The contact process [1] models activity spreading and dying out on a lattice: every site is either Active or Inactive. Unlike the Ising model, there is no Hamiltonian here — the dynamics are defined directly by per-site rates, not by an energy function.
Each attempt picks a random site $i$:
- if $i$ is Active, it heals to Inactive with probability $p$;
- if $i$ is Inactive, it adopts the current state of one uniformly random nearest neighbor — this is the only infection mechanism, there is no separate infection-probability parameter.
One sweep (one call to step) makes as many such attempts as there are sites.
This produces the classic directed-percolation phase transition: for large $p$ (fast healing), activity always dies out (an absorbing phase). For small $p$, activity can survive and spread indefinitely (a percolating phase). A critical $p_c$ separates the two — explore it with the slider below.
This simulation runs on an $L \times L$ toroidal lattice: the edges wrap around, so interactions cross the boundary seamlessly.
L
p =
Active fraction =
Active fraction
References
- T. E. Harris, Contact Interactions on a Lattice, The Annals of Probability 2, 969–988 (1974). doi:10.1214/aop/1176996493
- J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, (Cambridge University Press, 1999). doi:10.1017/CBO9780511524288
- H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Advances in Physics 49, 815–958 (2000). doi:10.1080/00018730050198152
- G. Grinstein, C. Jayaprakash, Y. He, Statistical Mechanics of Probabilistic Cellular Automata, Physical Review Letters 55, 2527–2530 (1985). doi:10.1103/PhysRevLett.55.2527
- T. Tomé, M. J. de Oliveira, Stochastic Dynamics and Irreversibility, (Springer, 2015). doi:10.1007/978-3-319-11770-6
- R. da Silva, E. Venites Filho, H. A. Fernandes, P. F. Gomes, Revisiting the Contact Model with Diffusion Beyond the Conventional Methods, Symmetry (2025). doi:10.3390/sym17050774