Fix the following MatLab code. Figure 4 doesn't display anything, and this line of code is wrong:  histogram(y_positions(numSteps_histogram(i), :));


Code:


% Define parameters
width = 5; % width of the pipe in mm
a = 0.01; % lattice constant in mm
tau = 1; % time step
pw = 0; % probability of waiting at each time step
numSteps = 10; % number of time steps to simulate
numWalks = 1000; % number of walks for histograms

% Calculate the number of lattice points in the width dimension
numPoints = width / a;

% Initialize arrays to store x and y positions
x_positions = zeros(numSteps, numWalks);
y_positions = zeros(numSteps, numWalks);

% Initialize the step counts in each direction
stepsRight = zeros(numSteps, 1);
stepsUp = zeros(numSteps, 1);

% Simulate the random walk for each walk
for w = 1:numWalks
    % Initialize the random walker’s position
    x = 0;
    y = floor(numPoints / 2); % starting in the middle of the pipe
    
    % Simulate the random walk
    for i = 1:numSteps
        % Record the position at each step
        x_positions(i, w) = x;
        y_positions(i, w) = y;
        
        % Check if the random walker makes a step or waits
        if rand > pw
            % Randomly choose the direction of the step
            direction = randi(4);
            % Make the step and update the position
            if direction == 1 % right
                x = x + a;
                stepsRight(i) = stepsRight(i) + 1;
            elseif direction == 2 % up
                y = y + 1;
                stepsUp(i) = stepsUp(i) + 1;
            elseif direction == 3 % left
                x = x - a;
                stepsRight(i) = stepsRight(i) - 1;
            else % down
                y = y - 1;
                stepsUp(i) = stepsUp(i) - 1;
            end
        end

        % Reflective boundary conditions in the y-direction
        if y < 0
            y = -y;
        elseif y >= numPoints
            y = 2 * numPoints - y - 1;
        end
    end
end

% Visualize the random walk trajectories
figure;
plot(x_positions, y_positions);
xlabel('x-position (mm)');
ylabel('y-position (mm)');
title('Random Walk Trajectories');

% Calculate the mean squared displacement in the x-direction
msd_x = mean(x_positions.^2, 2);
steps = (1:numSteps)';
figure;
plot(steps, msd_x);
xlabel('Number of steps');
ylabel('Mean Squared Displacement in x-direction (mm^2)');
title('Mean Squared Displacement in x-direction');

% Calculate the mean squared displacement in the y-direction
msd_y = mean(y_positions.^2, 2);
figure;
plot(steps, msd_y);
xlabel('Number of steps');
ylabel('Mean Squared Displacement in y-direction (mm^2)');
title('Mean Squared Displacement in y-direction');

% Plot histograms of x-positions for different numbers of steps
numSteps_histogram = [100, 1000, 10000, 100000];
figure;
for i = 1:length(numSteps_histogram)
    subplot(2, 3, i);
    histogram(x_positions(numSteps_histogram(i), :));
    xlabel('x-position (mm)');
    ylabel('Frequency');
    title(['Histogram after ', num2str(numSteps_histogram(i)), ' steps']);
end

% Plot histograms of y-positions for different numbers of steps
figure;
for i = 1:length(numSteps_histogram)
    subplot(2, 3, i);
    histogram(y_positions(numSteps_histogram(i), :));
    xlabel('y-position (mm)');
    ylabel('Frequency');
    title(['Histogram after ', num2str(numSteps_histogram(i)), ' steps']);
end

% Calculate the required time per step (tau) for a diffusion constant of 10^7 cm^2/s
required_D = 1e7; % cm^2/s
required_tau = a^2 / required_D; % s
fprintf('The average time per step (tau) should be %.4f seconds to obtain a diffusion constant of 10^7 cm^2/s.\n', required_tau);

% Calculate the diffusion constant in the x-direction by taking the slope of the mean squared displacement as a function of time
coefficients_x = polyfit(tau*steps, msd_x, 1);
D_x = coefficients_x(1) / 2;
fprintf('The diffusion constant (D) in the x-direction is %.4f mm^2/tau.\n', D_x);