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When is a system in thermal equilibrium?
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If a system has finitely many states $i=1, \ldots, n$ with energies $E_{i}$, we say it's in thermal equilibrium if it maximizes entropy subject to a constraint on its expected energy:
$\sum_{i=1}^{n} E_{i} p_{i}=E$
where $p_{i}$ is the probability it's in the $i$ th state. We then have
$p_{i} \propto \exp \left(-\beta E_{i}\right)$
where $\beta=1 / k T$ is the coolness, $k$ is Boltzmann's constant, and the temperature $T$ can be positive, negative, or even infinite:

by Diamond (55,129 points)

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