Random Systems

1.Random walks

The walker begins at the origin,x=0,and the first step is chosen at random to be either to the left or right,each with probability 1/2.In a physical process such as the motion of a molecule in solution,the time between steps is approximately a constant , so the step number is roughly proportional to time .Therefore,we often refer to the walker's position as a function of time.
Routine:We generate a random number in the range between 0 and 1 and compare its value to 1/2.If it is less than 1/2,our walker moves right ,otherwise it steps to the left.
If the total number of steps is n, and k steps are taken to right,then we can calculate in maths :

final19.gif final2.gif final3.gif
代码:http://www.jianshu.com/p/c3b6937c10ae final4.gif
代码:http://www.jianshu.com/p/b40283b37c41 final5.gif
代码:http://www.jianshu.com/p/ad9b5a52ce78

Conclusion:

We can see that the results of simulation are well approximately to the analyses in maths when the number of walkers are large enough.

When step length is random

we can change the code

length = 2*np.random.rand() - 1

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If the probability of moving to right is not equal to 1/2:

For example:

for i in range(100):
    for j in range(500):
        ruler = np.random.rand()
        if ruler<=0.75:
            x_now[j] = x_now[j] + 1
        else:
            x_now[j] = x_now[j] - 1
        x2_now[j] = x_now[j]**2

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Random walk 2D

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2.Random walks and diffusion

In our discussion of random walkers we have, up to this point, focused on the motion of individual walkers. An alternative way to describe the same physics involves the density of particles,which can be conveniently defined if the system contains a large number of particles.The density is then proportional to the probability per unit volume per unit time.P(i,j,k,n) is the probability to find the at the site(i,j,k) at time n.

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We can also get:

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Considerate problem of one dimension diffusion

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The finite-difference version of this is:

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We get:

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To guarantee numerical stability we must make sure that the space and time steps satisfy :

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代码:http://www.jianshu.com/p/63160fa9c5e1

final18.gif Diffusion_center.gif

Conclusion:

The graph of density is approximately to a Gaussian function. When time is enough long, the density is equal in the container.

3.Cluster growth models

The growth of clusters is closely related to random walks.
From Wikipedia, the free encyclopedia
The Eden growth model describes the growth of specific types of clusters such as bacterial colonies and deposition of materials. These clusters grow by random accumulation of material on their boundary. These are also an example of a surface fractal.The model is named after Murray Eden.
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory ,is applicable to aggregation in any system where diffusion is the primary means of transport in the system.
The basic idea is: First, an initial particle is uesd as the seed. A particle is randomly generated at random locations away from the seed to make a random walk until it becomes a part of the group; and then generates a random particle, repeat the above process, so that you can get enough DLA clusters.

DLA_Cluster.jpg Of7_p0001_15h.jpg
code:http://www.jianshu.com/p/c9f8c488e044 final9.gif final10.gif final11.gif