  Unraveling the complexities of random packing in three dimensions has long
puzzled physicists. While both experiments and simulations consistently show a
maximum density of 64 percent for tightly packed random spheres, we still lack
an unambiguous and universally accepted definition of random packing. This
paper introduces an innovative standpoint, depicting random packing as spheres
closest to a quenched Poisson field of random points. We furnish an efficacious
algorithm to probe this proposed model numerically. We unearth a unique
out-of-equilibrium thermodynamic phenomenon, akin to a `latent heat', that
emerges at $\phi_J \approx 0.65$ in three dimensions. This phenomenon is
accompanied by global and local structural rearrangements, marking a jamming
transition from an unjammed state to a jammed one. Notably, such a `jamming'
transition is absent for two-dimensional random packing. Our innovative
approach paves a new avenue for defining random packing and provides novel
insights into the behavior of amorphous materials.
