The interplay between stochastic chemical reactions and diffusion can generate rich spatiotemporal patterns. While the timescale for individual reaction or diffusion events may be very fast, the timescales for organization can be much longer. That separation of timescales makes it particularly challenging to anticipate how the rapid microscopic dynamics gives rise to macroscopic rates in the nonequilibrium dynamics of many reacting and diffusing chemical species. Within the regime of stochastic fluctuations, the standard approach is to employ Monte Carlo sampling to simulate realizations of random trajectories. Here, we present an alternative numerically tractable approach to extract macroscopic rates from the full ensemble evolution of many-body reaction-diffusion problems. The approach leverages the Doi-Peliti second-quantized representation of reaction-diffusion master equations along with compression and evolution algorithms from tensor networks. By focusing on a Schlögl model with one-dimensional diffusion between L otherwise well-mixed sites, we illustrate the potential of the tensor network approach to compute rates from many-body systems, here with approximately 3 × 10^15 microstates. Specifically, we compute the rate for switching between metastable macrostates, with the expense for computing those rates growing subexponentially in L. Because we directly work with ensemble evolutions, we crucially bypass many of the difficulties encountered by rare event sampling techniques—detailed balance and reaction coordinates are not needed.
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This content will become publicly available on July 1, 2025
Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations
Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a
straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation
results to experimental or observational data.
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- Award ID(s):
- 2128587
- NSF-PAR ID:
- 10531689
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review E
- Volume:
- 110
- Issue:
- 1
- ISSN:
- 2470-0045
- Subject(s) / Keyword(s):
- agent-based Monte Carlo simulations stochastic processes population dynamics reaction-diffusion systems Lotka-Volterra predator-prey model May-Leonard model
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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