Microcanonical Ensemble
The microcanonical ensemble describes an isolated system of fixed energy, volume, and particle number, assigning equal probability to every accessible microstate.
Definition
The microcanonical ensemble is the equilibrium probability distribution for an isolated system with fixed energy, volume, and particle number, in which every microstate compatible with these constraints is equally probable.
Scope
This topic covers the postulate of equal a priori probabilities, the density of states and phase-space volume, the Boltzmann definition of entropy as the logarithm of the number of accessible microstates, and the derivation of temperature, pressure, and chemical potential as entropy derivatives. The route from the microcanonical ensemble to the canonical ensemble through a heat bath is included.
Core questions
- Why is equal a priori probability the natural postulate for an isolated system?
- How does counting accessible microstates define the entropy?
- How do temperature, pressure, and chemical potential emerge as derivatives of the entropy?
- How does coupling to a large reservoir lead from the microcanonical to the canonical ensemble?
Key concepts
- Equal a priori probability postulate
- Density of states and phase-space volume
- Boltzmann entropy S = k log W
- Entropy derivatives and intensive variables
- Isolated systems at fixed energy
Key theories
- Boltzmann entropy of an isolated system
- The entropy of an isolated system equals Boltzmann's constant times the logarithm of the number of microstates consistent with its fixed energy, from which all thermodynamic quantities follow as entropy derivatives.
Clinical relevance
The microcanonical ensemble is the conceptual foundation of equilibrium statistical mechanics and the natural setting for molecular dynamics simulations of isolated systems, where total energy is conserved.
History
The microcanonical description grew from Boltzmann's combinatorial interpretation of entropy in the 1870s and was systematized within Gibbs's ensemble framework, becoming the starting point from which the other ensembles are derived.
Key figures
- Ludwig Boltzmann
- J. Willard Gibbs
Related topics
Seminal works
- boltzmann1877
- pathria2011
Frequently asked questions
- Why assume every accessible microstate is equally likely?
- For an isolated system in equilibrium there is no information distinguishing one compatible microstate from another, so the least-biased and dynamically motivated assumption is that all are equally probable; this postulate reproduces the known laws of thermodynamics.