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13 Combining Mathematical and Simulation Approaches to Understand... 311

0.12
50 000 simulation runs
Empirical relative
frequency distribution
0.10 Exact
prob. function
Error bars
Relative frequency 0.06
(std. error)
0.08

0.04

0.02

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Location of random walker after 100 time-steps

Fig. 13.12 Probability function of the position of the one-dimensional random walker in time-step
100, starting at an initial random location

To be sure, note that defining the state of the system in this way, it is true that there
is a fixed probability of going from any state to any other state, independent of time.
The transition matrix P D [p i,j ] corresponding to the model is:

2 3
0 1 0 0
:
6 : 7
6 0:5 0 0:5 0 : 7
6 7
0 0:5 0 0:5 0
6 7
6 7
0 0:5 0 0:5 0
6 7
6 7
6 7 (13.1)
P D P i;j D : : : : : :
6 : : : : : : : : : : : 7
6 : 7
0 0:5 0 0:5 0
6 7
6 7
:
6 7
6 : 7
4 : 0 0:5 0 0:5 5
0 0 1 0
where, as explained above, p i,j is the probability P(X nC1 D jjX n D i) that the system
will be in state j in the following time-step, knowing that it is currently in state i.
13.7.2 Transient Distributions of Finite THMCs
The analysis of the dynamics of THMCs is usually divided into two parts: transient
dynamics (finite time) and asymptotic dynamics (infinite time). The transient