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I keep forgetting the word describing the pairs of coupled quantities in stat. mech. e.g. inverse temperature $\beta$ and internal energy $E$ or chemical potential $\mu$ and particle number $N$.

I remember reading that $\beta$ and $E$ are "companion" quantities, or something along those lines, because they appear as a product in the exponential term of the Boltzmann distribution. One of the consequences of this structure is that knowing the distribution of energy $\mathbb{P}(E|\beta)$ at some point $\beta$ allows one to reweigh to a different point $\beta^\prime$ via: $$\frac{\mathbb{P}(E|\beta)}{\mathbb{P}(E|\beta^\prime)} \propto \exp[\Delta\beta E],$$ where $\Delta\beta$ is the difference between the two parameter points. There are also other consequences, like the $\beta$ derivative of the negative log of the partition function gives the average energy, with an analogous expression for $(\mu,N)$ since they also appear as a product inside the exponential. There are also similar expressions for the pair pressure and volume.

All I want to know is the word describing the pairs of these coupled quantities. I think it's an adjective but also could be a noun. Can't remember for the life of me and it seems quite tricky to find in textbooks and online.

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These are conjugate variables.

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    $\begingroup$ This seems a bit off because the conjugate of temperature is entropy. $\endgroup$ Apr 23, 2016 at 21:19
  • $\begingroup$ $E/kT$ is a dimensionless quantity; $ E\beta $ does not provide a conjugate pair. $\endgroup$ Apr 24, 2016 at 1:27

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