Random planting with harvest: A statistical-mechanical analysis

We formulate a statistical-mechanical description of a recently introduced random planting model in which plants are represented by growing hard disks. Seedlings of negligible size are introduced at random positions in a field, grow at a prescribed rate, and are harvested upon reaching a fixed maturity diameter. Planting attempts that would lead to an overlap at any time during growth are rejected. Starting from an empty field, this simple dynamical rule drives the system to a nonequilibrium steady state in which the mean planting and harvesting rates coincide. We show that the steady state can be mapped onto a nonadditive polydisperse hard-disk fluid, and we exploit this mapping to develop analytical predictions based on a low-density virial expansion and on scaled particle theory. The resulting description yields an effective adsorption isotherm for the steady-state plant density as a function of the planting rate and compares favorably with numerical simulations over a wide range of parameters. At large planting rates, the density approaches the optimal value achieved by desynchronized regular planting, and the data are consistent with an algebraic approach to this limit with an exponent close to 1/3. Beyond density and yield, we show that the spatial organization of the field at high planting rates exhibits clear signatures of the same underlying geometric constraints that characterize optimal desynchronized planting. This connection is revealed through both the conventional radial distribution function and a radius-resolved pair correlation g(z,r) which highlights strong size correlations associated with parent-child seeding events and whose structure can be interpreted as a dynamically broadened precursor of the corresponding ideal mixed-size lattice. Finally, we extend the theory to sigmoidal growth laws and compute the associated virial coefficient.