Metastable state

A long-lived, locally stable state (often corresponding to a local minimum of a thermodynamic potential) that eventually decays to the globally stable equilibrium via rare fluctuations.

Metastable state

A metastable state is a state that is locally stable under small perturbations but is not the globally stable equilibrium at the same control parameters. In statistical mechanics, metastability typically appears near first-order phase coexistence and is tied to free-energy barriers and nucleation.

Static characterization

Fix control parameters (e.g., inverse temperature $\beta$ and field $h$ ) and consider an order parameter (e.g., magnetization) $m$ as in order parameter .

A metastable state is commonly modeled by a local minimum of an effective thermodynamic function of $m$ , such as:

a constrained free-energy density (canonical viewpoint; see statistical free energy ), or
a large-deviation rate function for $m$ (see rate function for magnetization ).

In a Landau description (see Landau free-energy functional ), metastability corresponds to multiple local minima of the Landau free energy; the globally stable phase is the global minimizer, while the metastable phase is a local minimizer separated by a barrier.

Dynamical viewpoint and lifetime scale

When a microscopic dynamics is specified (e.g., Glauber-type dynamics for spins), a metastable state manifests as:

rapid relaxation to a quasi-equilibrium within a basin of attraction, followed by
a long waiting time before a rare transition to the stable phase.

Heuristically, the mean exit time from the metastable basin often has an Arrhenius-type form

\mathbb{E}[\tau] \sim \exp(\beta \,\Delta F^\ddagger),

where $\Delta F^\ddagger$ is an effective free-energy barrier.

Nucleation barrier (classical droplet heuristic)

At a first-order transition, decay is often driven by nucleation of a droplet of the stable phase inside the metastable background. A standard heuristic for a droplet of linear size $R$ in $d$ dimensions is

\Delta F(R) \approx \sigma\, S_d\, R^{d-1} \;-\; \Delta f\, V_d\, R^d,

where:

$\sigma$ is the surface tension (see surface tension and interfaces ),
$\Delta f$ is the bulk free-energy density difference between the phases,
$S_d$ and $V_d$ are geometric constants (surface area and volume of the unit ball).

The competition between the positive surface term and negative bulk term yields a critical droplet size $R_\star$ and a barrier height $\Delta F(R_\star)$ controlling the metastable lifetime.

Where metastability lives in equilibrium theory

Metastable behavior is naturally discussed using finite- and infinite-volume Gibbs measures:

In finite volume, the Gibbs measure (see finite-volume Gibbs measure ) may exhibit multiple “preferred” macrostates separated by exponentially small probabilities.
In infinite volume (see infinite-volume Gibbs measures ), the globally stable phases correspond to equilibrium Gibbs states; metastability is often encoded in how boundary conditions or fields select among competing phases (see phase transitions via Gibbs measures ).

Metastability is closely tied to thermodynamic stability (local convexity/positivity conditions) but differs in that a metastable state can satisfy local stability conditions while failing global optimality; compare thermodynamic stability .