Grand canonical ensemble

A statistical-mechanical ensemble for systems that exchange energy and particles with a reservoir while temperature, volume, and chemical potential remain fixed.

Grand partition function: A generating function that sums statistical weights over all particle numbers and energy states. It normalizes the probabilities of states with different particle numbers.
Statistical ensemble: A probability distribution over possible microscopic states used to describe a macroscopic system. The grand canonical ensemble is one choice of probability distribution for thermodynamic conditions.
Ideal gas: A gas model whose particles have negligible interactions and occupy a volume with elastic collisions. Its grand partition function gives a direct description of fluctuating particle counts.
Compressibility: A measure of how much a system’s volume changes in response to pressure. Particle-number fluctuations are related to compressibility in equilibrium systems.
Canonical partition function: A sum of Boltzmann weights over states at fixed particle number and temperature. It sums over energies at one particle number, unlike the grand partition function’s number sum.
Grand potential: A thermodynamic potential equal to internal energy minus temperature times entropy and chemical potential times particle number. It is the thermodynamic potential naturally minimized under fixed temperature, volume, and chemical potential.
Microcanonical ensemble: An ensemble of isolated systems with fixed energy, volume, and particle number. It fixes all three quantities that the grand canonical ensemble can exchange or hold at reservoir values.
Bose–Einstein distribution: The mean occupation-number distribution for indistinguishable bosons in thermal equilibrium. Grand canonical calculations derive occupation probabilities for bosonic modes.
Fluctuation–dissipation theorem: A set of relations connecting spontaneous equilibrium fluctuations to responses to external perturbations. It links grand canonical number fluctuations to thermodynamic response functions.
Microcanonical entropy: The logarithm of the number of accessible microstates at fixed energy, volume, and particle number. It describes isolated systems rather than systems coupled to particle and energy reservoirs.

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