[ganzuul]

For the latter maybe this? https://en.wikipedia.org/wiki/Parallel_tempering

[ycreader]

Is parallel tempering related to https://en.wikipedia.org/wiki/Bennett_acceptance_ratio ?

[keithalewis]

Nyet, and nyet. This is why conscientious authors define the terms they use. A tempered distribution is a linear functional on a space of differentiable functions, for example, D_x(f) = f'(x), the derivative of f at x. This is why tempered distributions cannot be composed.

In general, the dual of a space of functions is a space of set functions, aka measures. https://keithalewis.github.io/math/dual.html