timing+regime

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 * TODO List**
 * REF: margolus diffusion
 * REF: hexagonal diffusion
 * REF: Kauffman
 * REF: Monte Carlo Method

=Timing regimes=

So far we have considered CAs where updating over the field each time step is done **[|synchronously]**. However, this is a particular choice of **timing regime**, with **[|asynchronous]** updating as an obvious alternative. Actually synchronicity was chosen for simplicity, but it can be considered an unfair short cut relative to finding out about how local rules lead to higher order complexity. This is because in the synchronous timing regime timing is actually **controlled globally** each time step (i.e. not independent). This is of course somewhat of a special case and could be considered artificial.

An important question is therefore whether such a timing regime choice is bad. If time steps are relatively small, meaning that there is a small probability for state changes (stochastic next state), in practice the CA does not behave very synchronously and the effect of synchronicity is negligible. If time steps are large in relation to the modelled system however, which is what is most common in modelling, one should envisage **two [|timescales]**. An interaction time scale within a neighbourhood and a time scale over which multi-interactions are integrated.

//Particle conservation//
Interestingly such timing regime issues play a role in modelling [|diffusion] since synchronicity generates a problem with respect to particle conservation. Why does this happen? Normally diffusion is a [|random walk], but in a CA particles are modelled by turning non-particles into particles without the opportunity to simultaneously removing the particle (neighbour) that was the source of the new particle. Moreover, there is a conflict problem in the synchronous case when there are several particles as neighbours. Attempts to solve this problem are as follows:
 * 1) **Using approximations**. Here the average number of particles is maintained, however this leads to a clumped pattern of diffusion which is non-local!
 * 2) **Margolus diffusion**: Here alternate 4-square tile contra-rotation is used. In this way particles are conserved and can pass each other (Toffoli & Margolus 1987?).
 * 3) **Hexagonal diffusion**. A 6 layered hexagonal grid is used where each layer is a direction which determines a next step position (REF). In the verticle axis particles can react (bounce etc). For this model it is possible to show that the analytical limit quite well approaches continuous [|diffusion algorithms].

(//Interestingly gene invasions are often conceptualized as diffusion, however since they are dependent on birth-death processes they are in fact much better conceptualized as a clumped pattern.)//

The role of timing regimes in particle conservation makes an important point of how choosing a modelling formalism can push conceptualization in certain directions. In synchronous updating diffusion becomes a problem and an interesting difference is discovered between particle conservation and birth-death diffusion.

Of course the boolean networks studied by Kauffman (REF) were also studied in the synchronous regime. With asynchrony the general results still hold although they become less extreme.

Generally asynchrony causes a loss of the unique next state assumption due to stochasticity, but it allows conflicts to be avoided. However, how does one decide the order that states are updated?
 * **random locations** (but not all locations may be dealt with)
 * **randomize order** and treat all locations
 * **reaction rates**, reaction at fastest time step happens first ([|Monte Carlo Method]) (REF)

In event-based models there is of course an explicit scheduling of events and this highlights a difference between space-centered approaches and particle-centered approaches ([|Individual based models]). In the latter case particles decide what they want to do and are not dependent on (empty) space, i.e. entity decides output other square and own square.

Of course conventionally diffusion is modelled in lattice maps or [|partial differential equations] (PDEs) where lattice maps are the discrete approximations of PDEs and space and variables are considered to be continuous.

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