 Entropy (source code)

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= Entropy
{title2=$S$}
{wiki}

= Thermodynamic entropy
{synonym}

Entropy is an <extensive quantity> and a <state function> of an equilibrium system, measured in $\mathrm{J\,K^{-1}}$. Differences between equilibrium states are calculated along a <reversible thermodynamic process>:
$$
S_2-S_1=\int_1^2\frac{\delta Q_{\rm rev}}{T}.
$$
Here <heat> entering the system is positive and $T$ is its absolute <thermodynamic temperature>. The reversible path used for calculation need not be the path actually taken. Dividing the actual <heat> by a single <temperature> generally fails for an irreversible process.

For example, put two rigid bodies, each with constant <heat capacity> $C=100\,\mathrm{J\,K^{-1}}$, at $600\,\mathrm K$ and $300\,\mathrm K$ in an otherwise <isolated system>. Assume each body has a uniform internal <temperature>, no <phase transition> occurs, and no <work> is extracted. Their final <thermal equilibrium> is at $450\,\mathrm K$, because the <internal energy> lost by one equals that gained by the other. Using <entropy from heat capacity>,
$$
\Delta S_h=C\log(450/600)=-28.77\,\mathrm{J\,K^{-1}},\qquad
\Delta S_c=C\log(450/300)=40.55\,\mathrm{J\,K^{-1}},
$$
so $\Delta S_{\rm tot}=C\log(9/8)=11.78\,\mathrm{J\,K^{-1}}$. One body's <entropy> decreases while the total increases. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node41.html[MIT, Entropy changes and the hot-brick problem]{ref}

\Image[/heat-exchange-entropy.png]
{title=Thermal equilibration of two finite bodies conserves energy and increases total entropy}

In <statistical mechanics>, <Boltzmann's entropy formula> counts equally likely <microstates> compatible with a <macrostate>; <Gibbs entropy> handles a specified <probability distribution> over <microstates>. If the number of equally likely <microstates> doubles, <entropy> increases by $k_B\log2$. The <logarithm> makes multiplicative counts additive for independent systems. These are precise statements about states and probabilities, not a universal measure of visible messiness. Calling <entropy> merely "disorder" obscures the thermal calculation above and hides which <microstates> are being counted. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node57.html[MIT, Statistical entropy and randomness]{ref}

The usefulness of <entropy> is that measurable <heat capacities> determine its changes, and the <Second law of thermodynamics> turns those changes into constraints on processes. It also connects macroscopic measurements to microscopic descriptions without requiring us to track every particle.

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