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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðàó ó óóó ó ó ó êîÂï  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€�‚ƒ„…½‡ˆ‰Š‹Œ�Ž��‘’“”•–—˜™š›œ�žŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´Ñ¶·¸¹º»¼å¾¿ÀÁÂÃõÅÆÇÈÉÊËÌïýÿÿÿýÿÿÿÐþÿÿÿÒÓÔÕÖרÙÚÛÜÝÞßàáâãäÄæçèéêëþÿÿÿþÿÿÿîEðñòóôö÷øùúûýýÿÿÿþÿRoot Entryÿÿÿÿÿÿÿÿ��d›Oφꪹ)è Áž&£fÈí€PicturesÿÿÿÿÿÿÿÿH)Current Userÿÿÿÿÿÿÿÿÿÿÿÿ 2SummaryInformation(ÿÿÿÿµ<PowerPoint Document(ÿÿÿÿÿÿÿÿÿÿÿÿ†Ú�DocumentSummaryInformation8ÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÕ ·DWingdingsRomanT©T©ƒ‘Ü–É 0Ü–Õ¤€@ÿÿ¥ .©  @£nÿý?" dÿd@ÿÿïÿÿÿÿÿÿ  @@``€€ �ðˆð €!v      /ðbð$ç~å…I�W—Ä‚-S‰ÿybð$câšHÉôÄ%]A_¶Ë­÷ÿpy2ð$µè�.àƒ/޹¦@}U%ÿÇé 2ð${ëÖ=8rö‡IbÔ…2ƒ‰xÿΰ2ð$ “Ý^!þg£¸¯Iza_ÿ\~ 2ð$åvÌÊ’†¹\xb\¥Aªÿ| Ú(bð$ ç÷¢ mÉ�×YU­£ÿXV42ð$ýaÅX`Ì�,ôd³’(‰Œÿ‘®;2ð$0¥â] H{Šëõ¹‹² Ïÿå?T2ð$³[ÿsäXÛ7äFÇaˆ%ÿ4 $dbð$8Ùáa³|%È¥AaUb¼Ëÿ´Xp2ð$¾òÙÎ¥S—©ýMp�ÿ  tbð$þ¦ÛÏŠ 5MŠ¡.µØ×îÿ ,†2ð$sÓ�#ö�)þÜ�6ƒvþ¬ÿ›Ì�bð$ ƒÒ½i ~iº!”NWGÿög¥bð$š�·5º7ÕÕá]ýç {Fÿç]©2ð$WÞµ§Îì-Þ2©øíê¦Ûyÿ"DÃ2ð$ qœ³¤5JÞ†ü·,ažÓÿp&Wåƒ ð0�ƒ†A¿ÀÅAÿ@ñ÷ðpó€ó€ó€ó€Ð3º“°öÊš;­”ÇÊš;úgþý4YdYd¼ ‘ô–É 0þÿÿ”ÿÿÿpûppû@ <ý4dddd —‘w 0T©ð‚‘ÿ ˆ8Š0º___PPT10‹ pµpµðNóŸ¨Physics of fusion powerŸ¨,Lecture 13 : Diffusion equation / transport ª" óóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒóóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtóóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ óóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ð4ó ó óóó ó ó ó ó óóêîvï € 0 æðÞ ð¬ðvð( ð ð¬ðr ð¬ S ð€ü¿ÿ|ð=Kè·ðà   ð žðr ð¬ S ð€À¿ÿ|ðàV0ðà  ð žðX² 𬠃 ð0A�ƒ¿Àÿ?ð/¯ÒÜðX² 𬠃 ð0A�ƒ¿Àÿ?ðªüÿ ÊðX² 𬠃 ð0A�ƒ¿Àÿ?ðËßA 2 ðX² 𬠃 ð0A�ƒ¿Àÿ?ðF t a ðX² 𬠃 ð0A�ƒ¿Àÿ?ðÁ ß- æðl ð ¬ “ ð6…‡�ƒ¿ÀÿËœ1ÿ"ñ¿`ðØ¢Ž <ðˆ¢ ð ¬ ƒ ð0€…Ý¿�ƒ¿Àÿð2žÿ ð(Ÿ¡ 2ðÉ¢ ð ¬ ƒ ð0€ÈSÝ¿�ƒ¿Àÿð2» Æ ðiŸ¨5The integral = 1 (integration over all probabilities)¡6 26ÿþðÉ¢ ð ¬ ƒ ð0€œ¿�ƒ¿ÀÿðIŽ âÝ ðiŸ¨5The integral = 1 (integration over all probabilities)¡6 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation óóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðPó ó óóó ó ó ó ó óóóêî¸ï € 0 (ð Àð ´ð¸ð( ð ð´ðr ð´ S ð€øE¿ÿ|ð=Kè·ðà   ð žðr ð´ S ð€xá¿ÿ|ðàV0ðà  ð žðX² ð´ ƒ ð0A�ƒ¿Àÿ?ð· ™ðX² ð´ ƒ ð0A�ƒ¿Àÿ?ðCd{ O ðX² ð´ ƒ ð0A!�ƒ¿Àÿ?ðCõ q^ ðl² ð ´ “ ð6A#�ƒ¿ÀÿËjJÿ?"ñ¿`ð) ‘ù SðH ð´ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.…fÈtåÂð ñ  0 Œð„Ðð¸ðð( ð ð¸ðX ð¸ C ð¿ÿ ð°Ð ðà  ð„ ð¸ S ð€à#¿ÿ ð° °0Ððà   🪠ðH 𸠃 ð0�ƒ“Þ½h”ŽŸ‹¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë.…fÈtåÂrÖ Ž N$õ ™²b&Å1èÅ!é(€àà€ òD/È 0ÒÕ|·DArialUnicode MST©T©ƒ‘Ü–É 0Ü–Õ"·DTimes New RomanT©T©ƒ‘Ü–É 0Ü–Õ ·DWingdingsRomanT©T©ƒ‘Ü–É 0Ü–Õ0·DSymbolgsR›ù¢þóæ&Ÿ@ɵ#Øtã19\Öq««X¬*ù’ü=Às»·Ý3ÝE>ï’/É›îϺîöO§Û®ë~ñÇõß¿þy×ýÝOþªûÛÏþù7¿þý/]wÓýÓ¿ë??ûøŸï¿»í¾¿ÜvúÕí~À×oºî÷Îÿÿòï»î¿~ª~äóký[ôÏܾþ¾OñŸ?îºÿP/üJ½ð߃׺ÿùQ÷«¿¸í~§þú/—ÛÉϲøƒú÷¯ÔŸ¿UüBñßÿ÷{ósŸuOýë_|ü_½~Ÿ×'ëßçâååB!„B!„B!„B!¬šãµ]÷I­À·¦V€ö‚B!„B!„B!„B©ø†Z!„B!„B!„B!„°©Z�/©€B!„B!„B!„B›ªxK­„B!„B!„nåZÐvbïBÓóááa•/¿\.õÔ |N­„­ù!l…ïÞ½[ìËŸŸŸi;±w!„ý¹“‡¼÷þþ>ø½BìBa|~øð¡;�N¯>}î½���Aïƒ&®¸‰_+€o€¶r=]'swwG›Aˆ½C!àÏ�Çã¬?g½BìB¡ êqöápxõçÚ·Ó&bï&­¸¼á\ø~½þ‚ï‡{‡B(ŸOOO¯c&ü9„Ø;„Â2hÏÕõ3œû !ö!„pA­ÀwòkôÞ5 ½þòþý{úЉ½CaùÔþ[ûqÖ^ ÄÞ!„–µî¢¡Ïz§M ÄÞ!„Ê‹sÇ"ŒV+ð-ç à „J¢^k±àž±wa�{Ù%œ ~>Ÿ©Ï� ¾/Q×Áèýjº¿æÚÑB¿½Áˆýcá>BJM¬ÞCÏ^úqÚúe»-´^F½Áˆýcï»Ö |C­þ¡Ýq ¤˜«k;ôw¢/?¥‹%~î½­ë Ê"ö�½Cl¾Dê³€RÌ7êö’Z«}6?^ÿ¢Ûe®Ït}Œ…Ö“û¤îbÿØ;„5Ú¼^+Öã•=?C·�n'ýYRÛAÏ£JñC¨ýsèÙºí,$Ô¤” 7ˆýcïØ;Äæ·Ô€ÆØWT+ðå|­€þ.Òóa;ÆÛ;ç-iZcnþÖ¾ÏBB}i zƒØ?ö޽Cl~ËøfÏ=H¶}JØO¤óu|À§í17n·wõhH#—¤7ˆýcïØ;Äæ·ÔEìUãgm½„ý¾z¾?ðí=tÛ‰³Â÷ó?swn[:^²ÞJÚÿûœ÷žßùMZcZ°ýî[;[csÙ­¹›${/ÝæSjDš½Õ ÜüP+°‡o(!?˜ËóÜZº­s—kã®olò››®Y+·íã~wØ^(i¯­{6|ÈX.tÜ^²ÞZ�§ ¯/ñiSñÈÖÍä°ÿÖóµFZ°÷œ6¿¦6nì¹CÎ�Ó9ø\›…޵ֿÕîÚùœ¦ÖÌoùþ0¼^vn̬í;ÄW•®·çÞ¬¯Œq»ÕI¬ù�\ö_cî&ÍÞK·ù©ÝÞK‰ù6÷�zî¹=µZÇs¹é–vpm}ª—äÞkæ´KÉÝ‚j.7EßAêÏí8ynìQº6[œ§sï Øzÿ¡­‡�u>;y^½ö^ºÍçÐHíö^J~`ÇSùÜ\}œ­÷òõãÖv°¹¨/Ÿ ­]Ðcƒ¥¹|«st±1Wï<7^˜Ûÿ_‹ÞRÖSƪ£Ü2nwû+F, ™sÉaÿÒëb[µwI6_JítÍö.5æó[w�u¬OCösè¼×7ÎXÚÃïèÎmOå‘ÖÎçroýóKósr·}}¹Û¾>ÛsÇj1ûKšÞRÖSÆZÒm¾vü¼f îó7înîà¥í0ìO7¿œÊõÝÜÛ7Ohûy‰æ¨�KSo17¦uë`cö—4½•܇KÇÐScð Z�ïÂjæÆ–1î‘f}µ.ßP“È©‘˜¾A’½KÍô`ìßë\-�¶1ßøÒýEcýdm}껸míӃݳ»d�º­×Œ±™ý«Óõ˜¶ffÌ/º¾unü[ºÞJ®‘[:n_[#£?Ëg�SýgçO4CÏm�iÿänùí½&›Ï©‘Zí=§Í‡Ô¸óÚs{ŠôÞÔs»ûPb�ë3ÖOî9)S¿#$÷žº�sãdŒ+l}ÊЦÝ1oÈØªd½µ2O·µFFÖ”ÖÝá¾ÇÕQèçÆÕeï5Ù|n�Ôhï’Ï�›{óÍëµÓ±v¶wã…änKÏÞÖßÏíwl5W»05nrïô-ál÷©ýçØ=ÙnÎRÿP²ÞZ™{Ûº/Íž[9Ö?¶ÿuŸºµêî|Oèzw;Ôgï)m>¨VàÛýÏÈ­§}ƒôü@ëÑÖX}OíÝË —æ¿KïÇÓßɽ;bî;†Ô.¸}¸¤V’ûWÓŸC¥ûYkJkÏÛ…Ž³JÖ[ ót±ö°Ùõ·}í\‹õ»úÿÝ<®I„î¿ãþÕúì½›— ‘Ú콄˜¯ŸQG­U­ý©½Dº |cýÚœN­æ—´Ÿþ\ý~û‡ã¸©Ïñ�ý3WÇ端]²¶ãèÓúä¥ë'%ë­–y_Û¹cð­gèÏÑ~Ûö›þsè�ì^.ýšþsI i.ûom¼˜ÚÞk°y ©ÍÞk‹ù¾ú¸%ç*–vO‘‹Ííí€û�É-–Ì_—ª·Z¨}_Èù©±Ïd¯Éþ[œëÇÞËÔH-ö^cÌŸª�[3T+ð�©(i¥�»•®ÍZz–¨O›-œRÒ<]ÌñzÍößbž‡½—©‘Zì½Æ˜?U·4W,­NÈú’R÷„–Ì�šäÚôVû<�;S‚Ìiÿ­ånØ{Ù©ÁÞkŒù¾ú¸%{JbžË—‚ú™cÞÕÃië]Öú¦õVûÜ›»¿¯„1xNûo-wÃÞËÖH ö^kÌßZçΟ–Їv¨y¨}ðÖ»’÷Ö[P­À—ûŸ+PÛ<�{æk ãõܾb¸OßPW,ªQ#%Û{ÍùAŒZ:;ßZBÛh? µHmU¾±RKz«}ž®´ñ:ö�½Ãvì½f›�QgsÀ­ã«½i÷ÜP—wí †½— ·Úç†~\úûÇÞa;ö^»ÍǨ�s�ä{ ¶˜×ŒÕ¿„žUX’Þj‹!} Þ¢ýÕ ¼•Q+€oÀ7àö}¾ý 9Ösó¬«Èè÷þ÷\õ¥g1ç’?.—2^Çþ±wØŽ½·bózî:–=ØutI9¹~6ý�¸ÇazY(bŒaìyC>¼;±T½µ8—ÜþØ?öÛ±·É±ô´Ê°Úë¹< ù®½¯�>œÞۯdz–Cß®ý«ûzÌv´¿s«N$é­ÅýÈ’ëͱìÖaïAµŸ/«À?l·i þ_�Ï­Ûë¯�Þ`éëØ?ö±yñåBìB!¾Bˆ½Caãç Üì{®~Bò<!„ør!ö!„_!ÄÞ!„_Ž/‡Âj.�ˆ;ð ø!Ä—C±w!„ør!ö!„_!öN{$ªøŽZ!~Bˆ½C!Ä—C±w!Ä—C±waSµßR+€ß‡bïBñåBìB!¾BìB!¾¼©Z�o¨€ø!„ør!ö!„_!ÄÞ!„°F^.—îùù_acöT+ðå§µø!,w|~<»óùÜ�N'|9„Ø;„Bác÷ÇÇÇWî‚¶�{‡B(›ør±wë9Wà-ç @èR�߇m!ö!„B!„B!„BXo­ÀçÔ @!„B!„B!„B!„MÕ ÜP+!„B!„B!„B!„ÖŸܘZ�7oÆk¾ÿø?Ô ÔÅ¿T}úË�ýüó�ÿ‘âíÍO»«?öF¿¢ÿöço~ß}ýæ¼éÞþì³î©ûªûÃëköÏ/¿~£ÿ_+çoŒZþïM7Ä ¶�x Oâ)@<ñ žÄS€x â)@<ˆ§ñÄS€x OA–þ mÐ<ÄS0…‡‡‡@ßr¹\h.€æ ž‚)¼{÷.Ä·´¥8�NÊó¢½áññÑó*³óâ)±ç/—ËåîÕ³óâ)±ç/Çã±çʽ0÷Ð<ÄS•“å^”Ÿ¡5š€x 6º—Óé„{hâ)Ø‚§§'Õï¸€æ ž‚�0{Þ�Ç#[Úš€x ¶àîîNõþápxÿþ=­Ð<ÄS°Ê¥(ÇÂ$@óOÁFô'�ÞßßÓÍ@<«ÓuÎ2hâ)ØŽÓéd| g¯4@õñôéé)ÁI&çó¹¾*…ÇÇÇÃáp<UN=µñ-ê=õõ)hPÿhä’½üxª¬#MñÀår©¯–ÞT_˜šÆÑ7¸—EÖ×§ Aý£y�KöÂã©I5“‰ðùùY}\M!µ÷³¾eê ¥÷)hMÿhä’}Üxzwwqì¬3}Í€²µš”o–ŠÔM5córš»6²ô)hJÿhä’}Üxz>ŸU&åW©GS¨~aú¶UÃÿj.rR®Ã¿/@=¬ñ- Vv2ö)hGÿhä’}Üxúôô«dNýž\y²É'c%Ù§2Ô³LM˜W Ô¢dìSÐŽþÑ<È%û¸ñÔÜ6ëÑ2Ö©D¥+PÏ2•®›ã×äÏÙû´£4²È>z=ÒétÚ¾°k4Ÿ±zßÜFQ‡!˜3Ö®ç ú]ªËÒh5oŸ9úWY÷®ã_4/¶kê–}ôxúðŠ�¶ á40³…­�¨ G9eÝÊRTάd“2K—Ó§@ˆþ$«h^l×T,ûèñTIw£„L™=G2E€S[ÂK„2“þd¥õ€Éj…ô)¢ÿdNÍOSÊ>VÈ$¢ ®ÔKæû˜/£^Z41nÆþœHÖCµžr¦1ÕUË„·§�>Í÷ØØXpiÓ;ªSL£õn\ý»‰)æßÕ§wÕ)N;™æåÈ>b×§ùí²�OM&¦¾¹Y9íÿb‚¬z®ð¨ª~0ü03�ãÎÞ¼{…jOÖ¯5®CõΔ;¾…Cõ Y�êÐìØ Y+—Ó§R߈pÇ>ƒvS­d|‚i´¾›”Iºž'—þ‹ˆ§É4/Jöq»¦,Ío—ýöxj‚©;Ç;XBUßP½¸faŽ. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜óóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡óóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðÀó ó óóó ó ó ó ó óóóó#ó$ó%ó'êîäï € 0 TðL`ðÜðäð( ð ðÜðr ðÜ S ð€À†¿ÿ|ð=Kè·ðà   ð žðr ðÜ S ð€øÁ¿ÿ|ðàV0ðà  ð žðX² ðÜ ƒ ð0A,�ƒ¿Àÿ?ð% tìæ ðX² ðÜ ƒ ð0A-�ƒ¿Àÿ?ðV G5ðH ðÜ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.‡fÈ€:R½ð ñ 0 Œð„pðàðð( ð ðàðX ðà C ð¿ÿ ð°Ð ðà  ð„ ðà S ð€Ì¿ÿ ð° °0Ððà   🪠ðH ðà ƒ ð0�ƒ“Þ½h”ŽŸ‹¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë.‡fÈ€:R½r\& éÕ‚õ™ö[é„'Å1èN&é(€àà€ òD/È 0ÒÕ|·DArialUnicode MST©T©ƒ‘Ü–É 0Ü–Õ"·DTimes New RomanT©T©ƒ‘Ü–É 0Ü–Õ ·DWingdingsRomanT©T©ƒ‘Ü–É 0Ü–Õ0·DSymbolgsRomanT©T©ƒ‘Ü–É 0Ü–Õ@·DArial Unicode MST©ƒ‘Ü–É 0Ü–Õ€"¤€@ÿÿ¥ .©  @£nÿý?" dÿd@ÿÿïÿÿÿÿÿÿ  @@``€€ „ ð| ðàè;Ý%      !#$&'( )*+, -. /014 56789ßð¼bð$ç~å…I�W—Ä‚-S‰ÿybð$câšHÉôÄ%]A_¶Ë­÷ÿpy2ð$µè�.àƒ/޹¦@}U%ÿÇé 2ð${ëÖ=8rö‡IbÔ…2ƒ‰xÿΰ2ð$ “Ý^!þg£¸¯Iza_ÿ\~ 2ð$åvÌÊ’†¹\xb\¥Aªÿ| Ú(bð$ ç÷¢ mÉ�×YU­£ÿXV42ð$ýaÅX`Ì�,ôd³’(‰Œÿ‘®;2ð$0¥â] H{Šëõ¹‹² Ïÿå?T2ð$³[ÿsäXÛ7äFÇaˆ%ÿ4 $dbð$8Ùáa³|%È¥AaUb¼Ëÿ´Xp2ð$¾òÙÎ¥S—©ýMp�ÿ  tbð$þ¦ÛÏŠ 5MŠ¡.µØ×îÿ ,†2ð$sÓ�#ö�)þÜ�6ƒvþ¬ÿ›Ì�bð$ ƒÒ½i ~iº!”NWGÿög¥bð$š�·5º7ÕÕá]ýç {Fÿç]©2ð$WÞµ§Îì-Þ2©øíê¦Ûyÿ"DÃ2ð$ qœ³¤5JÞ†ü·,ažÓÿp&Wå2ð$Š´e’h9A=ôËY¥‡tFÿÝžbð$mS¾¤Ä�E”†•gP)A�ÿÕÇ 2ð$H!0 ‰þVs<Ù xŸÿ6ÿ· (bð$Y6UçY´øäúIíË{DÆÿŒœ2ð$ÚñâOÒ‚¸7kû|÷Ôh=ÿÖ#ß%2ð$¶¢£WÈœ!òÊjÁ_bð$™ëä˜ £äpV1«ðñcÿÍ XÙ2ð$äÊŒ™éàÊJÏ=‰Õ³ÿÓ%å2ð$¨ö4S* 7P‰oÇ ¶òÿz®;2ð$²ÓEÍ2…6›ÔyÎlîÆlÿ:’ibð$Ï3:iÄQþbôçmÝ ÿÄ (V2ð$Ÿàõ¯C>ì`OÄÎëKÿV2Ì„bð$ò‰Õ%/úf”T÷–÷ Œ·ÿ¦ ì_2ð$ ª¤û¯«lÏõ|·5 h¿ÿy""·2ð$ylü‹:p9ú½šéÂs_�ÿ“+›Ù2ð$†r 9¯˜õuOÂŒ—O¦ÿ‡ .2ð$ [.<Ç—;¨¶èvøøÿ¿ µ%2ð$Èå�¿Aà4nQ¼·.–šÿ_tFbð$tñÝ^Ô–ã€ÆSgWS�ÿk ÓV2ð$4ù¬µ›tñø=¦T ­¿ÿ� >`2ð$"%ò@4lºÀ�…k V!šÿ Ûm2ð$þÀ'ñnåWº38e&02¡Mÿýæ�2ð$Å©(±§’=œ26j(üWÿ&ã‘“ ð–€�‚ƒ„†A‡Á¿ÀÿÁÂÿÿÿÄÅAË8cÍÎÐÑÒÓÔÕ×ÿ"ñ¿ ñÿ@ñ÷ðpó€ó€ó€ó€Ð3º“°öÊš;­”ÇÊš;úgþý4YdYd¼ ‘ô–É 0þÿÿ”ÿÿÿpûppû@ <ý4dddd —‘w 0T©ð‚‘ÿ ˆ8Š0º___PPT10‹ pµpµð»óŸ¨Physics of fusion powerŸ¨,Lecture 13 : Diffusion equation / transport óŸ¨Many body problem Ÿ¨The plasma has some 1022 particles. No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûªõóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðÀó óóó ó ó ó ó óóóó#ó$ó%ó'ó)êî$ï € 0 ”ðŒ€ðäð$ð( ð ðäðr ðä S ð€ ¿ÿ|ð=Kè·ðà   ð žðr ðä S ð€àð¿ÿ|ðàV0ðà  ð žðH ðä ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.ˆfÈÀÌð ñ 0 Œð„�ðèðð( ð ðèðX ðè C ð¿ÿ ð°Ð ðà  ð„ ðè S ð€àü¿ÿ ð° °0Ððà   🪠ðH ðè ƒ ð0�ƒ“Þ½h”ŽŸ‹¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë.ˆfÈÀÌr)…( ««­õ™…¿¯)Å1DD‚(‚(‚(BEEE¢¢¢¢QRZXEt?¢ �Æ"œ„@1Æ).HiÞQdØ“ÐÞç!Š rÊÇ'�(bŠSí¿CéçT Pú Eú8U“(D3¤Šñi(D RcþQ¤—TŒ(bŠ”˜°ÏCé!5eˆBQ{ªÆ§± E” $R#ÿQDJLÝç!Š(A=¨j*5ö¯Ešž^�¦€Bi‚’ÞÇC™*ŸD½¿¿DÔÈ Ez==�4˜·DÔ\Oˆ"(!æƒBy�21B!ŠTžæÏÉE$PµED‘¹  Pˆ"%(a ¢ˆ©)9¢H Êà…(bx„B”÷žÌPˆò QŽæX’º‹£O±ž'D¹šë£ûï�¯§×Þo(¡ Q®æ19?åCT¨q×AO#@!ÊÕ<©Hgˆz 4QŽ&}¸¤A�Îéµx<(D9›\A(�JNaxJò[çC˜¯(œ�ÇÇþî^.Ý Ä(Pˆr6§ÜÁéým¥é5Èp½ˆ\TT,¦ß“(H’$íò4¢œÍ]”/÷J_ÅBg¹:U}#¤Ÿ©=e#=!Êáœä!ªZ–z® ‹š QN“’z¦œ(]BQ5O#;Q~ä%*P‹ºµ=M å�¨óóëù %Z ²i 勨ra<üz‹ŠT¢â9{n7B QöævÃs9]J·Ã$ ŽEOGap©�ʦîñºDEbêAZˆÚsb T¢¢@L%…(wGC Þ[�·½¦¨´XKi»#ÎPEÔš ¸¬P&@µDEåi·7â UDé"0¾EžŒ€jŠJƒê«47×?CQ�¿Ì†-ñÔUgÚȃ”þªˆÚÄÓ´­/ª!*zŸºU#¥{†*¢6â4þ1žÌ•¯&*_‹zݖפtÏPETwO ±Ú#Môd¦€5QQX/Ü{�Ò?Ò}øˆdAisR¼È[Œ”=¢NõS#*Q"y~gå2Õ† âœ”»¹ž²;"*_åLÚ¢‚Ì^Q›�Ç©ëÐ�7÷v-ªq ûkŒ²WÔ6 â‘œz]é*¿+¢šÕµ]Ô âñœú^Ø Ò‚(‡AMÒ4ð¾Kǯ¬5ûHËE­zôoOä4Ø �ZDÔ£œŸy¦·zr+¦éir·¤8€ê’Xµœ¾¨x†&�'öàÁÁ6ˆú,9=ÿ·khÙ¨x¦¦Lg$UßÃ.QåøôY}qÑÔZi™£Isæ1xNÌæ¢*I¯¯k S+Nl–|(3˜´A©IéT¯ ŽiQ¤õE­„já«ýÀÔ9øhÞQS”ÒŽiQ/GŸÒ­E­€ÊðCÄ�™Ý“ïª/ªÍǰ¨7£Où檢FefóqŒ4Æt}£D5¹)jAT³6kÄX[ÌÐÆŠ’™Õþ ªMD-v¹Œ±›�µc´)fýÅQ/GëˆÚ,ñ²ÈSëc›dkíõ¶Œ¥Žò1¡ñOñ£I¿ÚTÇdÝ1QV™2ǦOT¶±§ZqFÕi¢ÜÛ«Mšãìpx*gt¬G-77wò¥©Õ[�“m¢–¨žÛ™'j©"mü.Ì’«QŽQ#+¸¼¦ yó�âluQ6´·û¤Và”Õ�fù\ùh–¥ß€ñÔ¨J®À)«vÄ’©•¯T¸|Ýœ'5JÔzåylÛ'{¯õ쥚‹wŠ“7 ¬«¨³Íî (ÛúÐ]QÞ€²ìÉãjÃ{4DYVYgEyÊ®Ú:{vž_¢,ª®›-ï(«*ì¨(Ï@ÙTc'ÛÞ»!J]åm®åæéÞ�RWz“ëA9)ÊCPêZoq=([ßË!J]í-®å^ë{ JÝ•\ÊAQž‚RW|ýëA!Ê횯=(çÚßÛ�žº/׿”{¢¼¥®ûê׃r­<¢::síëA9'ÊcP]ϱu¯åXx=D©³öõ \… fÖ¾”[]À¥ÈÊ׃rL~4ŸtˆÓZQTQ7éΠ׃rJbèê‚(‹«‚(D! 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðÀó óóó ó ó ó ó óóóó#ó$ó%ó'ó)êîçï €  0 WðOð„ðçð( ð ð„ðr ð„ S ð€\ÎÝ¿ÿ|ð=Kè·ðà  Ý ð žðx ð„ c ð$€T¥¿ÿ|ðàV“ ðà Ý ð žð¢² ð„ à ðb„…A.Á�ÿÿÿƒÿÿÿ¿Àÿ€€€|collision02dð·È  ðà ð ¢ ð„ “ ð6€h@¿�ƒ¿ÀÿË8cÿ"ñ¿ ðÞ # ´ 𗟨_Depending on the (unknown) initial conditions the outcome of a collision can be very different ¡`^ðH ð„ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.}fÈ�ë!î­ï €  0 ð0ð�ð­ð( ð ð�ðr ð� S ð€À ¿ÿ|ð=Kè·ðà   ð žðx ð� c ð$€ˆ¿ÿ|ðàV“ ðà  ð žðX² ð� ƒ ð0A�ƒ¿Àÿ?ð�t ÝðX² ð� ƒ ð0A�ƒ¿Àÿ?ðƒ t ?𨲠ð� Ó ðh„…A.Á�ÿÿÿƒÿÿÿ¿Àÿ €€€|ˆcollision02dðŠÈ ò ðà ð ¢ ð� “ ð6€H ¿�ƒ¿ÀÿË8cÿ"ñ¿ ð) È Yj 𗟨_Depending on the (unknown) initial conditions the outcome of a collision can be very different ¡`^ðH ð� ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.}fÈ�¾•Ìî4ï € 0 ¤ðœ@ð Pð4ð( ð ðPðX² ðP ƒ ð0A�ƒ¿Àÿ?ðPϵAðx ðP c ð$€¤-ý¿ÿ@ˆð=Kè·ðà  ý ð žðx ðP c ð$€x@K¿ÿ@ˆðàV“ ðà ý ð žðX² ðP ƒ ð0A�ƒ¿Àÿ?ð¾ ¢Òõ ðX² ðP ƒ ð0A�ƒ¿Àÿ?ð6 Z / ðf² ðP ƒ ð0A�ƒ¿ÀÿPPÿ?"ñ¿`ðÞ „<¯ðXB ð P ƒ ð0D¿ÀÿPPËjJÑÿðÎ ß ƒ ðXB ð P@ ƒ ð0D¿ÀÿPPËjJÑÿðÎ ÿ´ƒ 𮲠ð P à ðn„…A/Á&�ÿÿÿƒÿÿÿ¿Àÿ€€€@classicaltransportðmÈ d ðà ðH ðP ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.‘cÈàŒ¸ürÿ¯/ß ‹ÖzÚõ™Û¯kä)Å1è„&é(€àà€ òD/È 0ÒÕ|·DArialUnicode MST©T©ƒ‘Ü–É 0Ü–Õ"·DTimes New RomanT©T©ƒ‘Ü–É 0Ü–Õ ·DWingdingsRomanT©T©ƒ‘Ü–É 0Ü–Õ0·DSym     /!"#$%&'()*+,-.F0123456789:;<=>?@ABCD_þÿÿÿGHIJKLMNOPQRSTUVWXYZ[\]^þÿÿÿ`abcdefghijklmnopqrstuvwxyz{|}~þÿÿÿbolgsRomanT©T©ƒ‘Ü–É 0Ü–Õ@·DArial Unicode MST©ƒ‘Ü–É 0Ü–Õ€"¤€@ÿÿ¥ .©  @£nÿý?" dÿd@ÿÿïÿÿÿÿÿÿ  @@``€€ Ü ðÔ ðàè;ß%      !# $&'( )*+, -. /014 56789ÿðbð$ç~å…I�W—Ä‚-S‰ÿybð$câšHÉôÄ%]A_¶Ë­÷ÿpy2ð$µè�.àƒ/޹¦@}U%ÿÇé 2ð${ëÖ=8rö‡IbÔ…2ƒ‰xÿΰ2ð$ “Ý^!þg£¸¯Iza_ÿ\~ 2ð$åvÌÊ’†¹\xb\¥Aªÿ| Ú(bð$ ç÷¢ mÉ�×YU­£ÿXV42ð$ýaÅX`Ì�,ôd³’(‰Œÿ‘®;2ð$0¥â] H{Šëõ¹‹² Ïÿå?T2ð$³[ÿsäXÛ7äFÇaˆ%ÿ4 $dbð$8Ùáa³|%È¥AaUb¼Ëÿ´Xp2ð$¾òÙÎ¥S—©ýMp�ÿ  tbð$þ¦ÛÏŠ 5MŠ¡.µØ×îÿ ,†2ð$sÓ�#ö�)þÜ�6ƒvþ¬ÿ›Ì�bð$ ƒÒ½i ~iº!”NWGÿög¥bð$š�·5º7ÕÕá]ýç {Fÿç]©2ð$WÞµ§Îì-Þ2©øíê¦Ûyÿ"DÃ2ð$ qœ³¤5JÞ†ü·,ažÓÿp&Wå2ð$Š´e’h9A=ôËY¥‡tFÿÝžbð$mS¾¤Ä�E”†•gP)A�ÿÕÇ 2ð$H!0 ‰þVs<Ù xŸÿ6ÿ· (bð$Y6UçY´øäúIíË{DÆÿŒœ2ð$ÚñâOÒ‚¸7kû|÷Ôh=ÿÖ#ß%2ð$¶¢£WÈœ!òÊjÁ_bð$™ëä˜ £äpV1«ðñcÿÍ XÙ2ð$äÊŒ™éàÊJÏ=‰Õ³ÿÓ%å2ð$¨ö4S* 7P‰oÇ ¶òÿz®;2ð$²ÓEÍ2…6›ÔyÎlîÆlÿ:’ibð$Ï3:iÄQþbôçmÝ ÿÄ (V2ð$Ÿàõ¯C>ì`OÄÎëKÿV2Ì„bð$ò‰Õ%/úf”T÷–÷ Œ·ÿ¦ ì_2ð$ ª¤û¯«lÏõ|·5 h¿ÿy""·2ð$ylü‹:p9ú½šéÂs_�ÿ“+›Ù2ð$†r 9¯˜õuOÂŒ—O¦ÿ‡ .2ð$ [.<Ç—;¨¶èvøøÿ¿ µ%2ð$Èå�¿Aà4nQ¼·.–šÿ_tFbð$tñÝ^Ô–ã€ÆSgWS�ÿk ÓV2ð$4ù¬µ›tñø=¦T ­¿ÿ� >`2ð$"%ò@4lºÀ�…k V!šÿ Ûm2ð$þÀ'ñnåWº38e&02¡Mÿýæ�2ð$Å©(±§’=œ26j(üWÿ&ã‘bð$gÎNÞPoÄ·úä'3ÞCÉÿ¾ ¥bð$MšÛ@õ��2ì¦éP@9´ÿ„Ǽ“ ð–€�‚ƒ„†A‡Á¿ÀÿÁÂÿÿÿÄÅAË8cÍÎÐÑÒÓÔÕ×ÿ"ñ¿ ñÿ@ñ÷ðpó€ó€ó€ó€Ð3º“°öÊš;­”ÇÊš;úgþý4YdYd¼ ‘ô–É 0þÿÿ”ÿÿÿpûppû@ <ý4dddd —‘w 0T©ð‚‘ÿ ˆ8Š0º___PPT10‹ pµpµð™óŸ¨Physics of fusion powerŸ¨,Lecture 13 : Diffusion equation / transport óŸ¨Many body problem Ÿ¨The plasma has some 1022 particles. No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó Ÿ¨Particle motion in a tokamakªŸ¨¦Drift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ª,W5 ó Ÿ¨4Conservation of canonical toroidal angular momentum ªŸ¨.The toroidal angular momentum Is conserved ª#/ðÀó óóó ó ó ó ó óóóó#ó$ó%ó'ó)êr³äõ™�ä? )Å1è$é(€àà€ òD/È 0ÒÕ|·DArialUnicode MST©T©ƒ‘Ü–É 0Ü–Õ"·DTimes New RomanT©T©ƒ‘Ü–É 0Ü–Õ ·DWingdingsRomanT©T©ƒ‘Ü–É 0Ü–Õ0·DSymbolgsRomanT©T©ƒ‘Ü–É 0Ü–Õ@·DArial Unicode MST©ƒ‘Ü–É 0Ü–Õ€"¤€@ÿÿ¥ .©  @£nÿý?" dÿd@ÿÿïÿÿÿÿÿÿ  @@``€€ Ü ðÔ ðàè;Æ!      !# $&'( )*+, -. /014 56789ÿðbð$ç~å…I�W—Ä‚-S‰ÿybð$câšHÉôÄ%]A_¶Ë­÷ÿpy2ð$µè�.àƒ/޹¦@}U%ÿÇé 2ð${ëÖ=8rö‡IbÔ…2ƒ‰xÿΰ2ð$ “Ý^!þg£¸¯Iza_ÿ\~ 2ð$åvÌÊ’†¹\xb\¥Aªÿ| Ú(bð$ ç÷¢ mÉ�×YU­£ÿXV42ð$ýaÅX`Ì�,ôd³’(‰Œÿ‘®;2ð$0¥â] H{Šëõ¹‹² Ïÿå?T2ð$³[ÿsäXÛ7äFÇaˆ%ÿ4 $dbð$8Ùáa³|%È¥AaUb¼Ëÿ´Xpð$ÿð$ÿð$ÿð$ÿð$ÿð$ÿð$ÿ2ð$Š´e’h9A=ôËY¥‡tFÿÝžbð$mS¾¤Ä�E”†•gP)A�ÿÕÇ 2ð$H!0 ‰þVs<Ù xŸÿ6ÿ· (bð$Y6UçY´øäúIíË{DÆÿŒœ2ð$ÚñâOÒ‚¸7kû|÷Ôh=ÿÖ#ß%2ð$¶¢£WÈœ!òÊjÁ_bð$™ëä˜ £äpV1«ðñcÿÍ XÙ2ð$äÊŒ™éàÊJÏ=‰Õ³ÿÓ%å2ð$¨ö4S* 7P‰oÇ ¶òÿz®;2ð$²ÓEÍ2…6›ÔyÎlîÆlÿ:’ibð$Ï3:iÄQþbôçmÝ ÿÄ (V2ð$Ÿàõ¯C>ì`OÄÎëKÿV2Ì„bð$ò‰Õ%/úf”T÷–÷ Œ·ÿ¦ ì_2ð$ ª¤û¯«lÏõ|·5 h¿ÿy""·2ð$ylü‹:p9ú½šéÂs_�ÿ“+›Ù2ð$†r 9¯˜õuOÂŒ—O¦ÿ‡ .2ð$ [.<Ç—;¨¶èvøøÿ¿ µ%2ð$Èå�¿Aà4nQ¼·.–šÿ_tFbð$tñÝ^Ô–ã€ÆSgWS�ÿk ÓV2ð$4ù¬µ›tñø=¦T ­¿ÿ� >`2ð$"%ò@4lºÀ�…k V!šÿ Ûm2ð$þÀ'ñnåWº38e&02¡Mÿýæ�2ð$Å©(±§’=œ26j(üWÿ&ã‘bð$gÎNÞPoÄ·úä'3ÞCÉÿ¾ ¥bð$MšÛ@õ��2ì¦éP@9´ÿ„Ǽ“ ð–€�‚ƒ„†A‡Á¿ÀÿÁÂÿÿÿÄÅAË8cÍÎÐÑÒÓÔÕ×ÿ"ñ¿ ñÿ@ñ÷ðpó€ó€ó€ó€Ð3º“°öÊš;­”ÇÊš;úgþý4YdYd¼ ‘ô–É 0þÿÿ”ÿÿÿpûppû@ <ý4dddd —‘w 0T©ð‚‘ÿ ˆ8Š0º___PPT10‹ pµpµðSóŸ¨Physics of fusion powerŸ¨,Lecture 13 : Diffusion equation / transport óŸ¨Many body problem Ÿ¨The plasma has some 1022 particles. No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. 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No description is possible that allows for the determination of position and velocity of all these particles Only averaged quantities can be described. The evolution of the averaged velocity is however influenced by a microscopic process : the collisions Each collision can have a different outcome depending on the unknown initial conditions ¡,€Z hóŸ¨Statistical description Ÿ¨¼For the microscopic process one can introduce a probability distribution y Y gives the probability that the position of the particle will change by an amount w in a given time interval t¡0½IlóŸ¨Distribution function Ÿ lSince one deals with the probability of a certain change in position it is useful to also describe the particle distribution as a probability The evolution of the distribution is due to the process described by the function y. The distribution at time t can be found from the distribution at t  t, by multiplying the probability of finding a particle at the position x  w with the probability that its position changes by an amount w ¡@·âGŒó Ÿ¨Evolution equation Ÿ¨s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation ¡ rtó Ÿ¨Evolution equation Ÿ¨ ¡" ZZ ó Ÿ¨Evolution equation Ÿ¨K With the definitions Can be written as a convection diffusion equation ó  Ÿ Simple form & . Ÿ¨ŒPreviously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ¡��ó! Ÿ¨Gaussian distribution Ÿ¨ZIf the probability is Gaussian with half-width s one can use s as the typical step length ¡6[/ ó"Ÿ¨"Brief look back at the collisions Ÿ¨—Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient ¡˜˜ó&Ÿ¨Many particles Ÿ¨For many particles that are independent (uncorrelated) the probability for finding a particle in a certain position is the same for all particles For many particles the one particle probability distribution can be thought of as a distribution of density Consequently, ¡ó(Ÿ¨Key things to remember Ÿ¨úMany body problem can only be described in a statistical sense Outcome of a microscopic process often depends on unknown initial conditions The microscopic processes can be described by a probability distribution for a certain change If the time interval is short and the step size is small the time evolution can be cast in a convection / diffusion equation The convection is zero for many phenomena, and the diffusion coefficient is proportional to the step size squared over 2 times the typical time ¡ûûóŸ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨ˆParticles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency ¡‰‰ªS0ó Ÿ¨*Transport in a homogeneous magnetic field ¡++ Ÿ¨MTypical values for a reactor The particles satisfy a diffusion equation ¡NNó*Ÿ¨Particle orbit in a tokamakªŸ¨¹In a Tokamak the particles drift away form the surface The drift velocity of a thermal particle scales as The drift away from the surface occurs in a time This gives a step size ¡ºZºª®ó,Ÿ¨Particle orbit in a tokamakªŸ¨cStep size Yields a diffusion coefficient Which is still much smaller than the observed ó-Ÿ¨Key things to remember Ÿ¨øCollisional scattering leads to a random walk of the particles in space with a step size of the order of the Larmor radius and a typical time determined by the collision frequency The diffusion coefficient of this process is very small In a tokamak the drift leads to a large deviation from the flux surface and therefore a larger effective step size The diffusion coefficient in a tokamak is much larger than in an homogeneous magnetic field, but still at least a factor 4-5 times smaller than observed ¡ùZùªH b~†t/ðÜó óóó ó ó óóóó#ó$ó%ó'ó)ó+ó.ó/êî$ï € 0 ”ðŒ@ð Pð$ð( ð ðPðX² ðP ƒ ð0A�ƒ¿Àÿ?ðPϵAðx ðP c ð$€¤-ý¿ÿ@ˆð=Kè·ðà  ý ð žðx ðP c ð$€x@K¿ÿ@ˆðàV“ ðà ý ð žðX² ðP ƒ ð0A�ƒ¿Àÿ?ð¾ ¢Òõ ðX² ðP ƒ ð0A�ƒ¿Àÿ?ð6 Z / ðf² ðP ƒ ð0A�ƒ¿ÀÿPPÿ?"ñ¿`ðÞ „<¯ðXB ð P ƒ ð0D¿ÀÿPPËjJÑÿðÎ ß ƒ ðXB ð P@ ƒ ð0D¿ÀÿPPËjJÑÿðÎ ÿ´ƒ 𮲠ð P à ðn„…A/Á&�ÿÿÿƒÿÿÿ¿Àÿ€€€@classicaltransportðmÈ d ðà ðè¢ ð P “ ð6€ Ç¿�ƒ¿ÀÿË8cÿ"ñ¿ ðÎ › ,b  ðtŸ¨FCollisional scattering leads to a random walk of the particle in spaceª <ðH ðP ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.‘cÈàŒ¸üîBï € 0 ²ðª ð ìðBð( ð ðìðr ðì S ð€$ù¿ÿ@ð=Kè·ðà   ð žðr ðì S ð€dã¿ÿ@ðàV“ ðà  ð žðl² ðì “ ð6A �ƒ¿ÀÿË8cÿ?"ñ¿ ðt‡ž ðl² ð ì “ ð6A�ƒ¿ÀÿË8cÿ?"ñ¿ ðF t H ðl² ð ì “ ð6A�ƒ¿ÀÿË8cÿ?"ñ¿ ð¤tޛ𤲠ð ì à ðd„…A Á�ÿÿÿƒÿÿÿ¿Àÿ€€€@particleorbitð@õ € ðà ð¢ ð ì “ ð6€L{¿�ƒ¿ÀÿË8cÿ"ñ¿ ðû õ †é 𚟨†Due to the combined effect of the motion along the field line and the drift, the particle moves a finite distance (w) from the surfaceðH ðì ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.�fÈ�³ðÊî¶ï € 0 &ððð ð¶ð( ð ððf² ð  ƒ ð0A�ƒ¿ÀÿË8cÿ"ñ¿ ð€ 7# 7 ðx ð c ð$€4¹¿ÿ@ˆð=Kè·ðà   ð žðx ð c ð$€ x¿ÿ@ˆðàV“ ðà  ð žðl² ð “ ð6A�ƒ¿ÀÿË8cÿ?"ñ¿ ð"7¡ðª² ð Ó ðj„…A Á�ÿÿÿƒÿÿÿ¿Àÿ€€€@ˆparticleorbitð@P Û ðà ð¢ ð “ ð6€0ê¿�ƒ¿ÀÿË8cÿ"ñ¿ ð) ~  𚟨†Due to the combined effect of the motion along the field line and the drift, the particle moves a finite distance (w) from the surfaceðf² ð  ƒ ð0A�ƒ¿ÀÿË8cÿ"ñ¿ ðs 7 º ðf² ð  ƒ ð0A1�ƒ¿ÀÿË8cÿ"ñ¿ ð¤dY ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë.�fÈ�³ðÊî$ï € 0 ”ðŒÐðøð$ð( ð ðøðr ðø S ð€ Û¿ÿ@ð=Kè·ðà   ð žðr ðø S ð€„]¿ÿ@ðàV0ðà  ð žðH ðø ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ)))€€€Ì™ÌÌ™™™3²²²ˆ8Š0º___PPT10‹ë. fȰkß¿ð ñ 0 Œð„àðüðð( ð ðüðX ðü C ð¿ÿ ð°Ð ðà  ð„ ðü S ð€ôY¿ÿ ð° °0Ððà   🪠ðH ðü ƒ ð0�ƒ“Þ½h”ŽŸ‹¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë. fȰkß¿ðñ 0 ˜ð�ðð(ð( ð ðð^ ð S ð¿ÿ ˆð°Ð ðà  ðŠ ð c ð$€I¿ÿ ˆð° °0Ððà   🪠ðH ð ƒ ð0�ƒ“Þ½h”ŽŸ‹¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë.�fÈ�$óÊr,á[î…*Œ,@d‘"—N™b›õ™½[‚�/Å1