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.


1 Answer 1


These are conjugate variables.

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.