.. _sphx_glr_api_gallery_statphys_random_walk:

Random walks
------------

Diffusion, the Einstein relation, and the central limit theorem.


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    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;s 1905 analysis of Brownian motion related a microscopically invisible quantity -- the erratic jostling of a pollen grain by individual water molecules -- to a macroscopically measurable one, the diffusion coefficient D, via \langle r^2(t) \rangle = 2 d D t in d spatial dimensions. This was the first quantitative link between atomic-scale motion and observable matter, and it played a central role in convincing skeptics that atoms were real. This example simulates an ensemble of 500 independent walkers on a 2D lattice, each taking unit steps along a randomly chosen axis at every time step, recovers D from their mean squared displacement, and separately shows the central limit theorem in action: however non-Gaussian the individual steps are, the distribution of the walkers&#x27; displacement after many steps is Gaussian.">

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  :doc:`/api/gallery/statphys/random_walk/plot_diffusion_and_clt`

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.. toctree::
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   /api/gallery/statphys/random_walk/plot_diffusion_and_clt