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Section 1.5 Entropy Taxonomy
Note ID: 202604110002 |
Tags: thermodynamics,
statistical mechanics,
information theory
A taxonomy of entropy across various domains.
Subsection 1.5.1 Entropy in Thermodynamics
Definition 1.5.1 . Clausius Entropy.
The Clausius entropy is a change in the entropy of a system due to some reversible process where it absorbs some amount of heat \(Q_\text{in}\) at a constant temperature \(T\text{:}\)
\begin{equation*}
\Delta S_\text{system} := \int_\text{rev}{\frac{d \, Q_\text{in}}{T}} .
\end{equation*}
Subsection 1.5.2 Entropy in Statistical Mechanics
Definition 1.5.2 . Boltzmann Entropy.
The Boltzmann entropy of a macroscopic system in a state with multiplicity \(\Omega\) is given by:
\begin{equation*}
S_\text{Boltzmann} := k_B \ln{\Omega} .
\end{equation*}
Definition 1.5.3 . Gibbs Entropy.
The Gibbs entropy of a macroscopic system is defined in terms of the probabilities \(p_i\) of being in microstate \(i\text{:}\)
\begin{equation*}
S_\text{Gibbs} := -k_B \sum_i{p_i \ln{p_i}}
\end{equation*}
