.. _sphx_glr_api_gallery_statphys_stochastic_processes:

Stochastic processes
--------------------

Brownian motion and the Einstein relation, Langevin dynamics and the
fluctuation-dissipation balance, the Fokker-Planck equation for the
probability density, the Ornstein-Uhlenbeck process, and the
Euler-Maruyama and Milstein schemes that integrate Itô stochastic
differential equations.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;s 1905 paper made two separate claims about a suspended particle. First, its random wandering spreads as \langle\Delta x^2\rangle = 2Dt. Second, the diffusion coefficient is not a free parameter: it equals the particle&#x27;s mobility \mu, the drift speed per unit applied force, times the temperature,">

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  .. image:: /api/gallery/statphys/stochastic_processes/images/thumb/sphx_glr_plot_einstein_brownian_motion_mobility_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/stochastic_processes/plot_einstein_brownian_motion_mobility`

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      <div class="sphx-glr-thumbnail-title">Einstein's Brownian motion: diffusion equals mobility times temperature</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Adriaan Fokker (1914) and Max Planck (1917) replaced the random path of a Brownian particle with an equation for its probability density. For dX = A(X)\,dt + \sqrt{2D}\,dW,">

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  .. image:: /api/gallery/statphys/stochastic_processes/images/thumb/sphx_glr_plot_fokker_planck_equation_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/stochastic_processes/plot_fokker_planck_equation`

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      <div class="sphx-glr-thumbnail-title">The Fokker-Planck equation: from noisy trajectories to a deterministic density</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Kiyosi Itô (1944) gave meaning to integrals against Brownian motion, and with them to stochastic differential equations dX = a(X)\,dt + b(X)\,dW. The key rule is that dW^2 = dt, not zero. As a result, \ln X for geometric Brownian motion dX = \mu X\,dt + \sigma X\,dW drifts at \mu - \sigma^2/2 rather than \mu, and the exact solution is">

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  .. image:: /api/gallery/statphys/stochastic_processes/images/thumb/sphx_glr_plot_ito_euler_maruyama_milstein_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/stochastic_processes/plot_ito_euler_maruyama_milstein`

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      <div class="sphx-glr-thumbnail-title">Itô's stochastic calculus: the Euler-Maruyama and Milstein schemes</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Langevin&#x27;s 1908 equation for a Brownian particle,">

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  .. image:: /api/gallery/statphys/stochastic_processes/images/thumb/sphx_glr_plot_langevin_fluctuation_dissipation_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/stochastic_processes/plot_langevin_fluctuation_dissipation`

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      <div class="sphx-glr-thumbnail-title">The Langevin equation: friction and noise must balance</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In 1930 George Uhlenbeck and Leonard Ornstein solved Langevin&#x27;s equation for the velocity of a free Brownian particle exactly. The velocity is not white noise. It is a process pulled back toward its mean at a rate \theta while being kicked with strength \sigma,">

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  .. image:: /api/gallery/statphys/stochastic_processes/images/thumb/sphx_glr_plot_ornstein_uhlenbeck_process_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/stochastic_processes/plot_ornstein_uhlenbeck_process`

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      <div class="sphx-glr-thumbnail-title">The Ornstein-Uhlenbeck process: Gaussian noise with a memory</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/statphys/stochastic_processes/plot_einstein_brownian_motion_mobility
   /api/gallery/statphys/stochastic_processes/plot_fokker_planck_equation
   /api/gallery/statphys/stochastic_processes/plot_ito_euler_maruyama_milstein
   /api/gallery/statphys/stochastic_processes/plot_langevin_fluctuation_dissipation
   /api/gallery/statphys/stochastic_processes/plot_ornstein_uhlenbeck_process