Knudsen Diffusion and Random Billiards

Transcription

1 Knudsen Diffusion and Random Billiards R. Feres feres/publications.html November 11, 2004 Typeset by FoilTEX

2 Gas flow in the Knudsen regime Random Billiards [1] Large Knudsen number: Kn = mean free path diameter of container Low pressure: 10 3 to 10 4 torr. (Range of pressures between industrial and high vacuum.) Many collisions with channel wall before hitting other gas molecules. Figure 1: Random flight of a molecule in channel Typeset by FoilTEX 1

3 Random Billiards [2] Standard (textbook) surface-gas interaction model Knudsen s cosine law: ds = cos θ π dω n dω θ Figure 2: Standard interaction model assumes that molecule reflects according to Knudsen s cosine law. Pre-collision velocity is forgotten and post-collision velocity is distributed with probability proportional to cos θ. Typeset by FoilTEX 2

4 General questions Random Billiards [3] Under what conditions, and to what extent, is this cosine law valid? How does it relate to surface roughness? How should it be modified to incorporate information about microscopic details of solid surface? How are the fine details of gas-surface interaction reflected in transport properties of the gas (e.g., diffusivity)? Typeset by FoilTEX 3

5 Random Billiards [4] Informal description of interaction model Figure 3: Channel with periodic microgeometry: interaction is billiard-like (elastic) in each cell. Want to obtain information about cell geometry from properties of random walk and diffusion limit (diffusion constant, mean exit time from finite length channel, etc). How does cosine law arise in this model? Typeset by FoilTEX 4

6 Is a hard spheres model justified? Random Billiards [5] Knudsen (1907): [...] the energy exchange between the impinging molecules and the glass surface may be very far from being complete [...] Arya-Chang-Maginn (2003): More recent work on diffusion in nanopores shows that the thermalization time of gas molecules with surrounding lattice is often much larger than the collision time for the case of weakly adsorbing molecules with diameters much smaller than the pore width. [...] The stochastic forcing is due to chaotic Hamiltonian and deterministic trajectory of the constant-energy molecule in the pore. Typeset by FoilTEX 5

7 Topics Random Billiards [6] Infinitesimal billiard structure and random billiards; stationary measures and ergodicity; spectrum of the Markov operator; the associated random flight; diffusion limit and exit times; examples and questions. Typeset by FoilTEX 6

8 Random Billiards [7] Bouncing off wall with infinitesimal geometry q Sa S w v p Figure 4: Limit of deterministic billiards. S is fixed and S a varies so that bumps scale down to zero in size. As sets, S a S, but reflection law becomes probabilistic, represented by a scattering operator. Typeset by FoilTEX 7

9 Weak limit of billiards, S a S Random Billiards [8] Replace incoming velocity v with random V ɛ concentrated near v; Find µ v a,ɛ, distribution of post-reflection velocities for billiard S a ; Obtain µ v := lim ɛ 0 lim a 0 µ v a,ɛ. If Ψ x (θ) is the velocity component of first return map to open side of microcage, x [0, 1], θ [0, π], then µ v (h) = 1 0 h(ψ x (θ))dx. Typeset by FoilTEX 8

10 Random billiards Random Billiards [9] θout p x q θin Figure 5: Post-collision direction, θ out, is a random function of pre-collision direction, θ in. The former is obtained from the velocity component of the first return map to the open side of the cell of the deterministic billiard map, θ out = Ψ x (θ in ), where x is random variable uniformly distributed over [0, 1]. Typeset by FoilTEX 9

11 Iterations of random billiard map Random Billiards [10] The change of velocity as the molecule collides with the two parallel channel walls is described by iterations of the random billiard map. This defines a One-dimensional random dynamical system on interval [0, π] (dynamical system with noise ). Figure 6: Doubling of billiard cell with a random map. Typeset by FoilTEX 10

12 The associated Markov chain Random Billiards [11] Deterministic (first return) billiard map, T (x, θ) := (Y (x, θ), F θ (x)), where (x, θ) E = [0, 1] [0, π]. Push the Lebesgue measure, λ, on [0, 1] forward to [0, π], under F θ : [0, 1] [0, π]. I.e., for θ [0, π] and A [0, π], define P (A θ) := λ(f 1 θ (A)). Then P (A θ) is the probability that the particle that entered through the open side of the billiard cell with angle θ will leave with angle in A. Define a Markov chain with state space V = [0, π] and transition probabilities P ( θ), θ V. Typeset by FoilTEX 11

13 Invariance of dµ(θ) = 1 2 sin θdθ and ergodicity Random Billiards [12] A probability measure µ on V is stationary if µ(a) = π 0 P (A θ)dµ(θ). Proposition 1. The measure µ is a stationary measure of the associated Markov chain. If T (resp., T n for all n) is ergodic, then the Markov process is ergodic (resp., aperiodic). If the Markov process is aperiodic, then lim P n (A θ) = µ(a) n for all measurable A [0, π] and µ-a.e. θ. Proof. Invariance of Liouville measure under return map of ordinary billiard is not destroyed after averaging over x [0, 1]. Same for any µ such that λ µ is T -invariant. Typeset by FoilTEX 12

14 A simple ergodicity criterion Random Billiards [13] A point p of maximal height of the billiard cell is non-degenerate if the height function is non-degenerate at p. Proposition 2. Suppose that the billiard cell contains a non-degenerate point of maximal height. Then the associated random billiard is ergodic and aperiodic, and it admits a unique stationary probability measure, which is µ. Figure 7: Hair comb geometry is not ergodic, but becomes ergodic if billiard ball has positive radius. Typeset by FoilTEX 13

15 Example: Hair comb geometry Random Billiards [14] h b Figure 8: Hair comb microgeometry. Initial angle: θ (0, π). State space: {θ, π θ}. Define k and s by 2h b tan θ = k + s, where k Z and s [0, 1) are the integer and fractional parts of the left-hand side of the equation. Typeset by FoilTEX 14

16 Random Billiards [15] Define p = { 1 s if k is odd s if k is even. Then the transition probabilities are given by the diagram: 1-p p θ π θ p 1-p Figure 9: Markov chain diagram Standard random walk on R (p = 1/2) corresponds to projection of flight in channel of radius r = 1/2, θ = π/4, and b/h = 4. Typeset by FoilTEX 15

17 Approach to cosine law Random Billiards [16] Graph of P Figure 10: Graph of Markov operator for hair comb (positive particle radius). Typeset by FoilTEX 16

18 Random Billiards [17] Graph of P Figure 11: Graph of P 8. Typeset by FoilTEX 17

19 Random Billiards [18] Graph of P Figure 12: Graph of P 32. Angle after 32 collisions does not remember initial angle, and is distributed according to Knudsen law. Much faster convergence is seen for more complicated geometries. Typeset by FoilTEX 18

20 Cavities and effusion Random Billiards [19] o d Figure 13: Randomizing pore; o is the length of the open side and d is the length of the mirror-reflecting, flat side. The inside of the circular cavity is lined with some ergodic microgeometry. Define α = d/(o + d). Then P = (1 α)p cav + αi, where P cav is the Markov operator conditioned on particle falling into the cavity. Typeset by FoilTEX 19

21 Random Billiards [20] Let a = o/l, where l is the perimeter of the circular billiard cell. Denote by Ψ the random billiard map associated to the micro-bump geometry lining the inner side of the cavity. Let µ be the probability measure on [0, π] given by dµ(θ) = 1 2 sin θdθ. (This is the velocity part of the Liouville measure.) Proposition 3. Let P a ( θ) = P cav ( θ) be the conditional measures for a cavity with ratio a = o/l. Suppose that Ψ is ergodic with respect to µ and that the conditional probabilities for Ψ are absolutely continuous with respect to µ. Then, for µ-a.e. θ, lim a 0 P a( θ) = µ. I.e., particles that fall into cavity escape with random direction distributed very nearly according to sin θ, if the opening of the cavity is very small. Typeset by FoilTEX 20

22 Reversibility and Self-adjointness Random Billiards [21] The transition probabilities of the random billiard Markov chain define an operator, P, on H = L 2 ([0, π], µ). P is a bounded self-adjoint operator, P = 1; the Markov chain is reversible: P (dψ θ)dµ(θ) = P (dθ ψ)dµ(ψ). These are consequences of time reversibility of deterministic billiard: T J = J T 1 where J(x, θ) = (x, π θ). (Flip map.) Typeset by FoilTEX 21

23 Spectrum Random Billiards [22] Suppose now that the doubled billiard cell is a dispersing (Sinai) billiard. Figure 14: Dispersing billiard Theorem 1. If cell is dispersing, P is a Hilbert-Schmidt operator. Hence, its spectrum consists of eigenvalues λ i [ 1, 1], each of finite multiplicity, with 0 as the only accumulation point. In particular, There is a gap in the spectrum between 1 (which has multiplicity 1), and the second eigenvalue. Typeset by FoilTEX 22

24 Random Flight in Channel Random Billiards [23] Consider particle undergoing random flight inside a two dimensional channel of infinite length and radius r, with constant scalar velocity v. Write Ω = [0, π] N, with projections π k : Ω [0, π]. Fix reference values r 0 and v 0 and write r = r 0 /ζ, v = g(ζ)v 0, where g(ζ) is an unbounded monotone increasing function. Position of particle along the channel at time t is Xζ,t (ω) R, ω Ω. Wish to study limit of X ζ,t as ζ. Let P be a bounded self-adjoint operator on L 2 ([0, π], µ ), for some probability measure µ, with spectrum in [ 1, 1]. Let Π denote the spectral measure of P and define, for a given h L 2 ([0, π], µ ), the measure on [ 1, 1] given by ɛ h (dλ) := h, Π(dλ)h. Typeset by FoilTEX 23

25 Random Billiards [24] Theorem 2. Give Ω the probability measure associated to (P, µ ), where P is the Markov operator for a given random billiard and µ is an ergodic stationary measure. Let h(θ) = 2r 0 cot θ. Suppose that h is square integrable with respect to µ, has zero mean, and that δ 2 0 := λ 1 λ ɛ h(dλ) <. Also suppose that g(ζ) = ζ. Then, for any sequence ζ n, the process Xζ n,t converges to Brownian motion with variance (δ2 0/τ 0 )t, where τ 0 is the mean value of 2r 0 /(v 0 sin θ) with respect to µ. If (P, µ ) is replaced with with (P, δ θ ), where the initial probability distribution is a delta-measure concentrated at an angle θ, then the distribution of Xζ n,t will again converge to Brownian motion (with same variance), the convergence now being in measure as functions of θ, relative to µ. Typeset by FoilTEX 24

26 Random Billiards [25] This means that in this (non-ergodic, finite variance) case, the probability distribution u(x, t) of the particle in a long narrow channel is approximated by a solution of the diffusion equation u t = 1 2 σ2 2 u x 2, where σ 2 = δ 2 0/τ 0. Note: σ 2 (2 γ)γ 1 h 2 (γ =spectral gap.) Example: for the hair comb geometry (e.g., θ 0 = π/4, b > 2h) σ 2 = ( ) b 2r 0 v 0 2h 1. So geometry can be recovered from diffusion characteristic. Typeset by FoilTEX 25

27 3-D example Random Billiards [26] u2 n b2 h b1 α u1 e1 Figure 15: Microcubicles geometry Typeset by FoilTEX 26

28 Random Billiards [27] This example is the three-dimensional version of hair comb. Cells are now the surface of a parallelepiped without its top face, with sides h (height), b 1 and b 2. u2 e2 b2 b1 u1 α e1 Figure 16: Orientation of the base of cells over the cylindrical channel e1 q v e2 n e1 v n p e2 Figure 17: Standard frame over cylindrical surface Typeset by FoilTEX 27

29 Random Billiards [28] e2 u' 2(α β) u β e1 -u -u' Figure 18: Possible values of the orthogonal projection of the reflected velocity Initial velocity: v = u v 3 n. From any particular time and position, a particle moves a distance δ along cylinder axis, which can be one of the four values: δ 1, δ 2, δ 3 = δ 2, δ 4 = δ 1. A jump by ±δ i takes time τ i. Their values are given later. Typeset by FoilTEX 28

30 Random Billiards [29] Define k i, s i as the integral and fractional parts of the following numbers: 2hv 3 / 1 v 2 3 b 1 cos(β α) = k 1 + s 1 2hv 3 / 1 v 2 3 b 2 sin(β α) = k 2 + s 2. It can now be calculated that p i (u) = p i (u ) = p i ( u) = p i ( u ) =: p i, where { s i if k i is even p i = 1 s i if k i is odd. Setting a = p 1 p 2, b = p 1 (1 p 2 ), c = (1 p 1 )p 2, d = (1 p 1 )(1 p 2 ), the transition probabilities can now be seen to be as shown in Figure 19. Typeset by FoilTEX 29

31 Random Billiards [30] a b a u b u' d c c d d c c d a -u' b -u b a Figure 19: Transition probabilities for the velocity process Typeset by FoilTEX 30

32 Random Billiards [31] The values of jump lengths and times are: δ 1 = 2rv 3 u sin β v u 2 cos 2 β δ 2 = 2rv 3 u sin(2α β) v u 2 cos 2 (2α β) τ 1 = τ 2 = 2rv 3 v u 2 cos 2 β 2rv 3 v u 2 cos 2 (2α β). Typeset by FoilTEX 31

33 Random Billiards [32] Eigenvalues of transition matrix P : λ 1 = a + b + c + d = 1, (1, 1, 1, 1) λ 2 = a + b c d = 2p 1 1, (1, 1, 1, 1) λ 3 = a b + c d = 2p 2 1, (1, 1, 1, 1) λ 3 = a b c + d = (2p 1 1)(2p 2 1) (1, 1, 1, 1) In particular, stationary distribution gives equal probability to each δ i. The diffusion limit is Brownian motion with variance [ σ 2 p1 = (δ 1 + δ 2 ) 2 + p ] 2 (δ 1 δ 2 ) p 1 1 p 2 τ 1 + τ 2 Typeset by FoilTEX 32

34 The ergodic case: infinite variance Random Billiards [33] Theorem 2 does not apply to the generic case, where the unique stationary measure is µ. The probability distribution for a single jump (if angle probability is µ) has density: 1 (1 + x 2 ) 3/2. This is a Student-t 2 distribution. (Infinite variance) In the i.i.d. case, it is still possible to obtain Brownian motion, but velocity v must be scaled differently: g(ζ) = ζ/ ln ζ. (Distribution of jumps still in domain of attraction of normal law.) Typeset by FoilTEX 33

35 Example: ideal cavity Random Billiards [34] Let P µ + αp µ, where P µ is the orthogonal projection to the 1- dimensional subspace of H spanned by the stationary measure µ and P µ is orthogonal projection to the orthogonal complement. Recall: α is the probability that particle will bounce off the flat side in a specular way, and 1 α the probability that it will fall into the cavity and eventually exit with a probability distribution of angles given by µ. Proposition 4. Define X ζ,t as before, now for P = P µ + αp µ and g(ζ) = ζ/ ln ζ. Then for any ζ n and any fixed t > 0, X ζ n,t converges in distribution to normal distribution with variance σ 2 = const. 1 + α 1 α. Typeset by FoilTEX 34

36 Key idea Random Billiards [35] Suppose Z 1, Z 2, are i.i.d. with distribution (1 + x 2 ) 3/2 /2. Then Φ Zk (λ) := e iλx λ 2 2(1 + x 2 ) 3/2dx = 1 2 ln λ 1 + O( λ 2 ). Now let N = [n 2 / ln n] and X N = (Z Z N )/n. Then Φ XN (λ) = (1 λ 2 2 ) [n 2 / ln n] ln n n 2 + O(1/n2 ) e λ 2 /2. Same phenomenon: corridors in deterministic (Lorentz gas) billiards with infinite horizon, cf. P. M. Bleher, D. Szasz. Classical gas kinetics: Börgers- Greengard-Thomann.) Typeset by FoilTEX 35

37 Exit time from long (finite) channel Random Billiards [36] prob. density exit time Figure 20: Typical graph (histogram) for exit time from long channel. Typeset by FoilTEX 36

38 Mean exit time Random Billiards [37] Define τ(ζl 0, r 0, v 0 ), the mean exit time for channel of length ζl 0. In the finite variance case (if conditions of Theorem 2 hold) then τ(ζl 0, r 0, v 0 ) const. ζ 2. We expect that in the general case, where µ is ergodic, τ(ζl 0, r 0, v 0 ) const. ζ 2 / ln ζ. Typeset by FoilTEX 37

39 Random Billiards [38] Approximate diffusivity for finite channel length Numerical experiment (see graph of next page): Calculate mean exit time, τ(l), from a channel of length 2L and fixed radius. The scalar velocity of the molecules is fixed. The molecular radius varies as in the figure. r=1 a=0.7 a=0.4 a=0.1 Figure 21: Surface viewed by center of mass of molecules of varying radii. Typeset by FoilTEX 38

40 Graph of (L 2 / log L)/τ(L) versus L Random Billiards [39] 2.6 Approximate diffusivity versus length a=0.7 2 (L 2 / log L) / t(l) a=0.1 a= Ideal Knudsen Channel length L Figure 22: Asymptotes give diffusivity for varying radius. Bigger molecular radius flater effective surface faster diffusion. Typeset by FoilTEX 39

41 Random Billiards [40] Sets of invariants of random billiard: Geometry: scale invariant ratios of curvatures, areas, lengths, etc.; Deterministic billiard: entropies, Lyapunov exponent; Markov chain: spectrum {λ 1, λ 2, }; Diffusion limit: moments of exit time; diffusion constants: ( ) ζ 2 τ(ζl 0, r 0, v 0 ) (D 1 + D 2 ζ 1 + ). ln ζ Typeset by FoilTEX 40

42 Questions Random Billiards [41] Is microgeometry completely determined (up to scale) by Markov spectrum? How much information about geometry is contained in diffusion constants? Expect: τ(ζl, r, v) ζ 2 / ln ζ const.κ top o, where κ top is curvature of point of maximum height and o is the length of open side of billiard cell. Prove appropriate version of C.L.T. for the general ergodic case; Find estimates for spectral gap of P in terms of billiard geometry; Relate deterministic invariants (entropies) to spectrum and diffusivity. (Important for applications:) Relate τ(l, r, v) to microgeometry for short channels. Typeset by FoilTEX 41

43 Random Billiards [42] Examples of Markov kernels: triangular cell α α γ Figure 23: A shallow triangular cell. Suppose α < π/6. Define maps T i : [0, π] R by T 1 (θ) = θ + 2α T 2 (θ) = θ + 2π 4α T 3 (θ) = θ 2α T 4 (θ) = θ + 4α. Typeset by FoilTEX 42

44 Random Billiards [43] Random billiard map, T : [0, π] [0, π] Random map for triangular cell: T (θ) = T i (θ) with probability p i (θ). θout P1 P2 P3 P4 θin α 2α 3α π 3α π 2α π α π Figure 24: Graph of T. Define u α (θ) = 1 2 ( 1 + tan α tan θ ). The pi are functions of θ involving u α (θ) as shown in the next page. Typeset by FoilTEX 43

45 Random Billiards [44] 8 1 θ [0, α) >< uα(θ) θ [α, π 3α) p 1 (θ) = 2 cos(2α)u 2α (θ) θ [π 3α, π 2α) >: 0 θ [π 2α, π] 8 0 θ [0, π 3α) >< uα(θ) 2 cos(2α)u p 2 (θ) = 2α (θ) θ [π 3α, π 2α) uα(θ) θ [π 2α, π α) >: 0 θ [π α, π] 8 0 θ [0, 2α) >< 2 cos(2α)u p 3 (θ) = 2α ( θ) θ [2α, 3α) uα( θ) θ [3α, π α) >: 1 θ [π α, π] 8 0 θ [0, α) >< uα( θ) θ [α, 2α) p 4 (θ) = uα( θ) 2 cos(2α)u 2α ( θ) θ [2α, 3α) >: 0 θ [3α, π]. Typeset by FoilTEX 44

46 Numerical example 1: semicircle Random Billiards [45] Figure 25: Semicircular microcave. Typeset by FoilTEX 45

47 Numerical example 2: round bumps Random Billiards [46] Figure 26: Length of open side = 1; radius of circular bumps = 1. Typeset by FoilTEX 46