What is libertarian populism?

For an idea that is supposedly so clearly destined for failure, libertarian populism sure has drawn a lot of attention from critics on both the left (Josh Barro, Jonathan Chait, Paul Krugman, Michael Lind, Matt Yglesias) and right (Matt Lewis, Ramesh Ponnuru).

Unsurprisingly, none of these writers are particularly generous with their definitions of what libertarian populism is, or what it is trying to accomplish. But that is not really their fault. The term libertarian populism is still being defined, and while Sen. Rand Paul, R-Ky., is often associated with the term, he neither labels himself as such or even uses the phrase.