BoltzmannGibbs

Boltzmann-Gibbs refers to the standard framework of equilibrium statistical mechanics that unites Boltzmann's probabilistic counting with Gibbs's ensemble approach. It describes how macroscopic properties arise from microscopic configurations by assuming that systems in thermal equilibrium maximize the Boltzmann-Gibbs entropy S = -k_B sum p_i log p_i subject to normalization and energy constraints. The framework encompasses the microcanonical ensemble (fixed energy) and the canonical and grand canonical ensembles (systems in thermal and particle exchange with reservoirs).

In the canonical ensemble, the probability of a microstate with energy E_i is p_i = exp(-β E_i)/Z, where β = 1/(k_B T) and Z = sum_i exp(-β E_i) is the partition function. Thermodynamic quantities follow from Z, and the average energy and entropy can be derived. In quantum form, states are described by a density operator ρ = exp(-β H)/Z with S = -k_B Tr(ρ log ρ).

Boltzmann-Gibbs statistics underpins much of classical physics, chemistry, and materials science, and is linked to the maximum-entropy principle in inference. It is most reliable for systems near equilibrium with short-range interactions and large numbers of constituents. Limitations arise for strongly interacting, non-equilibrium, or small systems, where generalized or non-extensive frameworks (for example, alternative entropy forms) may be more appropriate. The approach remains foundational to modern statistical mechanics and to computational methods such as Boltzmann sampling and related algorithms.