File:Randomwalk msd.svg
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Size of this PNG preview of this SVG file: 279 × 600 pixels. Other resolutions: 111 × 240 pixels | 223 × 480 pixels | 357 × 768 pixels | 476 × 1,024 pixels | 953 × 2,048 pixels | 335 × 720 pixels.
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Summary
[edit]DescriptionRandomwalk msd.svg |
English: some trajectories and the mean squared displacement of a random walk with a step size of . the circles sshow the MSD at different times |
Date | |
Source | Own work |
Author | Jkrieger |
SVG development InfoField | |
Source code InfoField | MATLAB code% timesteps in each random walk
timesteps=10000;
plotstep=10;
% diffusion coefficient for the random walk
D=0.5;
observation_radius=10;
observation_steps=100;
lim_factor=1.4;
plots=10;
plotevery=1;
%calculate grid size
dr=sqrt(2*D);
t=1:timesteps;
% simulate random walks on grid
x=zeros(plots,timesteps);
y=x;
for k=1:plots
for ct=2:timesteps
x(k,ct)=x(k,ct-1)+dr*(randi(2)-1.5)*2;
y(k,ct)=y(k,ct-1)+dr*(randi(2)-1.5)*2;
end
end
% calculate MSD as ensemble average on a logarihmic lag time scale
msdtau=round(logspace(0,floor(log10(timesteps)),50));
msd(1:length(msdtau))=0;
for tau=1:length(msdtau)
msd(tau)=mean( (x(:,msdtau(tau))-x(:,1)).^2 + (y(:,msdtau(tau))-y(:,1)).^2 );
end
% plot random walk
figure(1)
subplot(2,1,1)
for k=1:plotevery:plots
plot(x(k,1:plotstep:timesteps),y(k,1:plotstep:timesteps), 'Color', hsv2rgb([k/(plots+1),1,1]));
if (k==1)
hold on;
end
end
for k=5:5:length(msdtau)
drr=sqrt(msd(k));
rectangle('Position',[-drr,-drr,2*drr,2*drr], 'Curvature',[1,1], 'LineWidth', 2);
end
daspect([1,1,1])
hold off
xlim([-lim_factor*drr lim_factor*drr]);
ylim([-lim_factor*drr lim_factor*drr]);
xlabel('coordinate x');
ylabel('coordinate y');
title('random walk trajectories');
% plot MSD
subplot(2,1,2)
loglog(msdtau, msd, 'LineWidth', 2);
hold on
loglog(msdtau, 4*D*msdtau, 'r--', 'LineWidth', 2);
hold off
xlabel('lag time t')
ylabel('MSD \langle r^2(\tau)\rangle');
M{1}='measured MSD';
M{2}='\langle r^2(\tau)\rangle=4D\cdot\tau';
legend(M,'Location', 'NorthWest');
title('mean squared displacement');
xlim([1 1e4]);
ylim([1 1e4]);
daspect([1,1,1])
|
Licensing
[edit]This file is licensed under the Creative Commons Attribution 3.0 Unported license.
Attribution: Jan Krieger
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 14:44, 5 August 2012 | 335 × 720 (181 KB) | Jkrieger (talk | contribs) | font problem solved | |
14:29, 5 August 2012 | 335 × 720 (197 KB) | Jkrieger (talk | contribs) | last simulation had wrong step size (factor sqrt(2)=sqrt(d)) | ||
12:25, 5 August 2012 | 335 × 720 (105 KB) | Jkrieger (talk | contribs) | formating | ||
11:29, 5 August 2012 | 335 × 720 (114 KB) | Jkrieger (talk | contribs) | {{Information |Description ={{en|1=some trajectories and the mean squared displacement of a random walk with a step size of <math>\sqrt(4\cdot 0.5}=\sqrt(2)</math>. the circles sshow the MSD at different times}} |Source ={{own}} |Author ... |
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