.. _sphx_glr_api_gallery_plasma_kinetic:

Kinetic theory (particle-in-cell)
---------------------------------

An electrostatic particle-in-cell (PIC) solution of the 1D1V
Vlasov-Poisson system: particles stream along exact single-particle
orbits and the self-consistent field is recovered by depositing their
charge onto a grid and solving Poisson's equation each step
(:func:`~physicskit.plasma.kinetic.pic_simulate`). The same
deposit-solve-gather-push pipeline reproduces collisionless Landau
damping (:func:`~physicskit.plasma.kinetic.landau_damping_ic`,
:func:`~physicskit.plasma.kinetic.landau_damping_rate`), its
mirror-image growth mechanism, the two-stream instability
(:func:`~physicskit.plasma.kinetic.two_stream_ic`), and a Langmuir wave
ringing in place at the plasma frequency
(:func:`~physicskit.plasma.kinetic.langmuir_wave_ic`), all without ever
assuming a collision operator. The Weibel/filamentation instability
(:func:`~physicskit.plasma.instabilities.weibel_growth_rate`,
:func:`~physicskit.plasma.instabilities.simulate_weibel_filamentation`)
is treated as a reduced quasi-linear model of the same current-driven
physics, at the opposite (temperature-anisotropy) end of the kinetic
spectrum from a beam-driven instability.


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1946, Lev Landau solved the linearized Vlasov equation for a wave on a Maxwellian plasma and found something that seemed to violate reversible mechanics: the wave damps, exponentially, with no collisions or dissipation anywhere in the equations. Particles moving at very nearly the wave&#x27;s phase velocity &quot;surf&quot; it, and because a thermal distribution always has slightly more slightly slower particles being accelerated than slightly faster particles being decelerated, the net exchange drains the wave.">

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  .. image:: /api/gallery/plasma/kinetic/images/thumb/sphx_glr_plot_01_landau_damping_thumb.png
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  :doc:`/api/gallery/plasma/kinetic/plot_01_landau_damping`

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      <div class="sphx-glr-thumbnail-title">Landau damping from first principles with particle-in-cell</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Studying velocity-modulated electron beams for microwave tubes, Andrew Haeff (1948) and John Pierce (1948) independently found that a beam of electrons streaming through a background plasma spontaneously breaks up into density bunches that grow exponentially. David Bohm and Eugene Gross (1949) showed the instability is the mirror image of Landau damping (plot_01_landau_damping): two counter-streaming beams create a region of positive velocity-space slope between their peaks, and it is precisely that positive slope -- formalized a decade later as the Penrose criterion -- that pumps energy from the particles into the wave instead of draining it.">

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  .. image:: /api/gallery/plasma/kinetic/images/thumb/sphx_glr_plot_02_two_stream_instability_thumb.png
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  :doc:`/api/gallery/plasma/kinetic/plot_02_two_stream_instability`

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      <div class="sphx-glr-thumbnail-title">The two-stream instability: exponential growth and phase-space vortices</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Erich Weibel (1959) found that a plasma whose velocity distribution is anisotropic (:math:`T_\perp&gt;T_\parallel`, no bulk streaming at all) is unstable to a purely growing transverse electromagnetic mode -- no beam, density gradient, or external field required. Burton Fried, the same year, supplied the physical mechanism: a random transverse magnetic fluctuation curves particle orbits enough to bunch them into current channels, and the growing channel current reinforces exactly the fluctuation that curved the orbits into existence -- a self-sustaining feedback loop with no threshold.">

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  .. image:: /api/gallery/plasma/kinetic/images/thumb/sphx_glr_plot_03_weibel_filamentation_thumb.png
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  :doc:`/api/gallery/plasma/kinetic/plot_03_weibel_filamentation`

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      <div class="sphx-glr-thumbnail-title">Weibel filamentation: growing current filaments from temperature anisotropy</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Irving Langmuir (1928) discovered that a plasma&#x27;s electron gas resonates at a sharply defined natural frequency: displace the electrons and their own restoring electric field snaps them back, overshoots, and rings, exactly like a mass on a spring built from nothing but the plasma&#x27;s own charge density. langmuir_wave_ic seeds a particle-in-cell plasma with a sinusoidal displacement and a coherent velocity kick,">

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  .. image:: /api/gallery/plasma/kinetic/images/thumb/sphx_glr_plot_04_langmuir_wave_thumb.png
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  :doc:`/api/gallery/plasma/kinetic/plot_04_langmuir_wave`

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      <div class="sphx-glr-thumbnail-title">Langmuir waves: electron density ringing at the plasma frequency</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/plasma/kinetic/plot_01_landau_damping
   /api/gallery/plasma/kinetic/plot_02_two_stream_instability
   /api/gallery/plasma/kinetic/plot_03_weibel_filamentation
   /api/gallery/plasma/kinetic/plot_04_langmuir_wave