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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture V In the previous lecture we finished the derivation of the BBGKY hierarchy and started considering conservation laws in the Boltzmann equation. These conservation laws will enable us to derive the hydrodynamic equations for a dilute gas, completing the kinetic theory program Liouville’s theorem → BBGKY hierarchy → Boltzmann eqn. → Fluid mechanics (1) Then we turn briefly to physics applications of the Navier-Stokes equation, and then move on to local response theory and fluctuation-dissipation theorems. Derivation of first-order hydrodynamics The local conservation law we derived last time is Z χ ( r , p ) ∂ ∂t + p m · ∇ x + F · ∇ p f ( t, r , p ) d p . (2) This came from showing that the collision term in the Boltzmann equation, considered alone, preserves the conservation law: Z χ ( r , p ) ∂f ∂t coll d p = 0 . (3) We can rewrite the statement of local conservation (2) in the form ∂ ∂t Z χf d p + ∇ x · Z χ p m f d p- Z ∇ x χ · p m f d p + Z ∇ p ( χ F f ) d p- Z ∇ p χ · F f d p- Z χf ( ∇ p · F ) = 0 . (4) Here we have used the time-independence of χ in the first term. The fourth term integrates to a boundary term, which we take to be zero for a bounded system. The remaining terms can be written in a simple form by introducing the notation hi for mo- mentum averages: h A i = R Af d p R f d p = R Af d p n ( r , t ) . (5) Note that, since n ( r , t ) is by definition momentum-independent, h nA i = n h A i . We arrive at the general conservation law (switching from momentum to velocity), ∂ ∂t h nχ i + ∇ x · h n v χ i- n h v · ∇ x χ i- n m h F · ∇ v χ i- n m h χ ∇ v · F i = 0 . (6) The last term will be zero if F is independent of velocity, as in the cases we consider. The idea of this law is very similar to that of the local conservation of systems used in the proof of Liouville’s theorem: any change in the energy or momentum in a volume must come from streaming of particles into or out of the volume, since collisions within the volume do not change the total energy or momentum. 1 Now we can simply obtain transport equations for various properties from the conservation law. Let χ = 1 (particle number conservation). Then ∂n ∂t + ∇· h n v i = 0 , (7) where ∇ henceforth denotes the spatial gradient. Defining u = h v i as the mean velocity, this becomes ∂n ∂t + ∇· ( n u ) = 0 , (8) the continuity equation. Now let χ be the momentum m v . Then, with ρ ≡ mn , ∂ ∂t h ρv i i + ∂ ∂x j h ρv i v j i- nF i = 0 . (9) We can separate the quadratic velocity term here by using the definition of the mean velocity above: h v i v j i = h ( v i- u i )( v j- u j ) i + h v i i u j + h v j i u i- u i u j = h ( v i- u i )( v j- u j ) i + u i u j . (10) We define the pressure tensor P ij as P ij = ρ h ( v i- u i )( v j- u j ) i . (11) To check that this definition makes sense, you may want to check that each component of...
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