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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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt*Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/         0  (  X  0A? dX  0A? 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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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/4       v  0 v(  r  S |=K   r  S |V0  X  0A?/X  0A? X  0A?A 2 X  0A?F t a X  0A? - l  61"` <  02 ( 2  0S2  i5The integral = 1 (integration over all probabilities)6 26  0I  i5The integral = 1 (integration over all probabilities)6 26dB  @ <D>? _ XB @ 0D># H  0޽h ? )))̙̙380___PPT10.fY$   0 (  X  C       S   0    H  0޽h ? 3380___PPT10.fY$r  1( 8y˯norK;? 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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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/P         0 (  (  r  S E|=K   r  S x|V0  X  0A? 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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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! Gaussian distribution ZIf the probability is Gaussian with half-width s one can use s as the typical step length 6[/ ""Brief look back at the collisions Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient *Transport in a homogeneous magnetic field ++ Particles 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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! Gaussian distribution ZIf the probability is Gaussian with half-width s one can use s as the typical step length 6[/ ""Brief look back at the collisions Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient &Many particles 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, *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/       #$%'  0 TL`(  r  S |=K   r  S |V0  X  0A,?% t X  0A-?V G5H  0޽h ? )))̙̙380___PPT10.f:R  0 p(  X  C       S   0    H  0޽h ? 3380___PPT10.f:Rr\& Ղ['1N&( D/ 0|DArialUnicode MSTTܖ 0ܖ"DTimes New RomanTTܖ 0ܖ DWingdingsRomanTTܖ 0ܖ0DSymbolgsRomanTTܖ 0ܖ@DArial Unicode MSTܖ 0ܖ"@ .  @n?" dd@  @@``  | ;%      !#$&'( )*+, -. /014 56789b$~IWĂ-Syb$cH%]A_˭py2$./@}U% 2${=8rIbԅ2x2$ ^!gIza_\~ 2$vʒ\xb\A| (b$ mɝYUXV42$aX`̍,d(;2$0] H{ ?T2$[sX7Fa%4 $db$8a|%ȥAaUbXp2$ΥSMp  tb$ϊ 5M.,2$sӍ#)ܐ6v̍b$ ҽi~i!NWGgb$57] {F]2$W޵-2y"D2$ q5Jކ,ap&W2$eh9A=YtFb$mSāEgP)A 2$H!0 Vs< x6 (b$Y6UYI{D2$O҂7k|h=#%2$WȜ!Êj_b$ pV1c X2$ʌJ=ճ%2$4S* 7Po z;2$E26yll:ib$3:iQbm (V2$C>`OKV2̄b$%/fT _2$l|5 hy""2$yl:p9s_+2$r 9uOO .2$ [.<;v %2$吿A4nQ._tFb$t^ԖSgWSk V2$4t=T >`2$"%@4lk V! m2$'nW38e&02M2$ũ(=26j(W& AA8c" @p3ʚ;ʚ;g4YdYd 0ppp@ <4dddd w 0T 80___PPT10 ppPhysics of fusion power,Lecture 13 : Diffusion equation / transport Many body problem 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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! Gaussian distribution ZIf the probability is Gaussian with half-width s one can use s as the typical step length 6[/ ""Brief look back at the collisions Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient &Many particles 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, (Key things to remember Many 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 *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/      #$%')$  0 $(  r  S |=K   r  S |V0  H  0޽h ? )))̙̙380___PPT10.f  0 (  X  C       S   0    H  0޽h ? 3380___PPT10.fr)( )1DD(((BEEEQRZXEt? 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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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! Gaussian distribution ZIf the probability is Gaussian with half-width s one can use s as the typical step length 6[/ ""Brief look back at the collisions Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient &Many particles 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, (Key things to remember Many 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 *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/      #$%')   0 WO(  r  S \|=K   x  c $T|V     bA.|collision02d      6h@8c"  #  _Depending on the (unknown) initial conditions the outcome of a collision can be very different `^H  0޽h ? )))̙̙380___PPT10.}f!   0 0(  r  S  |=K   x  c $|V   X  0A?t X  0A? t ?  hA. |collision02d      6H8c" ) Yj _Depending on the (unknown) initial conditions the outcome of a collision can be very different `^H  0޽h ? )))̙̙380___PPT10.}f4  0 @ P4(  PX P 0A?PAx P c $-@=K   x P c $x@K@V   X P 0A?  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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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! Gaussian distribution ZIf the probability is Gaussian with half-width s one can use s as the typical step length 6[/ ""Brief look back at the collisions Here the distribution was defined for the angle of the velocity This distribution satisfies a diffusion equation With the diffusion coefficient &Many particles 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, (Key things to remember Many 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 *Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0 *Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry ,W5  4Conservation of canonical toroidal angular momentum .The toroidal angular momentum Is conserved #/      #$%')r? )1$( D/ 0|DArialUnicode MSTTܖ 0ܖ"DTimes New RomanTTܖ 0ܖ DWingdingsRomanTTܖ 0ܖ0DSymbolgsRomanTTܖ 0ܖ@DArial Unicode MSTܖ 0ܖ"@ .  @n?" dd@  @@``   ;!      !# $&'( )*+, -. /014 56789b$~IWĂ-Syb$cH%]A_˭py2$./@}U% 2${=8rIbԅ2x2$ ^!gIza_\~ 2$vʒ\xb\A| (b$ mɝYUXV42$aX`̍,d(;2$0] H{ ?T2$[sX7Fa%4 $db$8a|%ȥAaUbXp$$$$$$$2$eh9A=YtFb$mSāEgP)A 2$H!0 Vs< x6 (b$Y6UYI{D2$O҂7k|h=#%2$WȜ!Êj_b$ pV1c X2$ʌJ=ճ%2$4S* 7Po z;2$E26yll:ib$3:iQbm (V2$C>`OKV2̄b$%/fT _2$l|5 hy""2$yl:p9s_+2$r 9uOO .2$ [.<;v %2$吿A4nQ._tFb$t^ԖSgWSk V2$4t=T >`2$"%@4lk V! m2$'nW38e&02M2$ũ(=26j(W&b$gNPoķ'3C b$M@2P@9Ǽ AA8c" @p3ʚ;ʚ;g4YdYd 0ppp@ <4dddd w 0T 80___PPT10 ppSPhysics of fusion power,Lecture 13 : Diffusion equation / transport Many body problem 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 hStatistical description For 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 t0IlDistribution function 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 @G Evolution equation s Assuming a small time interval t and small step w one can use a Taylor expansion This yields the equation  rt Evolution equation  " ZZ  Evolution equation K With the definitions Can be written as a convection diffusion equation   Simple form & . Previously in estimating diffusion coefficient we have used a fixed step which could be either positive or negative This directly yields ! 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AA8c" @p3ʚ;ʚ;g4YdYd 0ppp@ <4dddd w 0T 80___PPT10 ppnPhysics of fusion power,Lecture 13 : Diffusion equation / transport Many body problem 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 hStatistical description For 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 t0IlDistribution function 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. 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