Boltzmann Collision Operator With Inverse Power Intermolecular Potentials
Boltzmann Collision Operator With Inverse Power Intermolecular Potentials
Young Ping Pao
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Now for the derivatives of s(c, k) with respect to c and k, we use the relation 1(3^)"^ J^(x)l = |x-^ J^(x)| ib (130) where x denotes t. Sin 29|cxk| . We note that the derivatives of 2 X with respect to c and k can be easily calculated. The end result may be stated as |3^[C(c)s(c, k)]| b | c | ^ | k | ^"^ - g-N (13^) 60 2 2 and for c > k, inequality C65) Indicates that uCc) > |blcnk|2v ^ ib|cr+2v _ 1 ^ (^35) which together with (13^) yield Re[Cl-e)Sjj^(c, k) + |uCc)] > b(c^+k^)^ - g^ N (136) f...or all c and k. We next consider (111) and note that if lc| is large enough, then Re[(l-e)s^(c, k) + |u(c) + v(c, k)] > bCc^+k^)"^ - Jn (137) and we shall take c to be so large that (137) is valid for IcI > c .
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