Random systems
标签（空格分隔）： Final
物基一班 熊毅恒 2014301020065
![]()
Paste_Image.png
Abstract
Because the random systems can not be determined or predicted by the initial conditions, in order to have a rough understanding on the random systems such as diffusion processes, we explore various kinds of random walk model and investigate some properties of them which can be useful in some random systems as well.
Keywords
- Random Systems
- Random Walk
- Diffusion
- Entropy
Introduction
We mainly consider the random walk model with some different approaches. We have mainly calculated the main square distance varies with time. In the self-avoiding walk we also investigate the standard deviation of the square distance varies with time. After that we set up a partial differential equation diffusion model and compare it with the previous random walk model. Finally, consider the entropy varies with time by random walk approach in the cream in coffee problem.
Random walk
-
The walker begins at the origin, and the first step is chosen at random at either to the right or the left,each with the probablitiety $\frac {1} {2}$ .Then the next step was chosen, and agian the probablities for stepping left or right are both $\frac {1} {2}$ 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. We will, therefore, often refer to the walker's position as a function of time.
1.png
Also, more applicability，let the step length to be random value in [-1,1]which have same probalilities .We find the x increasing speed is smaller than the previous one.
Then we consider the average of square distance varies with time. Since we have built a list to store the location of one particle in different time, all we have to do now is produce a large number of random moving particles(repeat the previous process many times) and average the final results.due to the equation $<x^![Uploading 2_803612.png . . .]
2>=2Dt $, we will investigate the D value of two different approaches.
this is the line for step length fixed approach, there are two kinds of spots in the diagram. The green one is the average for 20000 times and the orange one is the average for 2000 times. we can say some fluctuation on the orange spots but dispear when the average sample become large. the fitting line is y=1.004*x-0.0337.So the D value is 0.502
this is the line for step length unfixed approach, it is obvious that the distance increasing speed is smaller than the previous one and the D value is approximately 0.1677.
When the probability for moving left and right not being same
(right=0.75,left=0.25), we can see clearly that the line has a trend to go up ,although there are some lightly fluctuations.
One Dimension Diffusion
Writing the position after n steps, as a sum of n sperate steps fives:
Since the steps are independent of each other, the terms with will be with equal probability. If we average over a large number of separate walks this will leave only the terms with i j or s?. Thus we find :
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 (walkers). The idea, known as coarse graining, is to consider regions of space that are big enough to contain a large number of particles so that the density ( =mass/ volume) can be meaningfully defined. The density is then proportional to the probability per unit volume per unit time, denoted by z, t), to find a particle at (t, y, z) at time t. Thus, p and P obey the same equation. To find this equation, we focus back on an individual random walker. We assume that it is confined to take steps on a simple-cubic lattice, and that it makes one "walking step" each time step. P(i, j, k, n) is the probability to find the particle at the side (i, j, k) at time n. Since we are on a simple cubic lattice, there are 6 different nearest neighbor sites. If the walker is on one of these sites at time n — 1, there is a probability of 1/6 that it will then move to site (i, j, k) at time n. Hence, the total probability to arrive at (i, j, k) is :
So:
Apart from a constant factor (), the left side of this equation is just the finite difference approximation for the time derivative of P, while the right-hand side is proportional to a second order space derivative. This suggests taking the continuum limit, which leads to :
fusion equation. For ease of notation we will assume that p is a function of only one spatial dimension, x, although everyt,hing we do below can readily be extended to two or three dimensions. We can then write , so that the first index corresponds to space and the second to time. Converting (2) to one dimension yields :
We know that With the increase of time, the peak density curve fell, the increase scope, with a total area of remain the same.
In 2D situation ,we can have:
We can draw a same conlusion with the 1D This behavior is in accordance with our intuitive sense of a diffusion process, namely in the initial moments in a drip into certain diffusion, then spread to that point as the center spread to the quartet, until the density is the same everywhere.
Cream in coffee
In this topic, we solve the problem which there is a cream in the center of the coffee.
we consider it as two dimension random walk problem. First we set a number of particle in the center of the whole area which is shaped as a square. Then let them do random walk to up,down,left and right four direction. The only limit is once they are reach the edge of the area they can not pass it. Then we observe the whole random particle picture in different time.
this time, we consider an area of 128*128, and only the center area (16,16) has been occupied by the particle initially. the time sequence for eight subplot is not even. they are (0,10,100,500,1000,2000,4000,8000)
we can see more directly with following pics (From internet).
At this point ,we can say that the cream has completely merged with the coffee.
Entropy diffusion
Now its time for us to derive some useful information from this seemingly disordered system. The entropy, to deal with this we have applied a different sumulation method with the above one.
First, we set one particle at the center of the area and let it do the random walk step by step. Each step we will do 5000 times which means there are 5000 particles in the center of the area. Then we add the times of being occupied for every point in this area in 5000 times, so we derive the probability for each state(point) and then use the equation:(the sum is over all point in the area)we get the result.
As we can see, as time increases, the entropy value increases, but its growth speed is reduced.Eventually it will converge to a constant value.
