Astrofysikaliska dynamiska processer, VT2008 The equations of fluid dynamics and their connection with the Boltzmann equation Lecture Notes Susanne H¨ofner Department of Physics and Astronomy Uppsala University 1 1 Relation of Kinetic Theory to Fluid Mechanics 1.1 Distribution Function and Boltzmann equation Distribution Function To describe the state of the gas statistically, we define the distribution function f(x; v; t) such that f d3x d3v is the average number of particles contained in a volume element d3x about x and a velocity-space element d3v about v at time t. We demand that f ≥ 0 and that f ! 0 as the components of the particle velocity ! 1, sufficiently rapidly to guarantee that a finite number of particles has a finite energy. Macroscopic properties of the gas can be computed from f(x; v; t), e.g.: 1 n(x; t) = f(x; v; t) d3v (1) −∞ Z ρ(x; t) = m n(x; t) (2) 1 u(x; t) = n−1 f(x; v; t) v d3v = hvi (3) −∞ Z i.e. the number density of the particles n, the mass density ρ (where m is the mass of a single particle) and the average velocity of an element of gas u (= macroscopic flow velocity). To study macroscopic properties of the gas it is useful to decompose the particle velocity v into the average velocity u and a random velocity relative to the mean flow w: v = u + w (4) Boltzmann Transport Equation We now search for an equation describing the changes of f(x; v; t) with time. Consider a group of particles located in a phase-space volume element (dx0; dv0) around (x0; v0). We assume that an external force F(x; t) is acting on the particles so that they experience an acceleration a(x; t) = F(x; t)=m. For the moment, we ignore collisions (and interactions) between the particles. The phase-space element will evolve into (dx; dv) around (x; v) where x = x0 + v0 dt (5) v = v0 + a dt (6) and the phase volume of the new element is related to the original one by 3 3 3 3 d x d v = J(x; v=x0; v0) d x0 d v0 (7) with J denoting the Jacobian of the transformation (note that the components of x and v are independent sets of variables). It can be easily shown that J = 1 + O(dt2). Consequently, to first order in dt the volume of the phase-space element remains constant, i.e. 3 3 3 3 d x d v = d x0 d v0: (8) 2 The number of particles inside the original phase-space element is 3 3 δN0 = f(x0; v0; t0) d x0 d v0: (9) In the absence of collisions all these particles must end up in the new element and therefore δN0 = δN where 3 3 3 3 δN = f(x; v; t) d x d v = f(x0 + v0 dt; v0 + a dt; t0 + dt) d x d v: (10) Together with Eq. 8 this implies that f(x0 + v0 dt; v0 + a dt; t0 + dt) = f(x0; v0; t0) (11) i.e. in the absence of collisions the phase-space density of a group of particles is invariant. By expanding to first order in dt we find @f @f @f + vi + ai = 0 (12) @t @xi @vi which is known as the collisionless Boltzmann equation, or Vlasov's equation. When particles suffer a collision during dt their velocities will be changed and they will not, in general, end up in the velocity element (dv) centered around v but in a different velocity element (dv)0 centered around v0 with v0 =6 v. On the other hand, particles not contained in the original element (dv0) may end up in (dv) due to collisions. Therefore, f is no longer invariant. To take these processes into account we have to add a term on the right-hand side of Eq. 12 which gives the net rate at which particles are entering into Df the phase-space element under consideration. We write this term symbolically as Dt coll and obtain the so-called Boltzmann transport equation @f i @f i @f Df + v i + a i = : (13) @t @x @v Dt !coll Note that f is a one-particle distribution function, i.e. we assume that the probability of finding a particle at a particular point in phase space is independent of the coordinates of all other particles in phase space. Such a description is only valid for a dilute gas. The most general treatment of a system of N particles involves using a N-particle distribution function which gives the joint probability of finding the N particles in a given 6-N- dimensional phase-space element. The time evolution of this function is described by Liouville's equation. Eq. 13 can be derived as a special case from this general approach. 1.2 The Collision Integral Eq. 13 is of little use until we specify the right-hand side. In this section we will derive an explicit expression. 3 Dynamics of Binary Collisions We consider elastic collisions between two classic point particles (no internal degrees of freedom, non-relativistic regime). Such a collision is subject to momentum conservation 0 0 m1v1 + m2v2 = m1v1 + m2v2 (14) and energy conservation 1 2 2 1 02 02 (m1v1 + m2v2) = (m1v1 + m2v2 ) (15) 2 2 where m1 and m2 are the masses of the two particles (labeled `1' and `2'), v1 and v2 0 0 are their velocities before the collision, and v1 and v2 their velocities after the collision. 0 0 These two equations together with mass conservation (m1 + m2 = m1 + m2) are known as the summational invariants of the collision. For the description of the collision it is useful to define the relative velocity of the two particles as q21 = v2 − v1: (16) Using momentum and energy conservation it can be shown that 0 q ≡ jq21j = jq21j; (17) i.e. the relative velocity of the two particles is changed only in direction but not in mag- nitude by the collision. Basic Form of the Collision Integral The right-hand side of Eq. 13 is defined as the net rate at which particles are entering into the phase-space element under consideration. Thus, we can write Df = Rin − Rout (18) Dt !coll where Rin and Rout are the rates at which particles are scattered in and out of a given phase-space element due to collisions. 3 3 First, we calculate the rate at which particles of type 1 are scattered out of d x1 d v1. For each particle of type 1 the number of particles of type 2 moving with velocities in the range (v2; v2 + dv2), incident within a range of impact parameters (b; b + db) and 3 within azimuth range dΦ in a unit time is (f2 d v2) q b db dΦ where q = jv2 − v1j and f2 ≡ f(x; v2; t). The total number of collisions is given by integrating over all impact parameters, azimuths and incident velocities, and then multiplying by the number of 3 3 particles of type 1, i.e. f1 d x1 d v1. We obtain 3 3 3 3 3 Rout d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1: (19) Z Z Z 4 Rin can be evaluated by considering the inverse enounters, i.e. collisions where initial 0 0 velocities v1 and v2 are changed to the final values v1 and v2 within the considered phase-space element. We find 3 3 0 0 0 3 0 3 3 0 Rin d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1 (20) Z Z Z 0 0 0 0 0 where f1 ≡ f(x; v1; t) and f2 ≡ f(x; v2; t). It can be shown that q = q and that 3 3 3 0 3 0 d v1 d v2 = d v1 d v2. Therefore we can re-write Rin as 3 3 0 0 3 3 3 Rin d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1: (21) Z Z Z Thus we obtain Df1 0 0 3 = (f1f2 − f1f2) q b db dΦ d v2: (22) Dt !coll Z Z Z Introducing the cross section σ(Ω) dΩ = b db dΦ instead of the impact parameter and azimuth we can write the Boltzmann equation, accounting for binary collisions, as @f1 i @f1 i @f1 0 0 3 1 2 Ω 2 + v1 i + a i = (f1f2 − f f ) q σ( ) dΩ d v : (23) @t @x @v1 Z Z Properties of the Collision Integral Let Q(v1) be any function of the particle velocity v1 and define 0 0 3 3 Df1 3 I ≡ Q(v1) (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2 = Q(v1) d v1: (24) Z Z Z Z Dt !coll If we just interchange the labeling 1 and 2 of the particles then obviously the value of I doesn't change. Adding these two expressions we get 1 0 0 3 3 I = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2: (25) 2 Z Z Z If we replace the collision by its inverse the integral still has the same value because for every collision exist an inverse collision with the same cross section. Therefore we get 1 0 0 0 0 0 0 0 3 0 3 0 I = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω ) dΩ d v1 d v2 (26) 2 Z Z Z 1 0 0 0 0 3 3 = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2 (27) 2 Z Z Z 0 0 0 3 3 3 0 3 0 where we have used that σ(Ω ) = σ(Ω), dΩ = dΩ, q = q and d v1 d v2 = d v1 d v2.