phys212ln2 - Physics 212 Statistical mechanics II Fall 2006...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture II The most famous example of how entropy increases in a real system is the dilute classical gas. The first assumption we will make is that λ = na 3 1 , (1) or that the “gaseousness parameter” is small . Here n is the density of particles per unit volume and a is the range of the interparticle interaction: λ is just the number of particles within an interaction range of a fixed particle. For hard spheres, a is just the sphere radius. For Van der Waals-type forces between neutral atoms or molecules, the interaction potential falls off sufficently rapidly (usually V ( r ) r - 6 ) that our approximations are still well justified. One reason why plasma physics is difficult is that there the interparticle potential is given by the rather strong Coulomb force, V ( r ) r - 1 , so that the physics is quite complicated and the Debye screening must be taken into account. The second assumption is that the particles lack internal rotational or vibrational struc- ture . At room temperatures, this is a reasonable assumption for the vibrational levels of many gases. Rotations may be quite low energy, but in practice molecules often rotate so fast (i.e., the rotational levels are at such low energy) that they are simply averaged over on the intercollision time scale. Now we need a model for how the particles in the gas collide. Let Γ 1 , Γ 1 , Γ 2 , and Γ 2 be the momentum variables of two incoming and two outgoing particles from a collision: Γ = ( p ). We write Γ instead of p so that the formulas can be generalized to cases where internal angular momenta, for example, are conserved in collisions as well as normal momenta; then Γ = ( p , L , . . . ). We consider collisions that take Γ 1 , Γ 2 to Γ 1 , Γ 2 . Then define w 1 , Γ 2 ; Γ 1 , Γ 2 ) so that the rate per spatial volume at which pairs of particles from Γ 1 , Γ 2 get transferred via collisions to Γ 1 , Γ 2 is R = ( f ( t, r , Γ 1 ) d Γ 1 )( f ( t, r , Γ 2 ) d Γ 2 ) w 1 , Γ 2 ; Γ 1 , Γ 2 )( d Γ 1 d Γ 2 ) . (2) In the above we ignored the position variables in w and are assuming that the collision process is translation invariant. There will be a bit more said about the assumptions which go into the definition of f below. The picture is that we are obtaining the collision function w from some microscopic physics like quantum mechanics. That physics will probably be time-reversal invariant , in which case w 1 , Γ 2 ; Γ 1 , Γ 2 ) = w ((Γ 1 ) T , 2 ) T ; (Γ 1 ) T , 2 ) T ) . (3) For the monatomic gas to be considered below, which has no additional angular momentum or other internal variables so Γ T = ( - p
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