Random planting with harvest: A statistical-mechanical analysis

TL;DR

This work develops a statistical-mechanical description of the random planting model (RPM), where growing plants are represented as expanding hard disks that are harvested at a fixed maturity. By mapping the RPM steady state to a nonadditive, polydisperse hard-disk fluid, the authors reinterpret the steady-state planting probability as an insertion probability and connect it to an excess chemical potential, yielding an adsorption-isotherm framework. They derive a low-density virial expansion and implement a scaled-particle-theory closure with an effective diameter to predict steady-state densities across planting rates, obtaining near-perfect agreement with simulations and identifying a high-rate approach to the desynchronized optimum with exponent $b\approx1/3$. The structure of the field is probed via the radial distribution function $g(r)$ and a radius-resolved pair correlation $g(z,r)$, revealing parent–child size correlations and a dynamically broadened precursor of a mixed-size lattice, with extensions to sigmoidal growth and a reported second-virial coefficient for general growth laws. These results deepen the understanding of nonequilibrium packing with dynamic exclusion and point to broader applications in tissue growth and deposition processes.

Abstract

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 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 mixe--size lattice. Finally, we extend the theory to sigmoidal growth laws and compute the associated virial coefficient.

Figures (10)

• Figure 1: Left: Spacetime diagram of the random planting model. The older plant (left) was planted at $t_{p1}$ and will harvested at $t=t_{p1}+\tau$. In order to avoid overlap the younger plant must be placed at a distance of at least $d_{\rm min}=\sigma-| s_2-s_1|$. Middle: Optimum planting configuration for linear growth that is obtained for a desynchronisation factor of $\Delta t/\tau=1/2$. The density is $\rho_{\rm max}\sigma^{2}=64/(27\sqrt{3})\approx 1.3685$. The spacetime representation shows three consecutive layers; Right: overhead view of the field at time $t=\tau/2$. The plants in the first layer (green) have reached maturity and are about to be harvested, while the second layer plants (blue) are half way to maturity ($s=\sigma/4$).
• Figure 2: Configuration of plants in the steady state for $L/\sigma = 8$ and $k_p=10$. Left: plant disks only. Right: same configuration showing the exclusion circles (light gray) associated with each plant and the instantaneous available surface (blue) into which a new seedling can be inserted without causing future overlaps.
• Figure 3: Transient evolution of the plant density $\rho(t)$ from an initially empty field for planting rates $k_p=1,10,100,500$ (bottom to top). Time is measured in units of the plant lifetime $\tau$. The dashed horizontal line marks the theoretical upper bound $\rho_{\max}=64/(27\sqrt{3})$.
• Figure 4: Snapshots of the field morphology for $k_p=500$ at different times, illustrating the transient dynamics. Colors distinguish young plants (red), intermediate ages (green) and nearly mature plants (brown).
• Figure 5: Steady--state density $\rho$ as a function of the planting rate $k_p$ for $L/\sigma=20$. Points: simulation data. Dashed line: low--density prediction \ref{['eq:rho-low-density']}. Solid lines: SPT--based models described in Sec. \ref{['subsec:SPT']}.