  The union-closed sets conjecture, attributed to P\'eter Frankl from 1979,
states that for any non-empty finite union-closed family of finite sets not
consisting of only the empty set, there is an element that is in at least half
of the sets in the family. We prove a version of Frankl's conjecture for
families distributed according to any one of infinitely many distributions. As
a corollary, in the intersection-closed reformulation of Frankl's conjecture,
we obtain that it is true for families distributed according to any one of
infinitely many Maxwell--Boltzmann distributions with inverse temperatures
bounded below by a positive universal constant. Frankl's original conjecture
corresponds to zero inverse temperature.
