1. MODELING OF A FOREST FIRE AS PERCOLATION PROCESS.

A nearly-square square lattice is filled randomly with trees to a specified
density.  Then a forest fire is started by torching every pixel down one
column of the lattice, along the left edge.  Because the lattice is wrapped
left to right and top to bottom, the line of fires at the left is echoed on
the right. (The wrapping is made obvious by including the leftmost line of
trees on the right.)

cd ~jrupley/585/xtoys

[read xburn.txt for instructions, if you wish]

The command is of the form:

	xburn p block_size

		p = initial fraction lattice sites filled
		  = number of trees per unit area before burn

		block_size = number of pixels representing one lattice site
			   set for comfortable viewing (I suggest 4)

To see slow burn at critical tree density (pc), with tortuous burn paths
but with paths possibly connecting all edges, run

	xburn .593 4

To see a fast burn at high density, with highly interconnected filled
sites (percolation probability = order parameter P, large), and thus with a
vertical front of burning trees that moves fast, run

	xburn .9 4

To see a sequence of burns, run

	for i in `cat xburn-seq.txt`; do xburn $i 4 ; done

After each run of xburn, click on quit -- next burn will come up until
sequence in xburn-seq.txt is completed.

Notice the fast burn at low and high density of trees; the slow rate in the
critical region near .593 fraction filled, the establishment of edge-to-edge
connectivity in the critical region; the strong change in cluster size in
the critical region; and other characteristics of a phase transition.


2. TWO-DIMENSIONAL ISING MODEL - FERROMAGNETIC PHASE TRANSITION

A magnet is represented by a system of spins arranged on a square lattice.
Each spin can be in one of two states, A or B, up or down.  Adjacent
spins in the same spin state make a favorable (negative) contribution to the
energy, and adjacent spins in opposite states make an unfavorable
contribution.  

A magnetic field, if present, favors one spin state over the other.  We
operate first with no magnetic field.

The statistical weight of the collection spins represeted by the
lattice is determined by a Boltzmann factor in the energy,
exp(-beta*Ei), where beta = 1/kT.  Thus at low temperature the summed
favorable contributions of AA or BB interactions are important, and
the system can separate into phases of all A or all B type spins.  At
high temperature, the favorable/unfavorable contributions of adjacency
are washed out and thermal randomness appears.

The cluster size changes explosively at the critical temperature.

Spontaneous magnetization is a consequence of the disproportionation into
phases, one dominant, at temperatures below the critical.


To see the above, run:

xising

and play with the heat and cool buttons.


The cluster algorithm quickly gives the "correct" end result of cluster
size, etc.



To see more complicated situations, run:

xpotts

Select cell size 4 and select q=2 for the Ising model.

Play with the heat and cool buttons, as for xising.
Note at beta > beta critical, the system is phase separated.
Note that with phase separation the magnetization is far from zero.
Heat the system up and watch the pure phases break up again, and the
magnetization drop.


At beta > beta critical, turn on the magnetization, and watch the phases
go appropriately all A or all B (increase M -> all red)


For more information, you should read the information files:

xising.txt
xpotts.txt

(and perhaps also a text that includes a section on the Ising magnet).


3. CELLULAR AUTOMATA -- CONWAY'S GAME OF LIFE AND OTHERS

Cellular automata have been used to model physical, biological, and
other (computer) problems.  Elements on a lattice replicate, survive
and die according to rules that govern the transition from one state to
the next.  The patterns can be used to infer behavior, to construct a
complete computer (a Turing machine), or simply to play games.

For a classic, Conway's life, run:

xautomalab

	select pause
	select all eight neighbors on
	select births for 3 only
	select survivors for 2 or 3 only (xautomalab.txt is wrong wrt survivors)
	select periodic and tracer off
		or select open if want stable gun pattern
	set cell size 1 and expand window
	then restore the pattern and start the run
		before run xautomalab, need to have set up xautomalab.gif: 
			cp fiveguns.gif xautomalab.gif


There is an information file, xautomalab.txt

4. SANDPILES

The sandpile model, devised by Per Bak and colleagues, focuses on
"organized criticality", which itself is explained by considering how
a sandpile behaves - random addition, catastrophic relaxation, then
restoration by further random addition.

Run

xsand

select run off == pause
select cell size 8 or at least > 1, to speed up changes
set auto-d off
set tracer off
set boundary open

Then read the information sheet, xsand.txt, and run the suggested
experiments.