Generalized Gibbs ensemble (GGE)

Maximum-entropy equilibrium state constrained by multiple (often extensive) conserved quantities.

Generalized Gibbs ensemble (GGE)

A generalized Gibbs ensemble (GGE) is an equilibrium distribution obtained by maximizing entropy subject to several expectation-value constraints, typically associated with conserved quantities. It generalizes the familiar canonical ensemble (energy constraint) and grand-canonical ensemble (energy and particle-number constraints).

Let $\{Q_i\}_{i\in I}$ be a collection of observables (often commuting conserved quantities in the dynamics). The GGE is defined as the entropy maximizer among all states with prescribed expectations $\langle Q_i\rangle = q_i$ for all $i\in I$ , where entropy is taken as the Gibbs/Shannon entropy (compare the probabilistic Shannon entropy ).

Maximum-entropy construction

The variational principle (see entropy maximization construction ) yields a state of exponential form. In quantum language, the density operator is

\rho_{\mathrm{GGE}} = \frac{1}{Z_{\mathrm{GGE}}}\exp\!\Big(-\sum_{i\in I}\lambda_i Q_i\Big),

where the $\lambda_i$ are Lagrange multipliers fixed by the constraints, and the normalization is

Z_{\mathrm{GGE}} = \mathrm{Tr}\,\exp\!\Big(-\sum_{i\in I}\lambda_i Q_i\Big).

In a classical phase-space description, the same structure appears as a probability density on phase space with weight $\exp(-\sum_i \lambda_i Q_i(x))$ relative to the phase-space volume element .

The multipliers $\lambda_i$ can be interpreted as generalized inverse temperatures conjugate to the conserved quantities $Q_i$ . The ordinary inverse temperature $\beta$ (see inverse temperature ) is recovered when one of the constraints is the energy $H$ .

Special cases and interpretation

If the only constrained quantity is the energy $Q_1 = H$ , then $\rho_{\mathrm{GGE}}$ reduces to the canonical Gibbs state (see canonical ensemble ), with $\lambda_1=\beta$ .
If energy and particle number are constrained, $Q_1=H$ and $Q_2=N$ , the GGE reduces to the grand-canonical state (see grand-canonical ensemble ), with multipliers $(\beta,-\beta\mu)$ where $\mu$ is the chemical potential .

Physically, the GGE is useful when there are many relevant constraints (for example in integrable models with an extensive family of conserved quantities), in which case a small set of thermodynamic parameters is insufficient to characterize equilibrium.

Partition function, convexity, and generating relations

The logarithm of the GGE normalization,

\psi(\lambda) = \ln Z_{\mathrm{GGE}}(\{\lambda_i\}),

plays the role of a cumulant-generating function (see cumulant generating functions ). Differentiation generates constrained expectations:

\langle Q_i\rangle_{\mathrm{GGE}} = -\frac{\partial \ln Z_{\mathrm{GGE}}}{\partial \lambda_i}, \qquad \mathrm{Cov}(Q_i,Q_j)_{\mathrm{GGE}} = \frac{\partial^2 \ln Z_{\mathrm{GGE}}}{\partial \lambda_i\,\partial \lambda_j}.

This expresses fluctuation structure in the same way as in standard Gibbs ensembles (see connected correlations and cumulants ).

From a mathematical standpoint, $\psi(\lambda)$ is convex in the multipliers, reflecting general properties of log-partition functions connected to Legendre transforms and thermodynamic duality.