Equilibrium as a large-deviation minimizer

If macroscopic variables satisfy a large-deviation principle, equilibrium states are characterized as minimizers of the rate function (or maximizers of an entropy functional).

Equilibrium as a large-deviation minimizer

Statement (LDP characterization of equilibrium)

Let $(M_N)_{N\ge 1}$ be a sequence of macroscopic observables (order parameters, empirical measures, energy density, etc.) taking values in a topological space $\mathcal M$ . Suppose under a sequence of probability measures $(\mathbb P_N)$ (e.g. finite-volume Gibbs measures) the variables $M_N$ satisfy a large deviation principle with speed $N$ and good rate function $I:\mathcal M\to[0,\infty]$ :

\mathbb P_N(M_N\in B)\approx \exp\big(-N\inf_{m\in B} I(m)\big).

Then:

Any limit point of $M_N$ (in probability) lies in the set of global minimizers of $I$ .
If $I$ has a unique minimizer $m^*$ , then $M_N\to m^*$ in probability and equilibrium is unique at the macroscopic level.
Multiple minimizers of $I$ correspond to macroscopic coexistence (possible symmetry breaking / phase coexistence).

In canonical settings, if the LDP can be written in the form

I_\beta(m)=\beta\,\Phi(m)-S(m)+\text{constant},

then equilibrium macrostates maximize $S(m)-\beta\Phi(m)$ (a “maximum entropy minus energy” principle), matching the variational structure of the Gibbs variational principle .

Key hypotheses

Existence of an LDP for $M_N$ under the measures of interest.
Exponential tightness / compactness conditions ensuring a good rate function.
When using the variational representation of thermodynamic potentials: applicability of Varadhan’s lemma (or an appropriate Laplace principle).

Conclusions

Equilibrium = minimizers: typical macrostates are precisely the minimizers of the rate function.
Fluctuation scale: deviations away from equilibrium have probabilities decaying like $\exp(-N\times\text{cost})$ .
Phase transitions (macro): nonuniqueness or non-smooth changes in minimizers signal phase transitions.

Cross-links to definitions

Large deviations: large deviation principle , rate function , Varadhan’s lemma .
Statistical mechanics context: canonical ensemble , free energy (statistical) .

Proof idea / significance

The LDP upper/lower bounds imply that any neighborhood not containing minimizers of $I$ has exponentially small probability, forcing concentration near $\arg\min I$ . When a partition function or moment generating object is present, Varadhan’s lemma converts logarithmic asymptotics of integrals into a supremum over $\mathcal M$ , yielding the variational characterization of equilibrium.