Non-equilibrium systems are abundant in the world and are studied in many engineering and scientific areas. These typically involve large numbers of interacting agents, which can be heterogeneous or homogeneous. While often not susceptible to a closed-form mathematical treatment, we can simulate such systems and see what emerges.
Active walkers — a term coined by Lam — are a way to model dynamics and pattern formation. They change their environment as they walk, and can be used to study percolation, the water cycle, geophysical elements, and many phenomena related to biodiversity, evolution, and population biology.
Each active walker (AW) obeys the coupled Langevin equations: see
for the mathematical background. Here the focus is more on modeling AW and how they can be used. There are similarities and differences compared with Intelligent agents and with the Multi agent systems (MAS) in Artificial Intelligence, so it might be feasible to reuse some of the software for prototyping them.
A model for a simple Active Walker contains a i-lattice for the potential $V(i,n)$, for time $n \in 0,1,2...$,the landscaping function $W$. After $V$ and $W$ are updated the walker takes one step according to a probability step function $P_ij$, the chance that a walker will move from $i$ to $j$.
$P_ij$ can be deterministic, probabilistic, Boltzmann like or fuzzy, which has been proposed by Lam. Update rules:
$V(i,n) = V_0(i,n) + V_1(i,n)$
$V_1(i,n+1) = V_1(i,n) + W(r_i - R(n))$
$V_0$ is the external background, and at time 0 the AW is placed at $V_0$. $V_1$ is the landscaping rule, $V_1(i,0)=0$ and $r_i$ the position vector of site i and $R(n)$ is the walker position at time $n$.
For an $i$-lattice where each point has i neighbors, the set of neighbors $A_i$ are the values used for $j$ in $P_ij$.
A deterministic active walker moves to the j site with minimum $V$. Probabilistic active walkers: has a$P_ij = \lfloor V(i,n)- V(j,n) \rfloor^\eta$ if $V(i,n) \gt V(j,n)$ and else 0. $\eta$ is a parameter.
An active walker can branch (clone) or split into two active walkers.
Liu Lam, editor, Introduction to Nonlinear Physics, Springer, 2005.
K. Besnarda, J.-R. de Dreuzy, P. Davya and L. Aquilina, A modified Lagrangian-volumes method to simulate nonlinearly and kinetically sorbing solute transport in heterogeneous porous media, Journal of Contaminant Hydrology, 2010.
Abstract: Transport in subsurface environments is conditioned by physical and chemical processes in interaction, with advection and dispersion being the most common physical processes and sorption the most common chemical reaction. Existing numerical approaches become time-consuming in highly-heterogeneous porous media. In this paper, we discuss a new efficient Lagrangian method for advection-dominated transport conditions. Modified from the active-walker approach, this method comprises dividing the aqueous phase into elementary volumes moving with the flow and interacting with the solid phase. Avoiding numerical diffusion, the method remains efficient whatever the velocity field by adapting the elementary volume transit times to the local velocity so that mesh cells are crossed in a single numerical time step. The method is flexible since a decoupling of the physical and chemical processes at the elementary volume scale, i.e. at the lowest scale considered, is achieved. We implement and validate the approach to the specific case of the nonlinear Freundlich kinetic sorption. The method is relevant as long as the kinetic sorption-induced spreading remains much larger than the dispersion-induced spreading. The variability of the surface-to-volume ratio, a key parameter in sorption reactions, is explicitly accounted for by deforming the shape of the elementary volumes.
The tuning process of a large apparatus of many components could be represented and quantiﬁed by constructing parameter-tuning networks. The experimental tuning of the ion source of the neutral beam injector of HT-7 Tokamak is presented as an example. Stretched-exponential cumulative degree distributions are found in the parameter-tuning networks. An active walk model with eight walkers is constructed. Each active walker is a particle moving with friction in an energy landscape; the landscape is modiﬁed by the collective action of all the walkers. Numerical simulations show that the parameter-tuning networks generated by the model also give stretched exponential functions, in good agreement with experiments. Our methods provide a new way and a new insight to understand the action of humans in the parameter-tuning of experimental processes, is helpful for experimental research and other optimization problems.
Simulations based on the active walker model are used successfully to reconstruct the dielectric breakdown patterns observed in a cell with parallel-plate electrodes. Different types of patterns can be obtained with suitable parameters. These parameters correspond to the electrical and environmental conditions during the breakdown.