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I know that it often occurs that we need to take a derivitive with respect to $\beta$ in statistical mechanics. However, I think it is poor style to use equations with both T and $\beta$ in them especially since in most of the theory we take $\beta = \frac{1}{T}$. I see this abuse of notation frequently in textbooks, how to correct this?

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up vote 2 down vote accepted

This isn't abuse of notation. $\beta$ is the Lagrange multiplier for energy; it first shows up when you try to find maximum entropy distributions compatible with energy consevation. $T$ is the temperature; it first shows up when you're defining temperature via Carnot engines. It is a theorem that $\beta = 1/T$, i.e., that these two definitions are capturing the same physical concept.

Once you have proven that $\beta = 1/T$, you can use either symbol, depending on your taste in typesetting.

Abuse of notation is a different thing, when you use the same symbol to stand for two or more different things, letting the context make it 'clear' which meaning is intended. For example, many physicists use the symbol $\phi(x)$ to denote

  1. the map $\phi: X \to Y$,
  2. the value $\phi(x)$ of $\phi$ at $x$,
  3. the evaluation map $ev_x: \phi \mapsto \phi(x)$.
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i guess I should have done some of those derivations ha! never like those stat mech. proofs. Always seemed like magic tricks to me. – Timtam Jul 29 '13 at 22:36
To each their own, I guess. I like the stat mech arguments, and dislike the thermodynamic ones. – user1504 Jul 29 '13 at 22:39
I don't know the difference – Timtam Jul 29 '13 at 22:42

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