PlanetPhysics/Statistical Entropy

Statistical Entropy is a definition of [[../ThermodynamicLaws/|entropy]] based on statistical [[../Thermodynamics/|Thermodynamics]]. The definition is

$S=k_B\ln \mathrm{\Omega }$

where $k_B$ is [[../BoltzmannConstant/|Boltzmann's constant]], $1.38066\times 10^{-23}J{K}^{-1}$, and $\mathrm{\Omega }$ is the number of microstates corresponding to the observed thermodynamic macrostate.

A microstate of the thermodynamic [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] is one possible complete microscopic description of the system. For example, for an ideal gas, this would contain one possible set of values for all the [[../Position/|positions]] and [[../Velocity/|velocities]] of all the [[../Particle/|particles]] on the gas.

A macrostate of the thermodynamic system is one possible set of values for the externally measurable information about the system, such as the [[../BoltzmannConstant/|temperature]], pressure, and [[../Volume/|volume]].

The definition above assumes that all the microstates are equally probable. If they are not, the equation is

$S=-k_B\ln \sum _i{p}_i\ln p_i$

where the microstates are indexed by $i$ and $p_i$ is the probability that the system is in microstate $i$.

The equation was first introduced by Ludwig Boltzmann in 1877.