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By Gerard G. Emch

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Extra resources for Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Physics & Astronomical Monograph)

Example text

This p o i n t is discussed further in A p p e n d i x B; see also Fig. 4. W h e n the expression in Eq. 5) is inserted into the general e q u a t i o n of c h a n g e in Eq. 7), a n d when the definition of the Liouville o p e r a t o r in Eq. 8) 2 C .... 4 Fig. 4. Sketch showing why p~ and n~ are not simply related by p, = m~,n~, (a) The mass concentration p~ gives the mass of beads (from any chains) in a unit volume - here shown as a box - at some point in space, regardless of the locations of the centers of mass "c" of the chains.

This idea can be expressed by allowing the friction coefficient to be a secondorder tensor (DPL Sects. 1); this introduces additional undetermined parameters into the theory [9, 20a, 20b, 20c]. iii. The inclusion of pairwisefrictionaljbrces: Eq. 6) involves the difference between the averaged bead velocity and the fluid velocity. As seen in Appendix B this empiricism leads to an inconsistency, in that the mass fluxes j, in Eq. t7) do not sum to zero as required by the definition in Eq. 4). This inconsistency has led to replacing Eq.

R , - ~ 0 2 ~-,, t~Ju\ cry ~k~ (E(f:i~(e)off, A(rtti. 9) We now manipulate each of the "source terms" by using Eqs. 19-21) in order to obtain Eq. 1), with the heat-flux vector being the sum of four contributions: q = q(k) + q(O)+ q(a) + q(~). 1 T h e K i n e t i c C o n t r i b u t i o n to the H e a t F l u x Vector First, in the kinetic contribution Q(k) we replace both of the/~i in the dot product by i~i - v and put in compensating terms; the mass-average velocity v is a function of r and t.

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