LABORATORY 2: RANDOM WALKS AND MONTE-CARLO SIMULATIONS

 

Mathematics:

A molecule moves as a result of collisions with other molecules. Suppose an average time between two collisions equals tot = 1. Also suppose that a single collision displaces a molecule by x = s in the direction of the x-axis or in the opposite direction. If collisions are equally likely in both directions, the probabilities of the molecule's shift to the right and to the left after an individual collision are the same: p = q = . This is the simplest model for a one-dimensional random walk of molecules called the Brownian motion. If x is a position of a molecule, then all possible moves (trajectories) of x as a result of n collisions with other molecules are shown on the picture called a binary tree (shown for n = 6):

 

Consider an ensemble of N molecules thrown in a single point, say x = 0. As a result of molecular collisions, they start to spread or to diffuse. Let xk(n) denote a displacement of the k-th molecule after n collisions. The diffusion (spread) of molecules can be characterized by two mathematical quantities:

 

          mean (average displacement): = xk(n)

 

          variance (average squared displacement):*2 = ( xk(n) - )2

 

The theory of diffusion works for large N and predicts that = 0 and 2 = D n, where D is the coefficient of diffusion.

 

Objectives:

         visualize individual trajectories of molecules in the ensemble of N molecules

         plot the mean and the variance2 as functions of n for an ensemble of N molecules and find the coefficient of diffusion D

         plot the mean and the variance2 as functions of N for the same number of n collisions and compare with the theoretical values for and2

         understand limits of applicability of the theory of diffusion

MATLAB script "Individual trajectories of molecules in Brownian motion"

 

A typical pattern of N = 15 molecules for n = 200 collisions is shown here:

 

Steps in writing the MATLAB script:
  1. Initialize values of s, N, and n.
  2. Create a matrix of N by n random numbers equally distributed in the interval [-0.5, 0.5].
  3. Transform the entries of the random matrix to the binary entries: +s and s.
  4. Compute an N by n matrix of total displacements of molecules by summing columns of the binary matrix in (3) between the first column (a starting time of collisions) and an intermediate column (the current time of collisions).
  5. Plot each row of the matrix of total displacements at the same graph.

 

Exploiting the MATLAB script:

  1. Run the script with N = 10; 20; 50 and n = 100; 200; 300.
  2. Check that the trajectories are more spreaded for larger values of n
  3. Check that the trajectories are symmetric about the zero mean for large values of N

 

 

 

 

 

 

 

 

MATLAB script "Evolution of statistical parameters in a random walk of molecules (diffusion)"

 

The diffusion theory predicts that the mean stays at zero in time (with larger number of collisions n), while the variance2 grows linearly in time. Evolution of the mean and the variance2 in time depends on the number of molecules N in a statistical ensemble, as well as on each particular random data realization. Such numerical computations are called the Monte-Carlo simulations. A typical pattern for evolution of the mean and the variance2 is shown here for ten ensembles with different values of N.

 

We can compute the coefficient of diffusion D (the slope of the line2 versus time) and plot it as functions of number of molecules N. The theory predicts that the Monte-Carlo approximation of D should approach to the theoretical value of D with large N.

Steps in writing the MATLAB script:

 

  1. Initialize values for s, n, and a vector of values of N.
  2. Loop through the values of N computing the trajectories of molecules as in the previous script.
  3. Find the values of the mean and the variance2 at each instance of time
  4. Plot the mean and the variance2 as functions of time.
  5. Compute an average value for the coefficient of diffusion D for each value of N
  6. Plot the values of D as a function of N.

 

 

Exploiting the MATLAB script:

  1. Run the script with the same values of N = 100 : 100 : 1000 and n = 200 several times and check that each time Monte-Carlo simulations produce a different but similar picture.
  2. Comment the part of the script where plots of figure(1) and figure(2) are computed (this operation takes a lot of computational time). Run the script for N = 500 : 500 : 10000. Is there a better convergence for the coefficient of diffusion D as a function of N?

 

 

 

QUIZ: MATLAB script "Comparison of the average theory with individual trajectories of molecules"

 

The theory predicts that the standard deviation of an average molecule grows as a parabola with respect to time: = Due to the symmetry, molecules spread to the left and to the right equally likely. We can combine the parabolic walk of an average molecule in time with individual random trajectories of molecules in a given ensemble. A typical pattern for N = 20 and n = 1000 is shown here:

 

 

 
Steps in writing the MATLAB script:

 

1.       Compute the random matrix of total displacement similar to the first script

2.       Compute the mean, variance, and the coefficient of diffusion similar to the second script.

3.       Plot each row of the matrix of total displacements at the same graph by blue color.

4.       Compute two vectors for positive and negative trajectories of an average molecule.

5.       Plot the two vectors at the same graph by red color.

 

 

Exploiting the MATLAB script:

  1. Run the script with N = 20; 50 and n = 200; 500.