Cellular Automaton and traffic Flow 2
The model thus obtained is an example of mathematical object known by the name of "cellular automaton". The general definition of cellular automaton is a set of system with autonomous temporal evolution that has the following properties:
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* There are "cells" which takes finite states connected to each other
* The time flows in discrete, step-wise fashion
* The evolution of a cell is determined by current states of itself and neighboring cells
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(3.4)
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Our traffic model with single road is an example of one-dimensional cellular atomaton where cells are connected in linear sequence.
Consider the following particular rule among one dimensional cellular automaton
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* When the neighbor to the left is in the state"0", and the neighbor to the right in "1", the state of a cell in the next time step is "0".
* When the neighbor to the left is "0" and the neighbor to the right "1", a cell keeps the current state to the next time step.
* When the neighbor to the left is "1" and the current state of a cell is "0", the state in the next time step is "1".
* When the neighbor to the left is "1" and the current state is "1", the state in the next time step is identical to the current state of the neighbor to the right.
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(3.5)
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So the evolution in one time step of the state of each cell is determined by the current state of that cell and two nearest neighbors. Writing down the pattern of evolution explicitely, we have
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1 1 1
1
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1 1 0
0
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1 0 1
1
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1 0 0
1
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0 1 1
1
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0 1 0
0
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0 0 1
0
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0 0 0
0
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(3.6)
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If we consider "1" as the cell with a car and "0" the emptiness, this evolution can be thought of as representing the left-to-right traffic flow. For example, if we follow the evolution of the initial state "0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1 0 0", we obtain the flow diagram
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0 0 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 0
0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 0 1
1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0
0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1
1 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0
0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 1
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(3.7)
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A car moves in to the neighor to the right when it is empty, but stay in current position if the neighbor to the rightis occupied by another car. This turns out to be the simplest of discrete models of traffic flow.
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