Transcript Slide 1
Cellular Automata Modeling of Physical, Chemical and Biological Systems Peter HANTZ
Sapientia University, Department of Natural and Technical Sciences
Marine Genomics Europe Summer Course, Naples, 3 July 2007
space
Game on a String:
Two States: 0 (black) and 1 (white) Neighborhood: 3 cells A Transition Rule: 2 x 2 x 2=8 possibilities have to be specified Generic Rule:
3 Black = White 2 Black = Black 1 Black = Black 3 White = White
Denomination of the Rules: Wolfram Convention
current pattern: 111 110 101 100 011 010 001 000 new state for center cell 0 1 1 1 1 1 1 0 In binary notation: 0 + 2 6 + 2 5 + 2 4 + 2 3 + 2 2 + 2 1 + 0 = 126 Possible patterns: Outputs: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 =>256 possible general rules
A Game or Something More?
http://www.ifi.unizh.ch/ailab/teaching/FG05/script/FM3-CA.pdf
I. a trivial state II. simple or periodic structures Sierpinski triangle: a fractal III. chaotic structures IV. complex “migrating” structures
Game on a 2D grid
Neumann Neighborhood Moore Neighborhood NN - Possibilities to be specified: 2 5 = 32 => Number of Possible Rules: 2 32 = huge
Simple “totalistic” rules:
Conway’s Game of Life
(Moore Nbh.) Any live cell with fewer than 2 live neighbors dies, as if by loneliness. Any live cell with more than 3 live neighbors dies, as if by overcrowding. Any live cell with 2 or 3 live neighbors lives, unchanged, to the next generation. Any empty (dead) cell with exactly 3 live neighbors comes to life.
Features of the Game of Life
Steady objects Oscillators Gliders and Spaceships
CA in Biology/Chemistry:
Excitable Media
Greenberg-Hastings Model: cells with several states Q: E 1 …E e : R 1 …R r : one quiescent state
e
excited states
r
refractory states
Rules: Q →E1
when a NN is excited excited cells:
E k →E k+1 , E e
refractory cells:
R k →R k+1 , R r →R 1 →Q
Weimar, 1998 BZ Reaction Bub et al., 1998 http://psoup.math.wisc.edu/java/jgh.html
CA in Physics/Biology:
Growth Phenomena: diffusion-limited aggregation Particles
(blue) move randomly in the grid. When a blue particles touches a green solid particle, it also turns green and "sticks“.
Particle density (parameter):
a percentage, saying what fraction of sites will contain blue particles at the beginning.
The shape of the branching structure (~fractal) depends on the initial density of blue particles. http://germain.umemat.maine.edu/faculty/hiebeler/java/CA/DLA/DLA.html
Ben-Jacob et al., 1994
CA in Physics:
Traffic modeling
Nagel-Schreckenberg model: a probabilistic cellular automaton
The street is divided into cells, that may contain cars with velocity v
Step 1: acceleration
All cars that have not already reached the maximal velocity
v
max , accelerate by one unit:
v
->
v
+1
Step 2: safety distance
If a car has
d
empty cells in front of it, and its velocity is
v
larger then
d
, it reduces the velocity to
d
:
v
->min{
d
,
v
}
Step 3: randomization
With probability
p
, the velocity is reduced by one unit:
v
->
v
-1
Step 4: driving
Car
n
moves forward
v n
cells:
x n
->
x n
+
v n
.
Cyclic boundary conditions http://www.thp.uni-koeln.de/~as/Mypage/traffic.html
Traffic modeling continued
http://www.grad.hr/nastava/fizika/ole/simulation.html
green: v=4, khaki: v=3, brown: v=2, yellow: v=1,
red: v=0
Poore, 2006 Having p=0: second-order phase transition at a critical car density
Self-reproduction
The Idea of János NEUMANN, in 1948, before the discovery of the DNA!
=> development of the Cellular Automata Science Langton’s loop http://necsi.org/postdocs/sayama/sdsr/ 8 states, 29 rules: Tempesti, 1998
Self-reproduction with Evolution?
Artificial Life, Evoloop
“Collision” of two SR loops may lead to third, different “Hurted” SR loops may disappear SR loop Smaller individuals were naturally selected? Or just fragmentation?
http://necsi.org/postdocs/sayama/sdsr/
So, what are cellular automata?
Dynamical System
: a rule that describes the time evolution of the state (
x
) of an arbitrary system The
system state
at time t is a description of the system which is sufficient to predict the future states of the system without recourse to states prior to t.
Continuous
in state and time: differential equations
dx
(
t
)
dt f
(
x
(
t
))
Discrete
in state and time: cellular automata
x(t+1)=f(x(t))
Some Special Behavior of Dynamical Systems:
Fixed point Limit cycle
x(t+1)=x(t);
dx
(
t
) 0
dt
x(t+k)=x(t) for all t>t 0 for all t>t 0
Separating Length Scales: combining CA and Differential Equations
Hantz, 2002
a
(
x
,
y
,
t
)
t
b
(
x
,
y
,
t
)
t
c
(
x
,
y
,
t
)
t
D a
(
d
)
a
(
x
,
D D b c
( (
d d
) )
b
c
(
x
, (
x
,
y
,
t y
,
t
) )
r r
a
(
x
a
(
x
,
y
,
t
)
r
a
(
x
, ,
y
,
t
)
b
(
x
,
y
,
t
)
y
,
t
)
b
(
x
,
y
,
t
)
b
(
x
,
y y
,
t
) ,
t
) [
R
1 ]
R
0 : [
c
(
x
,
y
,
t
)
c
* *] [
d
(
x
,
y
,
t
)
empty
] [
d
(
x
,
y
,
t
t
)
active border
]
(formation of new active borders c * *
nucleation threshold)
R
[
d
1 : (
x
[
c
(
x
,
y
,
t
)
NN
,
y NN
,
t
c
*]
t
) [
d
(
x
,
active y
,
t
)
active border
] [
d
(
x NN
,
y NN
,
t
)
border
] [
d
(
x
,
y
,
t
t
)
bulk
empty precipitat e
] ] [
c
(
x
,
y
,
t
t
)
(progressi on of existing active borders /fronts/ c*
growth threshold)
0 ]
R
2 : [
c
(
x
,
y
,
t
)
c
*] [
d
(
x
,
y
,
t
)
active border
] [
T
(
x
,
y
,
t
) (
v
)] [
T
(
x
,
y
,
t
t
)
T
(
x
,
y
,
t
)
t
]
(aging of active borders the
(
) maximal lifetime may depend on the front speed)
R
3 : [
c
(
x
,
y
,
t
) [
d
(
x
,
y
,
t
t
)
c
*] [
d
(
x
,
passive y
,
t
)
border
]
active border
] [
T
(
x
,
y
,
t
) (
v
)]
(formation of passive borders)
R
4 : [
d
(
x
,
y
,
t
)
active border
] [
d
(
x NN
,
y NN
,
t
)
non
[
d
(
x
,
y
,
t
t
)
bulk precipitat e
]
enpty
(
x NN
,
y NN
)]
(only uncovered borders of the precipitat e can be active) PASSIVE BORDERS ARE UNPERMEABL E