phys212ln4 - Physics 212 Statistical mechanics II Fall 2005...

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Physics 212: Statistical mechanics II, Fall 2005 Lecture IV : BBGKY, conservation laws of Boltzmann eqn. Our program for kinetic theory in the last lecture and this lecture can be expressed in the following series of steps, from most exact and general to most specific and approximate (but also useful!): Liouville’s theorem BBGKY hierarchy Boltzmann eqn. Navier-Stokes eqn. (1) The first step, from Liouville’s theorem to the BBGKY hierarchy, will be completed in this lecture. The only assumption is that the system being dealt with is a gas with binary collisions, and the BBGKY hierarchy is still exact for this specific system. We will say a bit about the approximations required to get from the BBGKY hierarchy to the closed Boltzmann equation for f 1 toward the end of this lecture, and start deriving the Navier-Stokes equation in the next lecture. Let’s review what we learned last time. Suppose we have a problem of N particles moving in the Hamiltonian (switching back to the usual variables ( r, p ) rather than the generalized conjugate variables ( q, p )) H = N i =1 p 2 2 m + N i =1 U i + i<j v ij U i = U ( r i ) , v ij = v ji = v ( | r i - r j | ) . (2) One of the main results from last lecture is Liouville’s equation for the time evolution of g : ∂g ∂t + Nd i =1 ˙ p i ∂g ∂p i + ˙ r i ∂g ∂r i = 0 . (3) For the specific case of the dilute gas with binary collisions, Liouville’s equation becomes (where F = -∇ U , K ij = -∇ r i v ij ) ∂g ∂t = N i =1 - F i - N j =1 ,j = i K ij · ∇ p i g - p i m · ∇ r i g. (4) Here g is the distribution function in N -particle phase space. For compactness, in most of this lecture we will suppress the time and sometimes other variables appearing in g , when the meaning is clear. Now rewrite Liouville’s equation in the form ∂t + h N ( r 1 , p 1 , . . . , r N , p N ) g = 0 , (5) where the differential operator h N is defined as h N ( r 1 , p 1 , . . . , r N , p N ) = N i =1 p i m · ∇ r i + F i · ∇ p i 1
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+ 1 2 N i,j =1 K ij · ( p i - ∇ p j ) . (6) Our goal for most of this lecture will be to find an exact equation for the evolution of the n -body distribution functions f n by doing a partial integration of Liouville’s equation. The definition of the 1-body distribution function f 1 ( z 1 ), with z = ( p , r ), is f 1 ( z 1 ) = n i =1 δ ( z 1 - z i ) = N g ( z 1 , . . . , z n ) δ ( z 1 - z 1 ) dz 1 . . . dz N = N g ( z 1 , . . . , z n ) dz 2 . . . dz N . (7) The first equality is the definition of f 1 (previously just called f ). The second equality depends on having g be symmetric under interchange, as it should be if it describes identical particles; here the normalization factor N arose because the sum over i in the second form contains N identical terms. That is, the one-body distribution function f 1 is a particle density (normalized to N ), while the function g is normalized to 1 over all of phase space. This choice of normalization may seem strange, but it will make the BBGKY equation derived below simpler.
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