  We study the influence of thermal fluctuations on the buckling behavior of
thin elastic capsules with spherical rest shape. Thermal fluctuations affect
the buckling instability by two mechanisms. On the one hand, thermal
fluctuations can renormalize the capsule's elastic properties and its pressure
because of anharmonic couplings between normal displacement modes of different
wavelengths. This effectively lowers its critical buckling pressure [Kosmrlj
and Nelson, Phys. Rev. X 7, 011002 (2017)]. On the other hand, buckled shapes
are energetically favorable already at pressures below the classical buckling
pressure. At these pressures, however, buckling requires to overcome an energy
barrier, which only vanishes at the critical buckling pressure. In the presence
of thermal fluctuations the capsule can spontaneously overcome an energy
barrier of the order of the thermal energy by thermal activation already at
pressures below the critical buckling pressure. We revisit parameter
renormalization by thermal fluctuations and formulate a buckling criterion
based on scale-dependent renormalized parameters to obtain a
temperature-dependent critical buckling pressure. Then we quantify the
pressure-dependent energy barrier for buckling below the critical buckling
pressure using numerical energy minimization and analytical arguments. This
allows us to obtain the temperature-dependent critical pressure for buckling by
thermal activation over this energy barrier. Finally, we study the combined
effect of parameter renormalization and thermal activation by using
renormalized parameters for the energy barrier in thermal activation to obtain
our final result for the temperature-dependent critical pressure, which is
significantly below the results if only parameter renormalization or only
thermal activation is considered.
