Boltzmann equation and H-Theorem Pavel Popov May 29, 2020 Seminar on statistical physics - Lecturer: Prof. Dr. Georg Wolschin - Summer term 2020 1 / 37 Outline Introduction Microscopic description Density function Liouville equation BBGKY hierarchy and the Boltzmann equation Reduced distribution functions Grad’s derivation of the Boltzmann equation Stoßzahlansatz Boltzmann collision term Binary collisions Hard sphere molecules Cross section Symmetries of the collision term From microscopic to macroscopic physics Hydrodynamic equations H-Theorem Equilibrium distribution 2 / 37 Microscopic description I Generalized coordinates and conjugate momenta: qi , pi , i = 1, ..., N. In total: 6N variables. I Hamiltonian of the system H({q}, {p}). Equations of motion: q˙ = ∂H and p˙ = − ∂H for i = 1, ..., N. I i ∂pi i ∂qi Postulate of both equilibrium and nonequilibrium statistical mechanics All intensive macroscopic properties of a given system can be described in terms of the microscopic state of that system. 3 / 37 Density function I Number of points in the phase space goes to infinity. I The distribution of points throughout the phase space becomes continuous. I Describe this with a density function FN ({q}, {p}, t). FN should be symmetric in the qi and the pi (similarity of molecules). N R Y I Normalisation: dqi dpi FN ({q}, {p}, t) = 1. i=1 An ensemble average of a macroscopic property G({q}, {p}) can be defined as: N Z Y hG(t)i := dqi dpi G({q}, {p})FN ({q}, {p}, t) (1) i=1 4 / 37 Liouville equation I How does FN change with time? dF ∂F N ∂F ∂F N = N + X N · q˙ + N · p˙ = 0. (2) dt ∂t ∂q i ∂p i i=1 i i Idea of the proof: Use uniqueness of the solutions of the Hamilton equations. I If FN - constant along a trajectory in the phase space, it’s also every function of FN . Of specific interest will be the function R H(t) := Γ dxFN ln FN = const. 5 / 37 Introduction Microscopic description Density function Liouville equation BBGKY hierarchy and the Boltzmann equation Reduced distribution functions Grad’s derivation of the Boltzmann equation Stoßzahlansatz Boltzmann collision term Binary collisions Hard sphere molecules Cross section Symmetries of the collision term From microscopic to macroscopic physics Hydrodynamic equations H-Theorem Equilibrium distribution 6 / 37 Reduced distribution functions I We define the R-particle distribution function as follows: Z FR (x1, ..., xR , t) := dxR+1...dxN FN (x1, ..., xR , ..., xN , t) (3) I We also define the mass density as f (x1, t) := NmF1(x1, t), where m is the mass of a single particle. I How do the evolution equations for the reduced distribution functions look like? 7 / 37 I We start with the case R = N, where we have the Louiville equation from above: ∂F N ∂F p ∂F N + X N · i + N · F = 0, (4) ∂t ∂q m ∂p i i=1 i i where F i = p˙ i is the force, acting on the i-th particle. I For an arbitrary R we can integrate Eq.4 over the phases xR+1, ..., xN . We have: Z ∂F ∂ Z ∂F dx ...dx N = dx ...dx F = R (5) R+1 N ∂t ∂t R+1 N N ∂t Z N p ∂F R p ∂ Z dx ...dx X i · N = X i · dx ...dx F R+1 N m ∂q m ∂q R+1 N N i=1 i i=1 i Z N p ∂F + dx ...dx X i · N R+1 N m ∂q i=R+1 i (6) 8 / 37 The second term on the R.H.S. in Eq.6 can be made vanish, even if the volume of the system is finite, for example, when there are only elastic collisions of particles with the walls of the container. In that case, we have: FN (q1, ..., qN , p1, ..., pi , ..., pi,x , pi,y , pi,z , t) =FN (q1, ..., qN , p1, ..., pi , ..., −pi,x , pi,y , pi,z , t), ∀i : qi ∈ dS ⊥ x (7) and thus, when integrated over pi,x , the second term on the R.H.S. of Eq.6 becomes zero. We have: Z N p ∂F R p ∂F dx ...dx X i · N = X i · R (8) R+1 N m ∂q m ∂q i=1 i i=1 i 9 / 37 For conservative central intermolecular forces, we have: N ∂φ F = − X ij (9) i ∂q j=16=i i and the last term on the R.H.S. of the Liouville equation for FN , when integrated as above, becomes: Z N ∂φ ∂F R ∂φ ∂ Z − dx ...dx X ij · N = X ij · dx ...dx F R+1 N ∂q ∂p ∂q ∂p R+1 N N i,j=1 i i i,j=1 i i N Z X ∂φij ∂FN − dxR+1...dxN · 1≤i≤R ∂qi ∂pi R+1≤j≤N N Z X ∂φij ∂FN − dxR+1...dxN · R+1≤i≤N ∂qi ∂pi 1≤j≤N (10) 10 / 37 The last term on the R.H.S. vanishes. The second term can be written as: Z R ∂φ ∂F −(N − R) dx X iR+1 · R+1 (x , ..., x , t) (11) R+1 ∂q ∂p 1 R+1 i=1 i i When we combine all of the results, we get the equation for FR : ∂F R p ∂F R ∂φ ∂F R + X i · R − X ij · R ∂t m ∂q ∂q p i=1 i i,j=1 i i (12) Z R ∂φ ∂F =(N − R) dx X iR+1 · R+1 . R+1 ∂q ∂p i=1 i i Equation 12 is called the BBGKY (Bogoliubov, Born, H.S. Green, Kirkwood and Yvon) hierarchy of equations. 11 / 37 Boltzmann gas limit The Boltzmann equation is exact in the so called Boltzmann gas limit (BGL). This is described by three conditions: I the density is sufficiently low so that only binary collisions need to be considered I the spatial dependence of gas properties is sufficiently slow so that collisions can be thought of as being localised in the physical space I the interparticle potential is of sufficiently short range so that the first statement is meaningful. 12 / 37 Conditions on the parameters of the gas This means, for the particle number N, the mass of each particle m and for a parameter σ which characterises the range of the interparticle forces (σ2 would be then the cross section): N −→ ∞ m −→ 0 σ −→ 0 (13) Nσ2 = const Nm = const Note that this limit also describes ideal gas, because the total volume of the particles Nσ3 → 0. 13 / 37 Grad’s derivation Consider the truncated distributions: Z σ F1 := dy2...dyN FN D1 Z (14) σ F2 := dy3...dyN FN , D2 where Di is the part of the physical space where |qj − q1| ≥ σ for j = i, ..., N is fulfilled. σ σ In the BGL we can replace F1, F2 by F1 , F2 in the Liouville equation and we get: σ σ I ∂F1 ∂F1 σ + v1 · +(N − 1) dv2dS2 · (v1 − v2)F2 ∂t ∂q1 S2 Z (15) ∂ 0 σ −(N − 1) · dy2φ12F2 = 0, ∂v1 |q2−q1|>σ where S2 is the surface of the sphere |q2 − q1| = σ and φ0 := ∂φ12 . 12 ∂q1 14 / 37 If V = v2 − v1 is the relative velocity, one can introduce a plane + perpendicular to it with origin q1 so that the two half-spheres S2 − and S2 are projected onto a disc with area element dω = rdrdϕ. We exchange the integration over the sphere in Eq.15 for integration over the disc: Z ∂F1 ∂F1 + − +v1 · = N dωdv2V [F2(y1, y2 , t)−F2(y1, y2 , t)] (16) ∂t ∂q1 15 / 37 Now, we use the Stoßzahlansatz (collision number assumption), due to Boltzmann, which states that: F2(y1, y2, t) = F1(y1, t)F1(y2, t) (17) to get the final Boltzmann equation: ∂f ∂f + v1 · ∂t ∂q1 1 Z = dωdv V [f (q , v¯ , t)f (q , v¯ , t) − f (q , v , t)f (q , v , t)], m 2 1 1 1 2 1 1 1 2 (18) where we denoted with v¯1, v¯2 the velocities after the collision. Eq.18 is called the Boltzmann equation. The R.H.S. is a functional of f , which is called the collision term: 1 Z J[f ] := dωdv V [f¯ f¯ − f f ]. (19) m 2 1 2 1 2 16 / 37 A motivation for the assumptions The total number of molecules, which can target the surface element dS, is: 2 σ dω|V · e|∆t · f (q1, v2, t)dv2. (20) There are f (q1)dq1dv1 particles with velocity v1. The total number of binary collisions per given time interval and per dx1 , are: Z 2 ∼ σ |V · e|f (q, v1, t)f (q1, v2, t)dωdv2. (21) This consideration gives us a motivation for the loss term. The gain term is motivated analogously. 17 / 37 Introduction Microscopic description Density function Liouville equation BBGKY hierarchy and the Boltzmann equation Reduced distribution functions Grad’s derivation of the Boltzmann equation Stoßzahlansatz Boltzmann collision term Binary collisions Hard sphere molecules Cross section Symmetries of the collision term From microscopic to macroscopic physics Hydrodynamic equations H-Theorem Equilibrium distribution 18 / 37 Binary collisions Now we want to investigate the collision term. In classical binary collisions there is momentum and energy conservation: v1 + v2 = v¯1 + v¯2 2 2 2 2 (22) v1 + v2 =v ¯1 +v ¯2 . One can easily show that the modulus of the relative velocity doesn’t change. We have V = V¯ . We can also establish an explicit relation between the old and the new velocities.