Boltzmann's entropy formula

Boltzmann's entropy formula, S = k_B ln Ω, assigns entropy S to a macroscopic state according to the number Ω of microscopic states compatible with it, scaled by Boltzmann's constant k_B.

Boltzmann constant: A physical constant that relates temperature to energy per degree of freedom. It converts the dimensionless logarithm of the state count into entropy units.
Statistical mechanics: The framework that explains macroscopic physical behavior through the statistics of microscopic states. It provides the probabilistic bridge behind Boltzmann's relation.
Ludwig Boltzmann: An Austrian physicist who developed statistical mechanics and the kinetic theory of gases. His work established the connection between entropy and microscopic multiplicity.
Second law of thermodynamics: The law that the total entropy of an isolated system never decreases in spontaneous processes. Typical high-multiplicity macrostates explain why entropy tends to increase.
Gibbs paradox: An apparent discontinuity in classical entropy of mixing for identical gases, resolved by accounting for indistinguishability. It shows why counting arrangements requires a careful rule for identical particles.
Microstate: A complete microscopic specification of a system, such as its particles' positions and momenta. Each compatible microstate contributes to Ω in the formula.
Thermodynamic entropy: A state function measuring energy dispersal and the number of accessible microscopic arrangements. Boltzmann's formula gives a microscopic interpretation of this thermodynamic quantity.
H-theorem: Boltzmann's result that molecular collisions drive a gas distribution toward equilibrium under the molecular-chaos assumption. Its account of entropy increase sharpened questions about microscopic reversibility.
Thermodynamic equilibrium: A state in which macroscopic properties remain constant because no net thermodynamic driving forces remain. Equilibrium macrostates usually correspond to overwhelmingly many compatible microstates.
Quantum statistical mechanics: The statistical framework applying quantum states and quantum laws to ensembles of physical systems. Quantum state counting changes how microstates and their multiplicities are defined.

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