), while the sharper cost is governed by interfaces and surface tension; see surface tension and interfaces .

Explicit example: Curie–Weiss (mean-field) magnetization rate function

For the Curie–Weiss model (see Curie–Weiss model ), the magnetization obeys an LDP with an explicit rate function.

Let

I_0(m)=\frac{1+m}{2}\log\frac{1+m}{2}+\frac{1-m}{2}\log\frac{1-m}{2}+\log 2 \quad\text{for } m\in[-1,1],

the Cramér rate function of i.i.d. $\pm1$ spins (see Cramér's theorem ). Then for coupling $J$ and field $h$ , define the mean-field “free-energy-like” function

\phi_{\beta,h}(m)= I_0(m) - \frac{\beta J}{2}m^2 - \beta h\, m.

The magnetization rate function can be written as

I_{\beta,h}(m)=\phi_{\beta,h}(m)-\inf_{m'\in[-1,1]}\phi_{\beta,h}(m').

Its minimizers are exactly the stable equilibrium magnetizations, and its local minima encode mean-field metastability (compare mean-field approximation and Landau theory ).

Connection to the pressure (log-partition density)

Interpreting the shape of I(m)I(m)I(m)

Explicit example: Curie–Weiss (mean-field) magnetization rate function

Interpreting the shape of $I(m)$