Lecture 22 - i j Phase space: an Euclidean space based on q...

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Unformatted text preview: i j Phase space: an Euclidean space based on q and p of every particle for molecules,there are sN coordinates for each one time correlation-function method: study non-equilibrium (transport) systems { } { } ( 29 ( 29 & and define the phase space i i q p p p t q q t ( 29 ( 29 ( 29 ( 29 = 0 , 0 ; p t p p q t ( 29 ( 29 ( 29 ( 29 = 0 , 0 ; q t q p q t ( 29 ( 29 = = ' let s call p p q q { }{ } { } ( 29 ( 29 { } ( 29 ( 29 = = = , function in phase space A p p,q;t , , ; , ; i i A p q q p q t A p q t A t ( 29 ( 29 ( 29 = Definition of correlation function: C t A A t ( 29 ( 29 ( 29 ( 29 = 1 2 n 1 2 n , , ;0 , ; ... ... C t A p q A p q t dp dp dp dq dq dq f p q equilibrium phase space distribution ( 29 ( 29 ( 29 = u v uuuuur uuuur g for a vectorial function A C t A A t ( 29 { {- equivalent to equilibrium non equilibrium C t Q ( 29 it doesn't matter the model used to find , once we find it, we know all properties of NON-EQUILIBRIUM systems C t ( 29 there is ONE equilibrium condition Q there are many non-equilibrium processes many ONE C t ( 29 Gral. behavior of ? C t ( 29 initial conditions: equilibrium transport non-equilibrium v ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = = = uuuur uuuur g uuuur uuuur g at 0, 0 0 t C C f dp dq v v v v = = = equil. at equilibrium, is given by Maxwell-Boltmann distribution 3 1 for an ideal gas and 2 2 3 trans trans E kT E m kT m 2 v v v distribution of velocities ( 29 = = 3 kT C m 2 v ( 29 ( 29 as evolves, the particle will collide and , will change. After many collisions, there'll be between initial conditions 0 anf fi no corr nal con elati ditions t on t p q v v ( 29 = 0 for t C t ( 29 for thermal processes, (diffusion, viscosity,etc) X C t dt thermodynamic property thermodynamic property ( 29 ϖ ϖ- for perturbation by an electricl field,...
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This note was uploaded on 05/29/2011 for the course CHM 6461 taught by Professor Bowers during the Spring '08 term at University of Florida.

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Lecture 22 - i j Phase space: an Euclidean space based on q...

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