Modify the MatLab code below to answer the following problem:

Consider placing a uniform electric field in the x-direction by modifying the probabilities to step right, up, left or down. Instead of 1/4 probability to step in each of these directions, parameterize the probabilities as: ph + pv = 1 where ph is the probability to step in the horizontal direction pv is the probability to step in the vertical direction. Note that pv = 1 − ph . Let the probability to step up equal the probability to step down, such that pu = pd = pv/2. Let the probability to step right be increased as the electric field is increased. We will not try to get a functional relationship between field strength and probability. Instead, let us define a parameter called 𝛼 that has a range between [-1,1]. When 𝛼 = −1, the electric field causes the particle to always step left. When A = 1, the electric field causes the particle to always step right. When A = 0, the random walk has equal probability to step left and right given by pL = pR = ph/2. A reasonable parameterized model might look like:
ph = 1/2*(1+A^2)

Answer the following questions, for a variety of different 𝛼 for the electric field in the x-direction.
1) Visualize the random walk trajectories
2) In the x-direction, calculate the mean squared displacement as a function of number of steps.
3) In the y-direction, calculate the mean squared displacement as a function of number of steps.
4) For 1,000,000 walks, plot the histogram of locations of the random walker in the x-direction
after 100, 1000, 10000, 100000, 1000000 steps.
5) For 1,000,000 walks, plot the histogram of locations of the random walker in the y-direction
after 100, 1000, 10000, 100000, 1000000 steps.
Using the same model except with the electric field in the y-direction, answer these questions:
1) What should the new equations for the step-probabilities be?
2) Visualize the random walk trajectories
3) In the x-direction, calculate the mean squared displacement as a function of number of steps.
4) In the y-direction, calculate the mean squared displacement as a function of number of steps.
5) For 1,000,000 walks, plot the histogram of locations of the random walker in the x-direction
after 100, 1000, 10000, 100000, 1000000 steps.
6) For 1,000,000 walks, plot the histogram of locations of the random walker in the y-direction
after 100, 1000, 10000, 100000, 1000000 steps.






%% RANDOM WALKTS, BROWNIAN MOTION, AND DIFFUSION


% ————————— parameters
width = 5;                                           % width of pipe in mm
a = 0.01;                                         % lattice constant in mm
pw = 0;                                           % probability of waiting
num_steps = 1000000;                              % # steps for histograms
num_walks = 1000000;                       % # random walks for histograms

% ————————— calculate lattice size based on width
lattice_size = round(width/a);

% ————————— initialize random walker position
x = 0;
y = lattice_size/2;

% ————————— initialize mean squared displacement variables
msd_x = zeros(1,num_steps);
msd_y = zeros(1,num_steps);

% ————————— loop over the # of steps
for step = 2:num_steps+1
%           choice: does walker wait or make a step?
            if rand<=pw
%           walker waits
            continue;
            else
%           walker makes a step
%           calculate the step direction
            dx = randi([-1,1]);
            dy = randi([-1,1]);
%           update walker position
            x = x+dx;
            y = y+dy;
%           reflect at the y-direction boundary
            if y<1
                y = 2-y;
            elseif y>lattice_size
                y = 2*lattice_size - y - 1;
            end
%           calculate mean squared displacement in each direction
            msd_x(step) = (x*a)^2;
            msd_y(step) = ((y-(lattice_size/2))*a)^2;
      end
end

% —————————— 1) visualize the random walk trajectories
figure;
plot((0:num_steps)*a, (msd_y*a), 'b.')
title('Random Walk Trajectories');
xlabel('x (mm)');
ylabel('y (mm)');
xlim([0,num_steps*a]);

% ————————— 2) mean squared displacement in x-direction
figure;
n_steps = 1:num_steps;
msd_x = cumsum(msd_x) ./ (1:num_steps+1);
plot(n_steps(1:num_steps-1)*a, msd_x(2:num_steps)*a^2, 'b')
title('Mean Squared Displacement in the x-Direction');
xlabel('Number of Steps');
ylabel('Mean Squared Displacement');
xlim([0,num_steps*a]);

% ————————— 3) mean squared displacement in y-direction
figure;
msd_y = cumsum(msd_y) ./ (1:num_steps+1);
plot(n_steps(1:num_steps)*a, msd_y(1:num_steps)*a^2, 'b')
title('Mean Squared Displacement in the y-Direction');
xlabel('Number of Steps');
ylabel('Mean Squared Displacement');
xlim([0, num_steps*a]);

% ————————— 4) histogram of locations in x-direction
figure;
for num_steps = [100,1000,10000,100000,1000000]
    x_hist = zeros(1,lattice_size+1);
    for walk = 1:num_walks
        x=0;
        for step = 1:num_steps
            dx = randi([-1,1]);
            x = x+dx;
            if x<0
                x=0;
            elseif x>lattice_size
                x = lattice_size;
            end
        end
        x_hist(x+1) = x_hist(x+1) + 1;
    end
    subplot(2,3,find([100,1000,10000,100000,1000000] == num_steps));
    bar(0:lattice_size, x_hist / num_walks, 'b');
    title(['Histogram of x-locations after ', num2str(num_steps), ' steps']);
    xlabel('x');
    ylabel('Probability');
end

% ————————— 5) histogram of locations in y-direction
figure;
for num_steps = [100,1000,10000,100000,1000000]
    y_hist = zeros(1,lattice_size+1);
    for walk = 1:num_walks
        x = 0;
        y = lattice_size/2;
        for step = 1:num_steps
            dx = randi([-1,1]);
            dy = randi([-1,1]);
            x = x+dx;
            y = y+dy;
            if y<1
                y = 2-y;
            elseif y>lattice_size
                y = 2*lattice_size - y - 1;
            end
        end
        y_hist(y+1) = y_hist(y+1) + 1;
    end
    subplot(2,3,find([100,1000,10000,100000,1000000] == num_steps));
    bar(0:lattice_size, y_hist / num_walks, 'b');
    title(['Histogram of y-locations after ', num2str(num_steps), ' steps']);
    xlabel('y');
    ylabel('Probability');
end

% ————————— 6) calculate avg time per step for given diffusion constant
D_desired = 1e-7;                 % desired diffusion constant in cm^2/sec
t = D_desired / (a^2);                          % avg time per step in sec
%                     substitude the values, t = (10^-7) / (.01^2) = 10^-3
%                 so the average time per step (t) should be 10^-3 seconds

% ————————— 7) calculate diffusion constant in the x-direction
figure;
diffusion_x = gradient(msd_x*a^2) / (2*t);
plot(n_steps*a, diffusion_x);
title('Diffusion Constant in the x-Direction');
xlabel('Time (mm)');
ylabel('Diffusion Constant');
xlim([0, num_steps*a]);