Course Okt 28, Statistische Mechanik plus. Course Hartmut Ruhl, LMU, Munich. Personen. Hinweis. Rationale

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1 Okt 28, 2016

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3 ASC, room A239, phone , Patrick Böhl, ASC, room A205, phone ,

4 Das folgende Material ist nicht prüfungsrelevant. Das folgende Material dient der physikalischen Motivation der Betrachtung des Konzepts der Wahrscheinlichkeitsdichten. Das folgende Material dient der Wiederholung elementarer Konzepte aus der klassischen Vielteilchenmechanik.

5 Mehrteilchensysteme lassen sich mit dem Konzept der Wahrscheinlichkeitsdichten beschreiben. Die Wahrscheinlichkeitsdichte gehorcht der Liouville-Gleichung im klassischen Fall. Für praktische Anwendungen benutzt man das Konzept der reduzierten Wahrscheinlichkeitsdichten. Die dynamischen Gleichungen reduzierter Wahrscheinlichkeitsdichten bilden eine BBGKY-Hierarchie. Aus der BBGKY-Hierarchie lassen sich meist Einteilchenwahrscheinlichkeitsdichten im Gleichgewicht ableiten. In der vorliegenden Vorlesung spielen nur Wahrscheinlichkeitsdichten im Gleichgewicht eine Rolle.

6 In what follows we need the concept of probability density functions. To introduce them let us proceed in a rather intuitive way. A gas of N classical particles is represented in 6N-dimensional phase space. To describe N classical particles the probability density ρ N ( x 1, p 1,..., x N, p N, t) (1) is introduced. The probability of finding particle 1 in the phase space volume element d 3 x 1 d 3 p 1 at x 1, p 1, particle 2 in the phase space volume element d 3 x 2 d 3 p 2 at x 2, p 2 and so on is given by where the normalization condition is d 3N x d 3N p ρ N ( x 1, p 1,..., x N, p N, t), (2) N = d 3N x d 3N p ρ N ( x 1, p 1,..., x N, p N, t). (3) The temporal evolution of the probability density ρ N is governed by the dρ N dt ( x 1, p 1,..., x N, p N, t) = 0. (4) The quantity ρ N is a generalized function or distribution.

7 To derive the we start by assuming that we have complete information about a mechanical system of N particles. Hence, we require that the positions and momenta of all 6N particles are known to us leading to the definition of the following probability density C ( x 1, p 1,..., x N, p N, t) = Π N k=1 δ3 [ x k x k (t)] δ 3 [ p k p k (t)]. (5) The trajectories x k (t) and p k (t) are known. Hence, we have C ( x 1, p 1,..., x N, p N, t) t N [ = Π N k=1,k j δ3 [ x k x k (t)] δ 3 ] [ p k p k (t)] xj (t) δ3 [ x j x j (t)] x j=1 j δ 3 [ p j p j (t)] N [ Π N k=1,k j δ3 [ x k x k (t)] δ 3 ] [ p k p k (t)] δ 3 [ x j x j (t)] p j (t) δ3 [ p j p j (t)]. p j=1 j (6) Making use of m x(t) j δ 3 [ p j p j (t)] = p j δ 3 [ p j p j (t)] (7) we find

8 and C ( x 1, p 1,..., x N, p N, t) t N [ = Π N k=1,k j δ3 [ x k x k (t)] δ 3 ] p j [ p k p k (t)] m δ 3 [ xj x j (t)] x j=1 j δ 3 [ p j p j (t)] N [ Π N k=1,k j δ3 [ x k x k (t)] δ 3 ] [ p k p k (t)] δ 3 [ x j x j (t)] p j δ3 [ p j p j (t)] p j=1 j C N + v j C N + t x j=1 j j=1 (8) p j C p j = 0, (9) The function C is still a generalized function. Averaging C over all initial conditions compatible with an experiment we obtain a smooth probability density function for N particles. We call it ρ N.

9 Given N point particles Γ = (t, q(t), p(t)) defines the set of all points of the system. The space Γ is 6N + 1 dimensional. The variables q(t) and p(t) are each 3N dimensional. We define a density function ρ (q, p, t) : Γ R, where dqdp ρ (q, p, t) is the number of points at time t in the volume dqdp. Assuming that the number of points in the volume dv = dqdp is conserved it must hold t V v a = dv ρ (q, p, t) = (..., q i,..., ṗ j,... ds av a ρ (q, p, t) = V V ), a = In the limit V 0 we obtain the continuity equation (..., qi,..., pj,... dv a (v a ) ρ (q, p, t), (10) ). (11) t ρ (q, p, t) + a (v a ) ρ (q, p, t) = 0, (12) ( t ρ (q, p, t) + q i q i ) ( ρ (q, p, t) + p i ṗ i ) ρ (q, p, t) = 0. (13) Let H (q, p, t) be the Hamiltonian of the system then we have q i = p i H, ṗ i = q i H. (14) Making use of Eqs. (14) leads to t ρ + q i ) ) ( p i H ρ p i ( q i H ρ = 0. (15)

10 Further simplification leads to ) ) ) ) t ρ + ( p i ( H q i ρ ( q i ( H p i ρ = 0. (16) Introducing the Poisson bracket we obtain the ) ) ) ) t ρ = {H, ρ}, {H, ρ} = ( q i ( H p i ρ ( p i ( H q i ρ. (17)

11 The reduced s-particle distribution function is defined as f (s) N! (1...s) = (N s)! dq s+1 dp s+1... dq N dp N ρ (1...N). (18) We assume that the Hamiltionian H can be written as N N H (q, p, t) = T i + V ij. (19) i=1 i=1,j>i

12 We split the Hamiltonian into the following parts We define s s N N s N H (q, p, t) = T i + V ij + T i + V ij + V ij. (20) i=1 i=1,j>i i=s+1 i=s+1,j>i i=1 j>s H (q, p, t) = H s (q, p, t) + H r (q, p, t) + V I (q, p, t), (21) s s H s = T i + V ij, (22) i=1 i=1,j>i N N H r = T i + V ij, (23) i=s+1 i=s+1,j>i s N V I (q, p, t) = V ij. (24) i=1 j>s We obtain { t ρ = H s + H r + V I }, ρ. (25)

13 BBGKY hierarchy Partial integration yields It can shown Please proof this as an exercise. We finally obtain { } (N s)! t f (s) H (s) (N s)!, f (s) N! N! = dq s+1 dp s+1... dq N dp N { H r + V I }, ρ. (26) dq s+1 dp s+1... dq N dp N { H r }, ρ = 0. (27) { } (N s)! t f (s) H (s) (N s)!, f (s) N! N! = dq s+1 dp s+1... dq N dp N { V I }, ρ. (28) If all particles are equal and cannot be distinguished from each other we find t f (s) { H (s), f (s)} = s dq s+1 dp s+1 { V is+1, f (s+1)}. (29) i=1

14 A classical gas consisting of N identical hard with diameter σ is a sufficiently simple system. Since the do not interact as long as they are sufficiently far apart from each other the Liouville equation for the problem is given by t ρ N + N v i xi ρ N = 0, x i x j > σ, i, j {1,..., N}, i j. (30) i=1 The interaction of the is a contact interaction. It amounts to the formulation of appropriate boundary conditions, whenever two touch each other. We find ρ N (... x i, p i... x j, p j ) ( )... t c + ɛ = ρ N... x i, p i... x j, p j... t c ɛ, (31) p [ ] i = p i n ij nij ( p i p j ), p [ ] j = p j + n ij nij ( p i p j ), (32) v i = p i m n ij = x i x j x i x j (33) (34) at x i x j = σ n ij for i, j {1,..., N}, i j at collision time t c. The x i, v i, p i, i {1,..., N} are the pre-collision positions, velocities, and momenta of the. The time t c ɛ is the pre-collision time. The x i, v i, p i, i {1,..., N} are the post-collision positions, velocities, momenta and t c + ɛ the post-collision time. At the box boundaries ρ N is assumed to disappear. This is the case if we assume mirror reflection for the there.

15 For hard it can be shown that a hierarchy of equations for so called reduced probability density functions is obtained. The hierarchy is called the BBGKY hierarchy (Bogoliubov Born Green Kirkwood Yvon hierarchy). It is given by t f (s) ( x 1, p 1,..., x s, p s, t) + = s v i xi f (s) ( x 1, p 1,..., x s, p s, t) (35) i=1 s d 3 p s+1 ds is+1 n is+1 ( v i v s+1 ) f (s+1) ( x 1, p 1,..., x s+1, p s+1, t), i=1 where x s+1 = x i σ n is+1. The unit vector n is+1 points into the interior of sphere i and is normal to its surface. The quantity ρ (s) is the s-particle probability density defined by N f (s) ( x 1, p 1,..., x s, p s, t) = N! (N s)! d 3 p s+1 d 3 x s+1... d 3 x N d 3 p N (36) Π s ) i=1 ΠN j=s+1 ( x Θ i x j σ ρ N ( x 1, p 1,..., x N, p N, t), where ( ) { 1, xi x Θ x i x j σ = j > σ 0, else (37) blocks spherical volumes in configuration space that cannot be occupied by the hard.

16 F. Schwabl, Mechanik, Springer. Carlo Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press.