  A model for filament buckling at finite temperatures is presented. Starting
from the classical worm-like chain model under constant compression, we use a
mean-field approach for filament inextensibility to find the complete partition
function. We find that there is a simple interpolation formula that describes
the free energy of chains or filaments as a function of end-to-end separation,
which spans the whole range of filament stiffnesses. Using this formula we
study the buckling transition of semiflexible filaments and find that kinetics
plays an important role. We propose that the filament buckling is essentially
the first order transition governed by the kinetics of escaping a local free
energy minimum. A simple model for the kinetics is put forward, which shows the
critical buckling force for a filament is reduced by a fraction that has a
universal scaling with temperature with an exponent 0.56.
