In the causal set approach to quantum gravity, Poincar\'{e} symmetry is
modified by swerving in spacetime, induced by the random lattice discretization
of the space-time structure. The broken translational symmetry at short
distances is argued to lead to a residual diffusion in momentum space, whereby
a particle can acquire energy and momentum by drift along its mass shell and a
system in equilibrium can spontaneously heat up. We consider bounds on the rate
of momentum space diffusion coming from astrophysical molecular clouds, nuclear
stability and cosmological neutrino background. We find that the strongest
limits come from relic neutrinos, which we estimate to constrain the momentum
space diffusion constant by $k < 10^{-61} {\rm GeV}^3$ for neutrinos with
masses $m_\nu > 0.01 {\rm eV}$, improving the previously quoted bounds by
roughly 17 orders of magnitude.
