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CHAPTER 2. KINETIC THEORY 2.3 33 Lecture 7: Binary collisions 1. Without collisions: (a) A neighborhood dqdp moves to dqÕ dpÕ = dqdp since for Hamiltonian systems the flow is incompressible. (b) A molecule at (q, p, t) moves to (q + ”t, i.e. f (q + or p ”t, p m + F”t, t + ”t) in time p ”t, p + F”t, t + ”t)dqÕ dpÕ = f (q, p, t)dqdp m f (q + p ”t, p + F”t, t + ”t) = f (q, p, t) m (2.32) (2.33) 2. With collisions: A p ˆf f (q + ”t, p + F”t, t + ”t) = f (q, p, t) + m ˆt B ”t (2.34) coll By expanding the left hand site in powers of ”t and keeping only linear terms we get the transport equation ˆ p ˆ ˆ f (q, p, t) + · f (q, p, t) + F · f (q, p, t) = ˆt m ˆq ˆp 1 A ˆf ˆt B (2.35) coll 2 which is not very useful unless ˆf is specified. Of course, finding or modˆt coll eling the collision term is the biggest challenge in the kinetic theory. In the simplest model one only takes into account binary collisions and assumes that the colliding particles are uncorrelated (i.e. molecular chaos assumption). Consider an elastic collision of two spherically symmetric (spin-less) molecules with m1 , p1 and m2 , p2 . After collision their respective momenta are pÕ1 and pÕ2 . Then the following conservation laws apply: • Momentum conservation: • Energy conservation: p1 + p2 = pÕ1 + pÕ2 . (2.36) p21 p2 pÕ2 pÕ2 + 2 = 1 + 2. m1 m2 m1 m2 (2.37) CHAPTER 2. KINETIC THEORY 34 Define: • Reduced mass µ© • Relative momentum p© m1 m2 m1 + m2 3 m1 p1 + m2 p2 p1 p2 =µ + m1 + m2 m1 m2 (2.38) 4 (2.39) Then the conservation of energy can be written as |p| = |pÕ | , (2.40) i.e. collision simply rotates relative momentum. Such rotations can be specified by a unit vector in spherical coordinates (◊, „), where • inclination angle ◊ between p and pÕ • azimuthal angle „ of pÕ about p. The dynamical aspect of the collisions are described by differential cross section |d‡/d | - the cross-sectional area which scatters particles into solid angle d around . For scattering of two spheres of diameter D with impact parameter b the scattering angle is given by cos A B ◊ 2 = b D (2.41) and thus, the cross-sectional area is A B A B ◊ ◊ D2 D2 d‡ = bd„db = D cos D sin d◊d„ = sin ◊d◊d„ = d 2 2 4 4 (2.42) and d‡ D = d 4 The total cross-section is given by ‡total = which ‡total = ⁄ ⁄ -- d‡ -d -d - D2 d = fiD2 4 (2.43) (2.44) (2.45) CHAPTER 2. KINETIC THEORY 35 for an elastic scattering of hard spheres. In quantum mechanics the scattering probabilities Pp1 ,p2 æpÕ1 ,pÕ2 per unit time per unit volume are expressed through scattering amplitudes Ap1 ,p2 æpÕ1 ,pÕ2 using Born rule, - -2 - Pp1 ,p2 æpÕ1 ,pÕ2 = ”(p1 +p2 ≠pÕ ≠pÕ ) ”(p2 +p2 ≠pÕ2 ≠pÕ2 ) -Ap1 ,p2 æpÕ1 ,pÕ2 1 2 1 2 1 2 2 = ”(p1 +p2 ≠pÕ ≠pÕ ) ”(p2 +p2 ≠pÕ2 ≠pÕ2 ) |ÈpÕ1 , pÕ2 |T|p1 , p2 Í| (2.46) 1 2 1 2 1 2 where T is the quantum mechanical transition matrix and all of the factors of (2fi) where absorbed into definition of Ap1 ,p2 æpÕ1 ,pÕ2 . If we assume that the interactions are invariant under spatial rotations, reflections and time reversal (e.g. electromagnetic interactions), then the transition matrix T has the following symmetries ÈpÕ1 , pÕ2 |T|p1 , p2 Í = ÈRpÕ1 , RpÕ2 |T|Rp1 , Rp2 Í ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠p1 , ≠p2 |T| ≠ pÕ1 , ≠pÕ 2 Í, (2.47) ÈpÕ1 , pÕ2 |T|p1 , p2 Í = Èp1 , p2 |T|pÕ1 , pÕ2 Í. (2.48) where R is a matrix representing spatial rotations and/or reflections. Note that the vector p can also include spin d.o.f., then R rotates the spatial and spin d.o.f., but reflects only spatial d.o.f. It follows form (2.47) that This can be shown by: • time reversal ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠p1 , ≠p2 |T| ≠ pÕ1 , ≠pÕ2 Í, • rotation on angle fi around any axis perpendicular to total momentum, ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠R1 pÕ1 , ≠R1 pÕ2 |T| ≠ R1 p1 , ≠R1 p2 Í, • reflection to complete the interchange the initial and final relative momenta, ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠R2 R1 pÕ1 , ≠R2 R1 pÕ2 |T|≠R2 R1 p1 , ≠R2 R1 p2 Í = Èp1 , p2 |T|pÕ1 , pÕ2 Í, One can apply the ideas of kinetic theory to other systems with a large number of d.o.f. For example, a very large network of cosmic strings can be described by a transport equation analogous to the Boltzmann transport equation for particles. Question to Go: Think of a system with a large number of d.o.f. for which the ideas of the kinetic theory might be useful.