Cooperative Education Program
Cooperative Education Program M E 1
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This 3 page Class Notes was uploaded by Reginald Hane on Thursday September 17, 2015. The Class Notes belongs to M E 1 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/205264/m-e-1-university-of-wisconsin-madison in Mechanical Engineering at University of Wisconsin - Madison.
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Date Created: 09/17/15
Ohm s Law and Hall MHD by E Alec Johnson October 2007 1 Boltzmann equation The Boltzmann equation for species 3 is an evolution equation for particle density in phase space Phase space is a collection of coordinates which bijectively specify the state of a particle For point particles such as electrons and protons points in phase space are represented by X V the position and proper velocity of the particle Ignore the blue text and wide tildes if you do not care about the special relativistic regime For molecules with other degrees of freedom phase space must also include other variables which specify not only translational modes but such state quantities as rotational and vibrational modes The rate of motion of a particle through phase space is X 6 V a The Boltzmann equation speci es particle balance in phase space 812 V m V asfs 0 Here 3 is the species index X is position in space V is particle velocity v W is proper velocity f5 f9pt X V is the particle density of species 3 ie f t X V dX dv is the number of particles in the in nitesimal box dde as 7E V x B g divS is the rate of change of the proper velocity of a particle Where qs is charge per particle m5 is mass per particle E is electric eld B is magnetic eld g is gravitational eld and CS CE known as the collision operator is the rate of change in particle density due purely to collisions within species 3 or with other species Note that Cs is an operator which depends on the distribution functions of each species regarded as a function of velocity but not of position or time Csv gt gt f t X V 62 where E is the set of all species indices 11 Notes on the relatistic Boltzmann equation We use 1739 to denote the elapse of proper time The Lorentz factor V as a function of velocity and as a function of proper velocity is 39yv lt17 32W W1212 111 Differentation of the Lorentz factor To express derivatives of the Lorentz factor in terms of derivatives of the proper velocity we differentiate the relation 39yz 1 62 We get 39yd39y d v ie dam do 112 Derivation of the Boltzmann equation at Vx dtxm W dtvsfs cs 8tf9vx d XfsVV y f9Cs atfs V vfs W asfs 0 135 113 Veri cation of Lorentzinvariance of the Boltzmann equation To verify its physical correctness we express the Boltzmann equation in a manifestly Lorentz invariant form a 3553 1 8143533 CS7 80 a flm t V39 3a 8450995 where 5 is the second proper derivative of four position and thus a Lorentz invariant tensor To see that the canceled term is indeed zero and can thus be inserted recall that W is a function of v and therefore any quantity that is a function of ct x 0 v can be de ned to be independent of the formal parameter 0 Indeed 50 160 d7c39y dTV vc dtV Nfc a Vc Likewise is a function of txv The quantity a is a Lorentz invariant scalar called the proper density ie the rest mass per rest volume in phase space It takes a little thought to see this Lorentz transforms have determinant 1 in four space and the spatial components of the velocity suffer length contraction introducing a single factor of 39y 114 Conventions of interpretation Note that by multiplying by mS or qs and respectively by making the rede nitions f5 f msfsp and CS C mSCE or f5 fsq qs and CS Cg qSCE we can also regard the Boltzmann equation as a statement of conservation of mass or charge Henceforth we drop the default species index 3 except as a reminder particularly when making de nitions until we consider multiple species So there is an implicit species index 3 on most variables except for the independent variables 75 x v and xv below and the eld variables E B g 2 Species balance laws Henceforth we View the Boltzmann equation by default as mass conservation in phase space Taking moments of the Boltzmann equation yields balance laws for density momentum and energy De ne i vas I ma De ne ps Ms ltMgts f the average velocity us ltv9 and the thermal velocity