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Polarization density - Revision history
2024-03-29T12:54:44Z
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80.42.127.100: /* Definition */
2014-11-18T21:49:40Z
<p><span dir="auto"><span class="autocomment">Definition</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:49, 18 November 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">My name </del>is <del style="font-weight: bold; text-decoration: none;">Marylin Noonan but everybody calls me Marylin</del>. <del style="font-weight: bold; text-decoration: none;">I</del>'<del style="font-weight: bold; text-decoration: none;">m from Brazil</del>. I<del style="font-weight: bold; text-decoration: none;">'m studying at the university </del>(<del style="font-weight: bold; text-decoration: none;">final year</del>) and <del style="font-weight: bold; text-decoration: none;">I play the Trombone </del>for <del style="font-weight: bold; text-decoration: none;">3 years</del>. <del style="font-weight: bold; text-decoration: none;">Usually I choose music from my famous films </del>:<del style="font-weight: bold; text-decoration: none;">D</del>. <br><del style="font-weight: bold; text-decoration: none;">I </del>have <del style="font-weight: bold; text-decoration: none;">two sister</del>. <del style="font-weight: bold; text-decoration: none;">I like Sculpting</del>, <del style="font-weight: bold; text-decoration: none;">watching TV </del>(<del style="font-weight: bold; text-decoration: none;">Modern Family</del>) and <del style="font-weight: bold; text-decoration: none;">Vintage Books</del>.<br><br><del style="font-weight: bold; text-decoration: none;">Look into my weblog [http://fachuraodnetu</del>.<del style="font-weight: bold; text-decoration: none;">pl pozycjonowanie stron]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">More mature video games ought to be discarded. They happen to be worth some money at several video retailers. Step buy and sell many game titles, you might even get your upcoming bill at no cost!<br><br>The underside line </ins>is<ins style="font-weight: bold; text-decoration: none;">, this turns out to be worth exploring if get strategy games, especially when you're keen on Clash to do with Clans. Want recognize what opinions you possess, when you do</ins>.<ins style="font-weight: bold; text-decoration: none;"><br><br>Throughout the clash of clans Cheats (a secret popular social architecture quite possibly arresting bold by Supercell) participants can acceleration mass popularity accomplishments for example building, advance or training members of the military with gems that are usually now being sold for absolute money. 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They're basically monetizing the valid player's impatience. Every amusing architecture daring Which i apperceive of manages to find a deal</ins>.<br><ins style="font-weight: bold; text-decoration: none;"><br>The nfl season is here as well as , going strong, and along the lines of many fans we in total for Sunday afternoon when the games begin. If you </ins>have <ins style="font-weight: bold; text-decoration: none;">held and liked Soul Caliber, you will love this particular game. The upcoming best is the Microchip Cell which will with little thought fill in some sqs. Defeating players as if that by any means that necessary can be each of our reason that pushes themselves to use Words now with Friends Cheat. 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80.42.127.100
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en>Hhhippo: Reverted 1 edit by 2001:610:1908:1400:4932:9d10:cb8e:a8a1 (talk): Units are not part of the definition. (TW)
2014-03-04T17:45:51Z
<p>Reverted 1 edit by <a href="/wiki/Special:Contributions/2001:610:1908:1400:4932:9d10:cb8e:a8a1" title="Special:Contributions/2001:610:1908:1400:4932:9d10:cb8e:a8a1">2001:610:1908:1400:4932:9d10:cb8e:a8a1</a> (<a href="/index.php?title=User_talk:2001:610:1908:1400:4932:9D10:CB8E:A8A1&action=edit&redlink=1" class="new" title="User talk:2001:610:1908:1400:4932:9D10:CB8E:A8A1 (page does not exist)">talk</a>): Units are not part of the definition. (<a href="/index.php?title=WP:TW&action=edit&redlink=1" class="new" title="WP:TW (page does not exist)">TW</a>)</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:45, 4 March 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[physics]], '''Landau damping''', named after its discoverer,<ref </del>name<del style="font-weight: bold; text-decoration: none;">="orig">Landau, L. ''On the vibration of the electronic plasma''. J. Phys. USSR 10 (1946), 25. English translation in JETP 16, 574. Reproduced in Collected papers of L.D. Landau, edited and with an introduction by D. ter Haar, Pergamon Press, 1965, pp. 445–460; and in Men of Physics: L.D. Landau, Vol. 2, Pergamon Press, D. ter Haar, ed. (1965).</ref> the eminent [[Soviet Union|Soviet]] physicist [[Lev Davidovich Landau]], </del>is <del style="font-weight: bold; text-decoration: none;">the effect of [[damping]] ([[exponential decay|exponential decrease]] as a function of time) of [[plasma oscillation|longitudinal space charge waves]] in [[Plasma (physics)|plasma]] or a similar environment.<ref>Chen, Francis F</del>. '<del style="font-weight: bold; text-decoration: none;">'Introduction to Plasma Physics and Controlled Fusion''. Second Ed., 1984 Plenum Press, New York.</ref> This phenomenon prevents an instability </del>from <del style="font-weight: bold; text-decoration: none;">developing, and creates a region of stability in the [[parameter space]]. It was later argued by [[Donald Lynden-Bell]] that a similar phenomenon was occurring in galactic dynamics,<ref>Lynden-Bell, D</del>. '<del style="font-weight: bold; text-decoration: none;">'The stability and vibrations of a gas of stars''. Mon. Not. R. astr. Soc. 124,</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">My </ins>name is <ins style="font-weight: bold; text-decoration: none;">Marylin Noonan but everybody calls me Marylin</ins>. <ins style="font-weight: bold; text-decoration: none;">I</ins>'<ins style="font-weight: bold; text-decoration: none;">m </ins>from <ins style="font-weight: bold; text-decoration: none;">Brazil</ins>. <ins style="font-weight: bold; text-decoration: none;">I</ins>'<ins style="font-weight: bold; text-decoration: none;">m studying </ins>at the <ins style="font-weight: bold; text-decoration: none;">university </ins>(<ins style="font-weight: bold; text-decoration: none;">final year</ins>) and <ins style="font-weight: bold; text-decoration: none;">I play </ins>the <ins style="font-weight: bold; text-decoration: none;">Trombone for </ins>3 <ins style="font-weight: bold; text-decoration: none;">years</ins>. <ins style="font-weight: bold; text-decoration: none;">Usually I choose music from my famous films </ins>:<ins style="font-weight: bold; text-decoration: none;">D</ins>. <<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">I </ins>have <ins style="font-weight: bold; text-decoration: none;">two sister</ins>. <ins style="font-weight: bold; text-decoration: none;">I like Sculpting</ins>, <ins style="font-weight: bold; text-decoration: none;">watching TV </ins>(<ins style="font-weight: bold; text-decoration: none;">Modern Family</ins>) and <ins style="font-weight: bold; text-decoration: none;">Vintage Books</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Look into my weblog </ins>[http://<ins style="font-weight: bold; text-decoration: none;">fachuraodnetu</ins>.<ins style="font-weight: bold; text-decoration: none;">pl pozycjonowanie stron</ins>]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">4 (1962), 279–296.</ref> where the gas of electrons interacting by electro-static forces is replaced by a "gas of stars" interacting by gravitation forces.<ref>Binney, J., and Tremaine, S. ''Galactic Dynamics'', second ed. Princeton Series in Astrophysics. Princeton University Press, 2008.</ref></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Wave-particle interactions<ref>Tsurutani, B., and Lakhina, G. ''Some basic concepts of wave-particle interactions in collisionless plasmas''. Reviews of Geophysics 35(4), p.491-502. [http://download.scientificcommons.org/442719 Download]</ref> ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Landau damping occurs due to the energy exchange between an electromagnetic [[wave]] with phase velocity <math>v_{ph}</math> and particles in the plasma with velocity approximately equal to <math>v_{ph}</math>, which can interact strongly with the wave. Those particles having velocities slightly less than <math>v_{ph}</math> will be accelerated by the wave electric field to move with the wave phase velocity, while those particles with velocities slightly greater than <math>v_{ph}</math> will be decelerated by the wave electric field, losing energy to the wave.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[File:Maxwell dist ress partic landau.svg|thumb]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In a collisionless plasma the particle velocities are often taken to be approximately a [[Maxwell-Boltzmann_distribution|Maxwellian distribution function]]. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">If the slope of the function is negative, the number of particles with velocities slightly less than the wave phase velocity is greater than the number of particles with velocities slightly greater. Hence, there are more particles gaining energy from the wave than losing to the wave, which leads to wave damping.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">If, however, the slope of the function is positive, the number of particles with velocities slightly less than the wave phase velocity is smaller than the number of particles with velocities slightly greater. Hence, there are more particles losing energy to the wave than gaining from the wave, which leads to a resultant increase in the wave energy.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Physical interpretation ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Mathematical theory of Landau damping is somewhat involved--see the section below. However, there is a simple physical interpretation which, though not strictly correct, helps to visualize this phenomenon.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[File:Phys interp landau damp.svg|thumb]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">It is possible to imagine [[Plasma oscillation|Langmuir waves]] as waves in the sea, and the particles as surfers trying to catch the wave, all moving in the same direction. If the surfer is moving on the water surface </del>at <del style="font-weight: bold; text-decoration: none;">a velocity slightly less than the waves he will eventually be caught and pushed along the wave (gaining energy), while a surfer moving slightly faster than a wave will be pushing on the wave as he moves uphill (losing energy to the wave).</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">It is worth noting that only the surfers are playing an important role in this energy interactions with the waves; a beachball floating on </del>the <del style="font-weight: bold; text-decoration: none;">water </del>(<del style="font-weight: bold; text-decoration: none;">zero velocity</del>) <del style="font-weight: bold; text-decoration: none;">will go up </del>and <del style="font-weight: bold; text-decoration: none;">down as </del>the <del style="font-weight: bold; text-decoration: none;">wave goes by, not gaining energy at all. Also, a boat that moves very fast (faster than the waves) does not exchange much energy with the wave.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Theoretical physics: perturbation theory ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Theoretical treatment starts with [[Vlasov equation]] in the non-relativistic zero-magnetic field limit, Vlasov-Poisson set of equations. Explicit solutions are obtained in the limit of small <math>E</math>-field. The distribution function <math>f</math> and field <math>E</math> are expanded in series: <math> f=f_0(v)+f_1(x,v,t)+f_2(x,v,t)</math>, <math>E=E_1(x,t)+E_2(x,t)</math> and terms of equal order are collected.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">To '''first order''' the Vlasov-Poisson equations read</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>(\partial_t + v\partial_x)f_1 + {e\over m}E_1 f'_0 = 0, \quad </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> \partial_x E_1 = {e\over \epsilon_0} \int f_1 {\rm d}v</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Landau calculated<ref name="orig" /> the wave caused by an initial disturbance <math>f_1(x,v,0) = g(v)\exp(ikx)</math> and found by aid of [[Laplace transform]] and [[contour integration]] a damped travelling wave of the form <math>\exp[ik(x-v_{ph}t)-\gamma t]</math> with [[wave number]] <math>k</math> and damping decrement</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\gamma\approx-{\pi\omega_p^</del>3 <del style="font-weight: bold; text-decoration: none;">\over 2k^2N} f'_0(v_{ph}), \quad</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> N = \int f_0 {\rm d}v</math></del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Here <math>\omega_p</math> is the [[plasma oscillation]] frequency and <math>N</math> is the electron density. Later [[Nico van Kampen]] proved<ref>van Kampen, N.G., ''On the theory of stationary waves in plasma'', Physica 21 (1955), 949–963. See http</del>:<del style="font-weight: bold; text-decoration: none;">//theor</del>.<del style="font-weight: bold; text-decoration: none;">jinr.ru/~kuzemsky/kampenbio.html</del><<del style="font-weight: bold; text-decoration: none;">/ref</del>> <del style="font-weight: bold; text-decoration: none;">that the same result can be obtained with [[Fourier transform]]. He showed that the linearized Vlasov-Poisson equations </del>have <del style="font-weight: bold; text-decoration: none;">a continuous spectrum of singular normal modes, now known as '''van Kampen modes'''</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\frac{\omega_p^2}{kN} f'_0 \frac{\mathcal P}{kv-\omega} + \epsilon \delta(v-\frac{\omega}{k}) </math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">in which <math>\mathcal P</math> signifies principal value, <math>\delta</math> is the delta function (see [[generalized function]]) and</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\epsilon = 1 + \frac{\omega_p^2}{kN} \int f'_0 \frac{\mathcal P}{\omega-kv} {\rm d}v</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">is the plasma permittivity</del>. <del style="font-weight: bold; text-decoration: none;">Decomposing the initial disturbance in these modes he obtained the Fourier spectrum of the resulting wave. Damping is explained by phase-mixing of these Fourier modes with slightly different frequencies near <math>\omega_p</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">It was not clear how damping could occur in a collisionless plasma: where does the wave energy go? In fluid theory</del>, <del style="font-weight: bold; text-decoration: none;">in which the plasma is modeled as a dispersive dielectric medium,<ref>Landau, L.D. and Lifshitz, E.M., ''Electrodynamics of continuous media'' §80, Pergamon Press (1984).</ref> the energy of Langmuir waves is known: field energy multiplied by the Brillouin factor <math>\partial_\omega</del>(<del style="font-weight: bold; text-decoration: none;">\omega\epsilon</del>)<del style="font-weight: bold; text-decoration: none;"></math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">But damping cannot be derived in this model. To calculate energy exchange of the wave with resonant electrons, Vlasov plasma theory has to be expanded to '''second order''' and problems about suitable initial conditions </del>and <del style="font-weight: bold; text-decoration: none;">secular terms arise</del>. </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[File:Ldamp2.jpg|thumb]] In</del><<del style="font-weight: bold; text-decoration: none;">ref</del>><del style="font-weight: bold; text-decoration: none;">Best, Robert W.B., ''Energy and momentum density of a Landau-damped wave packet'', J. Plasma Phys. 63 (2000), 371-391</del><<del style="font-weight: bold; text-decoration: none;">/ref</del>> <del style="font-weight: bold; text-decoration: none;">these problems are studied. Because calculations for an infinite wave are deficient in second order, a </del>[<del style="font-weight: bold; text-decoration: none;">[wave packet]] is analysed. Second-order initial conditions are found that suppress secular behavior and excite a wave packet of which the energy agrees with fluid theory. The figure shows the energy density of a wave packet traveling at the [[group velocity]], its energy being carried away by electrons moving at the phase velocity. Total energy, the area under the curves, is conserved.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Mathematical theory: the Cauchy problem for perturbative solutions ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The rigorous mathematical theory is based on solving the [[Cauchy problem]] for the evolution equation (here the partial differential Vlasov-Poisson equation) and proving estimates on the solution. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">First a rather complete linearized mathematical theory has been developed since [[Landau]].<ref>See for instance Backus, G. ''Linearized plasma oscillations in arbitrary electron distributions''. J. Math. Phys. 1 (1960), 178–191, 559. Degond, P. Spectral theory of the linearized Vlasov–Poisson equation. Trans. Amer. Math. Soc. 294, 2 (1986), 435–453. Maslov, V. P., and Fedoryuk, M. V. The linear theory of Landau damping. Mat. Sb. (N.S.) 127(169), 4 (1985), 445–475, 559.</ref></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Going beyond the linearized equation and dealing with the nonlinearity has been a longstanding problem in the mathematical theory of Landau damping. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Previously one mathematical result at the non-linear level was the existence of a class of exponentially damped solutions of the Vlasov-Poisson equation in a circle which had been proved in<ref>Caglioti, E. and Maffei, C. "Time asymptotics for solutions of Vlasov-Poisson equation in a circle", J. Statist. Phys. 92, 1-2, 301-323 (1998)</ref> by means of a scattering technique (this result has been recently extended in<ref>Hwang, H. J. and Velasquez J. J. L. "On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem" preprint http://arxiv.org/abs/0810.3456</ref>). However these existence results do not say anything about ''which'' initial data could lead to such damped solutions.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In the recent paper<ref>Mouhot, C., and Villani, C. ''On Landau damping'', preprint </del>http://<del style="font-weight: bold; text-decoration: none;">fr.arxiv.org/abs/0904.2760 (quoted for the [[Fields Medal]] awarded to [[Cédric Villani]] in 2010)</ref> the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation</del>. <del style="font-weight: bold; text-decoration: none;">It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time. The damping phenomenon is reinterpreted in terms of transfer of regularity of <math>f</math> as a function of <math>x</math> and <math>v</math>, respectively, rather than exchanges of energy. Large scale variations pass into variations of smaller and smaller scale in velocity space, corresponding to a shift of the Fourier spectrum of <math>f</math> as a function of <math>v</math>. This shift, well known in linear theory, proves to hold in the non-linear case.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==See also==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[List of plasma (physics) articles]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Notes and references ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><references /></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Landau Damping}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Plasma physics]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr>
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en>Hhhippo
https://en.formulasearchengine.com/index.php?title=Polarization_density&diff=6262&oldid=prev
en>Danh: undo
2013-10-27T19:17:28Z
<p>undo</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Roberto </del>is the <del style="font-weight: bold; text-decoration: none;">name Simply put i love </del>to be <del style="font-weight: bold; text-decoration: none;">titled </del>with though <del style="font-weight: bold; text-decoration: none;">I try </del>not to <del style="font-weight: bold; text-decoration: none;">really like being named like that</del>. <del style="font-weight: bold; text-decoration: none;">My very good say it's not really for me but possibilities I love doing </del>is <del style="font-weight: bold; text-decoration: none;">also </del>to <del style="font-weight: bold; text-decoration: none;">bake but I'm so thinking </del>on <del style="font-weight: bold; text-decoration: none;">starting new things</del>. <del style="font-weight: bold; text-decoration: none;">South Carolina </del>is <del style="font-weight: bold; text-decoration: none;">where some </del>of <del style="font-weight: bold; text-decoration: none;">my home is</del>. <del style="font-weight: bold; text-decoration: none;">Software developing is how My personal support my family</del>. <del style="font-weight: bold; text-decoration: none;">You can consider my website here</del>: <del style="font-weight: bold; text-decoration: none;">http</del>://<del style="font-weight: bold; text-decoration: none;">circuspartypanama</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Look into my webpage </del>... <del style="font-weight: bold; text-decoration: none;">clash </del>of <del style="font-weight: bold; text-decoration: none;">[</del>http://<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">dict</del>.<del style="font-weight: bold; text-decoration: none;">cc</del>/<del style="font-weight: bold; text-decoration: none;">englisch</del>-<del style="font-weight: bold; text-decoration: none;">deutsch</del>/<del style="font-weight: bold; text-decoration: none;">clans</del>+<del style="font-weight: bold; text-decoration: none;">hack</del>.<del style="font-weight: bold; text-decoration: none;">html clans hack</del>] <del style="font-weight: bold; text-decoration: none;">cydia </del>([http://<del style="font-weight: bold; text-decoration: none;">circuspartypanama</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/ <del style="font-weight: bold; text-decoration: none;">Highly recommended Webpage</del>])</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[physics]], '''Landau damping''', named after its discoverer,<ref name="orig">Landau, L. ''On the vibration of the electronic plasma''. J. Phys. USSR 10 (1946), 25. English translation in JETP 16, 574. Reproduced in Collected papers of L.D. Landau, edited and with an introduction by D. ter Haar, Pergamon Press, 1965, pp. 445–460; and in Men of Physics: L.D. Landau, Vol. 2, Pergamon Press, D. ter Haar, ed. (1965).</ref> the eminent [[Soviet Union|Soviet]] physicist [[Lev Davidovich Landau]], </ins>is the <ins style="font-weight: bold; text-decoration: none;">effect of [[damping]] ([[exponential decay|exponential decrease]] as a function of time) of [[plasma oscillation|longitudinal space charge waves]] in [[Plasma (physics)|plasma]] or a similar environment.<ref>Chen, Francis F. ''Introduction to Plasma Physics and Controlled Fusion''. Second Ed., 1984 Plenum Press, New York.</ref> This phenomenon prevents an instability from developing, and creates a region of stability in the [[parameter space]]. It was later argued by [[Donald Lynden-Bell]] that a similar phenomenon was occurring in galactic dynamics,<ref>Lynden-Bell, D. ''The stability and vibrations of a gas of stars''. Mon. Not. R. astr. Soc. 124,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">4 (1962), 279–296.</ref> where the gas of electrons interacting by electro-static forces is replaced by a "gas of stars" interacting by gravitation forces.<ref>Binney, J., and Tremaine, S. ''Galactic Dynamics'', second ed. Princeton Series in Astrophysics. Princeton University Press, 2008.</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Wave-particle interactions<ref>Tsurutani, B., and Lakhina, G. ''Some basic concepts of wave-particle interactions in collisionless plasmas''. Reviews of Geophysics 35(4), p.491-502. [http://download.scientificcommons.org/442719 Download]</ref> ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Landau damping occurs due to the energy exchange between an electromagnetic [[wave]] with phase velocity <math>v_{ph}</math> and particles in the plasma with velocity approximately equal to <math>v_{ph}</math>, which can interact strongly with the wave. Those particles having velocities slightly less than <math>v_{ph}</math> will be accelerated by the wave electric field to move with the wave phase velocity, while those particles with velocities slightly greater than <math>v_{ph}</math> will be decelerated by the wave electric field, losing energy to the wave.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Maxwell dist ress partic landau.svg|thumb]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In a collisionless plasma the particle velocities are often taken </ins>to be <ins style="font-weight: bold; text-decoration: none;">approximately a [[Maxwell-Boltzmann_distribution|Maxwellian distribution function]]. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the slope of the function is negative, the number of particles with velocities slightly less than the wave phase velocity is greater than the number of particles </ins>with <ins style="font-weight: bold; text-decoration: none;">velocities slightly greater. Hence, there are more particles gaining energy from the wave than losing to the wave, which leads to wave damping.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If, however, the slope of the function is positive, the number of particles with velocities slightly less than the wave phase velocity is smaller than the number of particles with velocities slightly greater. Hence, there are more particles losing energy to the wave than gaining from the wave, which leads to a resultant increase in the wave energy.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Physical interpretation ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Mathematical theory of Landau damping is somewhat involved--see the section below. However, there is a simple physical interpretation which, </ins>though not <ins style="font-weight: bold; text-decoration: none;">strictly correct, helps </ins>to <ins style="font-weight: bold; text-decoration: none;">visualize this phenomenon.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Phys interp landau damp</ins>.<ins style="font-weight: bold; text-decoration: none;">svg|thumb]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It </ins>is <ins style="font-weight: bold; text-decoration: none;">possible to imagine [[Plasma oscillation|Langmuir waves]] as waves in the sea, and the particles as surfers trying </ins>to <ins style="font-weight: bold; text-decoration: none;">catch the wave, all moving in the same direction. If the surfer is moving on the water surface at a velocity slightly less than the waves he will eventually be caught and pushed along the wave (gaining energy), while a surfer moving slightly faster than a wave will be pushing </ins>on <ins style="font-weight: bold; text-decoration: none;">the wave as he moves uphill (losing energy to the wave)</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It </ins>is <ins style="font-weight: bold; text-decoration: none;">worth noting that only the surfers are playing an important role in this energy interactions with the waves; a beachball floating on the water (zero velocity) will go up and down as the wave goes by, not gaining energy at all. Also, a boat that moves very fast (faster than the waves) does not exchange much energy with the wave.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Theoretical physics: perturbation theory ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Theoretical treatment starts with [[Vlasov equation]] in the non-relativistic zero-magnetic field limit, Vlasov-Poisson set </ins>of <ins style="font-weight: bold; text-decoration: none;">equations</ins>. <ins style="font-weight: bold; text-decoration: none;">Explicit solutions are obtained in the limit of small <math>E</math>-field</ins>. <ins style="font-weight: bold; text-decoration: none;">The distribution function <math>f</math> and field <math>E</math> are expanded in series</ins>: <ins style="font-weight: bold; text-decoration: none;"><math> f=f_0(v)+f_1(x,v,t)+f_2(x,v,t)</math>, <math>E=E_1(x,t)+E_2(x,t)</math> and terms of equal order are collected.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">To '''first order''' the Vlasov-Poisson equations read</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins style="font-weight: bold; text-decoration: none;"><math>(\partial_t + v\partial_x)f_1 + {e\over m}E_1 f'_0 = 0, \quad </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \partial_x E_1 = {e\over \epsilon_0} \int f_1 {\rm d}v</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Landau calculated<ref name="orig" /> the wave caused by an initial disturbance <math>f_1(x,v,0) = g(v)\exp(ikx)</math> and found by aid of [[Laplace transform]] and [[contour integration]] a damped travelling wave of the form <math>\exp[ik(x-v_{ph}t)-\gamma t]</math> with [[wave number]] <math>k<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> and damping decrement</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\gamma\approx-{\pi\omega_p^3 \over 2k^2N} f'_0(v_{ph}), \quad</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> N = \int f_0 {\rm d}v<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Here <math>\omega_p<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> is the [[plasma oscillation]] frequency and </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">N</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">is the electron density</ins>. <ins style="font-weight: bold; text-decoration: none;">Later [[Nico van Kampen]] proved<ref>van Kampen, N</ins>.<ins style="font-weight: bold; text-decoration: none;">G</ins>.<ins style="font-weight: bold; text-decoration: none;">, ''On the theory </ins>of <ins style="font-weight: bold; text-decoration: none;">stationary waves in plasma'', Physica 21 (1955), 949–963. See </ins>http://<ins style="font-weight: bold; text-decoration: none;">theor</ins>.<ins style="font-weight: bold; text-decoration: none;">jinr</ins>.<ins style="font-weight: bold; text-decoration: none;">ru</ins>/<ins style="font-weight: bold; text-decoration: none;">~kuzemsky/kampenbio.html</ref> that the same result can be obtained with [[Fourier transform]]. He showed that the linearized Vlasov-Poisson equations have a continuous spectrum of singular normal modes, now known as '''van Kampen modes'''</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\frac{\omega_p^2}{kN} f'_0 \frac{\mathcal P}{kv-\omega} + \epsilon \delta(v</ins>-<ins style="font-weight: bold; text-decoration: none;">\frac{\omega}{k}) <</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">in which <math>\mathcal P</math> signifies principal value, <math>\delta</math> is the delta function (see [[generalized function]]) and</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\epsilon = 1 </ins>+ <ins style="font-weight: bold; text-decoration: none;">\frac{\omega_p^2}{kN} \int f'_0 \frac{\mathcal P}{\omega-kv} {\rm d}v</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">is the plasma permittivity. Decomposing the initial disturbance in these modes he obtained the Fourier spectrum of the resulting wave</ins>. <ins style="font-weight: bold; text-decoration: none;">Damping is explained by phase-mixing of these Fourier modes with slightly different frequencies near <math>\omega_p</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It was not clear how damping could occur in a collisionless plasma: where does the wave energy go? In fluid theory, in which the plasma is modeled as a dispersive dielectric medium,<ref>Landau, L.D. and Lifshitz, E.M., ''Electrodynamics of continuous media'' §80, Pergamon Press (1984).</ref> the energy of Langmuir waves is known: field energy multiplied by the Brillouin factor <math>\partial_\omega(\omega\epsilon)</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">But damping cannot be derived in this model. To calculate energy exchange of the wave with resonant electrons, Vlasov plasma theory has to be expanded to '''second order''' and problems about suitable initial conditions and secular terms arise. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Ldamp2.jpg|thumb]] In<ref>Best, Robert W.B., ''Energy and momentum density of a Landau-damped wave packet'', J. Plasma Phys. 63 (2000), 371-391</ref> these problems are studied. Because calculations for an infinite wave are deficient in second order, a [[wave packet]] is analysed. Second-order initial conditions are found that suppress secular behavior and excite a wave packet of which the energy agrees with fluid theory. The figure shows the energy density of a wave packet traveling at the [[group velocity]], its energy being carried away by electrons moving at the phase velocity. Total energy, the area under the curves, is conserved.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Mathematical theory: the Cauchy problem for perturbative solutions ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The rigorous mathematical theory is based on solving the [[Cauchy problem]</ins>] <ins style="font-weight: bold; text-decoration: none;">for the evolution equation </ins>(<ins style="font-weight: bold; text-decoration: none;">here the partial differential Vlasov-Poisson equation) and proving estimates on the solution. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">First a rather complete linearized mathematical theory has been developed since </ins>[<ins style="font-weight: bold; text-decoration: none;">[Landau]].<ref>See for instance Backus, G. ''Linearized plasma oscillations in arbitrary electron distributions''. J. Math. Phys. 1 (1960), 178–191, 559. Degond, P. Spectral theory of the linearized Vlasov–Poisson equation. Trans. Amer. Math. Soc. 294, 2 (1986), 435–453. Maslov, V. P., and Fedoryuk, M. V. The linear theory of Landau damping. Mat. Sb. (N.S.) 127(169), 4 (1985), 445–475, 559.</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Going beyond the linearized equation and dealing with the nonlinearity has been a longstanding problem in the mathematical theory of Landau damping. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Previously one mathematical result at the non-linear level was the existence of a class of exponentially damped solutions of the Vlasov-Poisson equation in a circle which had been proved in<ref>Caglioti, E. and Maffei, C. "Time asymptotics for solutions of Vlasov-Poisson equation in a circle", J. Statist. Phys. 92, 1-2, 301-323 (1998)</ref> by means of a scattering technique (this result has been recently extended in<ref>Hwang, H. J. and Velasquez J. J. L. "On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem" preprint </ins>http://<ins style="font-weight: bold; text-decoration: none;">arxiv</ins>.<ins style="font-weight: bold; text-decoration: none;">org/abs</ins>/<ins style="font-weight: bold; text-decoration: none;">0810.3456</ref>). However these existence results do not say anything about ''which'' initial data could lead to such damped solutions.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In the recent paper<ref>Mouhot, C., and Villani, C. ''On Landau damping'', preprint http://fr.arxiv.org/abs/0904.2760 (quoted for the [[Fields Medal]</ins>] <ins style="font-weight: bold; text-decoration: none;">awarded to [[Cédric Villani]] in 2010</ins>)<ins style="font-weight: bold; text-decoration: none;"></ref> the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation. It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time. The damping phenomenon is reinterpreted in terms of transfer of regularity of <math>f</math> as a function of <math>x</math> and <math>v</math>, respectively, rather than exchanges of energy. Large scale variations pass into variations of smaller and smaller scale in velocity space, corresponding to a shift of the Fourier spectrum of <math>f</math> as a function of <math>v</math>. This shift, well known in linear theory, proves to hold in the non-linear case.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[List of plasma (physics) articles]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Notes and references ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Landau Damping}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Plasma physics]]</ins></div></td></tr>
</table>
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en>Zorrobot: r2.7.3) (Robot: Adding nn:Elektrisk polarisasjon
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