Talk:PlanetPhysics/Statistical Entropy

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Statistical Entropy %%% Primary Category Code: 05.20.Gg %%% Filename: StatisticalEntropy.tex %%% Version: 1 %%% Owner: woodgrain %%% Author(s): woodgrain %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\emph{Statistical Entropy} is a definition of \htmladdnormallink{entropy}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html} based on statistical \htmladdnormallink{Thermodynamics}{http://planetphysics.us/encyclopedia/Thermodynamics.html}. The definition is

$$S = k_B \ln \Omega$$

where $k_B$ is \htmladdnormallink{Boltzmann's constant}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}, $1.38066 \times 10 ^ {-23} J K^{-1}$, and $\Omega$ is the number of microstates corresponding to the observed thermodynamic macrostate.

A \emph{microstate} of the thermodynamic \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} 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 \htmladdnormallink{positions}{http://planetphysics.us/encyclopedia/Position.html} and \htmladdnormallink{velocities}{http://planetphysics.us/encyclopedia/Velocity.html} of all the \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} on the gas.

A \emph{macrostate} of the thermodynamic system is one possible set of values for the externally measurable information about the system, such as the \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}, pressure, and \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html}.

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.

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