Lectures Notes on Boltzmann's Equation

Lectures notes on Boltzmann's equation Simone Calogero 1 Introduction Kinetic theory describes the statistical evolution in phase-space1 of systems composed by a large number of particles (of order 1020). The main goal of kinetic theory, as far as the physical applica- tions are concerned, is to predict the evolution of those quantities associated to the system which depend only on the local average dynamics of the particles. These are called macroscopic quantities. The most important examples of physical systems to which kinetic theory applies are dilute gases (where the molecules play the role of the particles) and in this case examples of macroscopic quantities are the temperature and the pressure of the gas. In fact, the temperature of a gas is a measure of the mean kinetic energy of the molecules in small regions of the gas, while the pressure is a macroscopic manifestation of the exchange of momentum between the particles and the walls of the gas container (of with any body inserted into the gas). The relation between kinetic and macroscopic quantities will be discussed in Section 7. The statistical description of the dynamics is given in terms of the one particle distribution function, denoted by f, which is a function of time, position and velocity (or momentum2), that is 3 3 f = f(t; x; v); t ≥ 0 ; x 2 R ; v 2 R : By definition, f(t; x; v) is the probability density to find a particle at time t, in the position x, with velocity v. Thus the integral Z Z f(t; x; v) dv dx V Ω gives the probability to find a particle in the region V at time t with velocities v 2 Ω. Since the number of particles is so large, the above quantity can also be interpreted as the relative number of particles in the region V at time t with velocities v 2 Ω. Moreover, denoting by M the total mass of the system, the above integral multiplied by M can be interpreted as the total mass of the system in the region V . We shall use freely all these interpretations of the function f, which lead naturally to require that 1 3 3 0 ≤ f(t; ·; ·) 2 L (R × R ) and that kf(t)k1 = const = 1. The Boltzmann equation is a (integro-)partial differential equation for the one particle distribution function f. Let us \derive" it in the most simple case, that is for a system of particles moving with constant velocities. Suppose (for notational simplicity) that all particles are identical (otherwise we should introduce a particles distribution for each species). If x(t) denotes the position vector at time t and v the (constant) velocity vector of a particle, the position at time t + will be given by x(t + ) = x(t) + v : 1That is the manifold of all possible positions and velocities 2In non-relativistic mechanics, the momenum and velocity of a particle differ only by a multiplicative constant| the mass of the particle. 1 Now if instead of knowing the exact position of the particle at time t we only know the probability f(t; x(t); v) to find the particle there, it is natural to assume that f(t + , x(t + ); v) = f(t; x(t); v) ; since we know with certainty that the particle has moved with constant velocity v in the -interval of time. Provided f is a sufficiently smooth function, the above identity is equivalent to the free transport equation @tf + v · rxf = 0 : (1) The solution of (1) with initial data f(0; x; v) = fin(x; v) is given by f(t; x; v) = fin(x − vt; v). Note that fin ≥ 0 implies f(t; x; v) ≥ 0 and that kf(t)k1 = kf0k1. This is consistent with the interpretation of f as a probability density (i.e., of f(t; x; v)dvdx as a probability measure). Next we make the assumption that in the (infinitesimal) interval of time [t; t + ] the particle undergoes a collision with the other particles of the system. Here the word \collision" is used to refer to a general interaction which takes place in such a short time interval and small region of space that we can safely say that it occurs at time t in the position x. If the result of the collision is to increase the number of particles with velocity v, equation (1) modifies to @tf + v · rxf = G: Here G = G(t; x; v) ≥ 0 is called the gain term and gives the probability density that a new particle with velocity v is gained after the collision. Likewise, if a particle with velocity v is lost after the collision with probability L, we have @tf + v · rxf = −L: The function L = L(t; x; v) ≥ 0 is called loss term and the minus sign indicates that it makes f decrease (obviously, if a particle looses velocity v, the probability to have a particle with this velocity after the collision will be smaller). In general we have @tf + v · rxf = G − L: (2) Hence the Boltzmann equation is a balance equation: It says us how the particle distribution f changes as a consequence of the collision. In order to give an explicit form to G and L we have to give more information on how collisions occur. We shall consider only the case of the Boltzmann equation for binary elastic collisions. Exercise 1. Consider a system of particles moving under the influence of an external force field F = F (t; x). What should the equation for the one particle distribution function be in this case? Prove that the resulting equation (Vlasov equation) is consistent with the interpretation of the solution as a probability density (i.e., the solution is non-negative and its L1 norm is constant). 2 The Boltzmann equation for binary elastic collisions Binary elastic collisions means that we take into account only collisions between pair of particles (binary), i.e. the simultaneous collisions between three or more particles are assumed to occur with negligible probability. Moreover the collisions are assumed to be elastic, meaning that, besides the total momentum of the particles, also the total kinetic energy is conserved during the collision3. 3If the kinetic energy is not conserved, the collision is called inelastic. 2 0 0 Let v; v∗ denote the velocities before the collision of two particles and v ; v∗ their velocities after the collision (see Fig. 1). We also assume the particles to have the same mass (which is also conserved in the collision). Then the conservation laws of momentum and kinetic energy take the form 0 0 v + v∗ = v + v∗ (cons. of momentum) ; (3) 2 2 0 2 0 2 jvj + jv∗j = jv j + jv∗j (cons. of kinetic energy) : (4) Suppose the pre-collisional velocities (v; v∗) are given and we want to derive the post-collisional 0 0 velocities (v ; v∗) from (3) and (4). Since we have 6 unknowns but only 4 equations, the above system is underdetermined. However one can prove that the manifold of the solution is a 2-sphere. Precisely we have the following 0 0 Lemma 1. A quadruple (v; v∗; v ; v∗) solves (3)-(4) if and only if 0 v = v − [(v − v∗) · !]!; (5) 0 v∗ = v∗ + [(v − v∗) · !]!; (6) for some ! 2 S2. 0 0 Proof. The claim is true for the trivial solution (v = v, v∗ = v∗) (since it is obtained for ! 0 0 orthogonal to v − v∗). Otherwise, set v∗ = v∗ + a !, v = v + b !; from the conservation of 0 0 momentum, equation (3), we have a = −b. Substituting v∗ = v∗ + a ! and v = v − a ! in (4) and solving for a we obtained the desired result with 0 0 v∗ − v∗ v − v ! = 0 = − 0 : jv∗ − v∗j jv − vj 3 The triple (v; v∗;!) is called a collision configuration. The direction of ! is the scattering direction of the collision. Some important geometrical properties of binary elastic collisions are collected in the following lemma, whose proof is left as exercise. Lemma 2. The following holds: 0 0 0 0 (i) jv − v∗j = jv − v∗j; j(v − v∗) · !j = j(v − v∗) · !j; 0 0 0 0 0 0 (ii) v = v − [(v − v∗) · !]!; v∗ = v∗ + [(v − v∗) · !]!; 0 0 (iii) The Jacobian of the transformation (v; v∗) ! (v ; v∗) is equal to 1. Exercise 2. Prove Lemma 2. The meaning of (ii) is that a binary elastic collision is a reversible process (see Fig. 2). Recall from the Introduction that the gain term measures the probability that a new particle with velocity v results from the collision of two particles. By (ii) of Lemma 2, this is the case if the 0 0 particles collide with velocities v = v − [(v − v∗) · !]! and v∗ = v∗ + [(v − v∗) · !]! (see Fig. 2 0 0 again). The probability to find a particle with velocity v (resp. v∗) in the point x at time t is 0 0 given by f(t; x; v ) (resp. f(t; x; v∗)). The probability to have both particles at the same time in the same position (so that they may collide) is given by the product 0 0 f(t; x; v )f(t; x; v∗) provided we assume that the occurrence of two particles in the same point at the same time with given velocities are two independent events. This is called the molecular chaos assumption. Now if the two particles had always the same probability (say one) to collide independently from their velocities and scattering angle, then the product above would already give the probability density to obtain a new particle with velocity v from the collision of two particles with velocities 0 0 v and v∗.

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