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Section 1.2 Entropy Taximony
Note ID: 202604110002 |
Tags: <thermodynamics>,
<statistical mechanics>,
<information theory>
A taxomony of entropy across various domains.
Subsection 1.2.1 Entropy in Thermodynamics
Definition 1.2.1 . Clausius Entropy.
The
Clausius entropy is a change in the entropy of a system due to some
reversible process where it absobes 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 1.2.2 Entropy in Statistical Mechanics
Definition 1.2.2 . 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 1.2.3 . Gibbs Entropy.
The
Gibbs entropy of a macroscopic system is defined in terms of the probabilities
\(\mathbb{p}_i\) of being in microstate
\(i\text{:}\)
\begin{equation*}
S_\text{Gibbs} := -k_B \sum_i{\mathbb{p}_i \ln{\mathbb{p}_i}}
\end{equation*}
