Simulate a 2D Ising landscape — simulate

Perform a numeric simulation using the landscape. The simulation is performed using a similar algorithm as Glauber dynamics, that the transition possibility is determined by the difference in the potential function, and the steady-state distribution is the same as the Boltzmann distribution (if not setting beta2). Note that, the conditional transition possibility of this simulation may be different from Glauber dynamics or other simulation methods.

Usage

simulate_Isingland(l, ...)

# S3 method for class '`2d_Isingland`'
simulate_Isingland(
  l,
  mode = "single",
  initial = 0,
  length = 100,
  beta2 = l$beta,
  ...
)

# S3 method for class '`2d_Isingland_matrix`'
simulate_Isingland(
  l,
  mode = "single",
  initial = 0,
  length = 100,
  beta2 = NULL,
  ...
)

Arguments

l: An Isingland object constructed with make_2d_Isingland() or make_2d_Isingland_matrix().
...: Not in use.
mode: One of "single", "distribution". Do you want to simulate the state of a single system stochastically or simulate the distribution of the states? "single" is used by default.
initial: An integer indicating the initial number of active nodes for the simulation. Float numbers will be converted to an integer automatically.
length: An integer indicating the simulation length.
beta2: The $beta$ value used for simulation. By default use the same value as for landscape construction. Manually setting this value can make the system easier or more difficult to transition to another state, but will alter the steady-state distribution as well.

Value

A sim_Isingland object with the following components:

output A tibble of the simulation output.
landscape The landscape object supplied to this function.
mode A character representing the mode of the simulation.

Details

In each simulation step, the system can have one more active node, one less active node, or the same number of active nodes (if possible; so if all nodes are already active then it is not possible to have one more active node). The possibility of the three cases is determined by their potential function:

$$P(n_{t}=b|n_{t-1}=a) = \frac{e^{-\beta U(b)}}{\sum_{i \in \{a-1,a,a+1\},0\leq i\leq N}e^{-\beta U(i)}}, \ \mathrm{if} \ b\in\{a-1,a,a+1\}\ \&\ 0\leq i\leq N; 0, \mathrm{otherwise},$$

where $n_{t}$ is the number of active nodes at the time $t$, and $U(n)$ is the potential function.

Examples

if (FALSE) { # interactive()
Nvar <- 10
m <- rep(0, Nvar)
w <- matrix(0.1, Nvar, Nvar)
diag(w) <- 0
result1 <- make_2d_Isingland(m, w)
plot(result1)

set.seed(1614)
sim1 <- simulate_Isingland(result1, initial = 5)
plot(sim1)
}