Bali’s Ancient Rice Terraces: A Hamiltonian Approach

We propose a Hamiltonian approach to reproduce the relevant elements of the centuries-old Subak irrigation system in Bali, showing a cluster-size distribution of rice-field patches that is a power-law with an exponent of $\sim 2$. Besides this exponent, the resulting system presents two equilibria. The first originates from a balance between energy and entropy contributions. The second arises from the specific energy contribution through a local Potts-type interaction in combination with a long-range antiferromagnetic interaction without attenuation. Finite-size scaling analysis shows that, as a result of the second equilibrium, the critical transition balancing energy and entropy contributions at the Potts (local ferromagnetic) regime is absorbed by the transition driven by the global-antiferromagnetic interactions, as the system size increases. The phase transition balancing energy and entropy contributions at the global-antiferromagnetic regime also shows signs of criticality. Our study extends the Hamiltonian framework to a new domain of coupled human-environmental interactions.

Figure 1

(a) The five regions where the data was taken: (i) Gianyar, (ii) Tabanan, (iii) Sukawati, (iv) Kusamba, and (v) Klungkung. (b) Log-log plot for the cluster-size distribution of the aggregated normalized values of all the regions and different years where photosynthetic activity was measured (13 in total). The corresponding exponent is $\tau \sim 1.87(2)$.

Figure 2

(a) Order parameter, (b) cluster-size entropy, (c) susceptibility, and (d) specific heat, for the Subak Hamiltonian on grids of linear size, $L=60$ with open boundary conditions. Pest stress is fixed to $a=1.0$. We show several values of water stress, $b$. Color code in (b)–(d) is the same as in (a).

Figure 3

Cluster-size entropy (a) and (b) and specific heat (c) and (d) for the Subak Hamiltonian with $a=1.0$ and two different values of $b$. Colors indicate system size $L=30$, 34, and 40. For $b=1$, $T_c$ moves (peak of specific heat) to high temperatures with increasing system size. For $b=0.05$, $T_c$ moves toward lower temperatures with increasing size. Color code in (b)–(d) is the same as in (a).

Figure 4

Fractal dimension derived from the way the percolating cluster fills the space (a). We find a slope of $D=1.927(4)$. In (b)–(d), we show an attempt to collapse the cluster-size distribution onto a single function for various values of $L$. We find a typical scenario for a weak first-order transition. Color code in (b) and (c) is the same as in (d).