Deterministic chaos and diffusion in maps and billiards

Transcription

1 Deterministic chaos and diffusion in maps and billiards Rainer Klages Queen Mary University of London, School of Mathematical Sciences Mathematics for the Fluid Earth Newton Institute, Cambridge, 14 November 2013 Deterministic chaos and diffusion Rainer Klages 1

2 Outline 1 diffusion in simple chaotic maps motivation: from stochastic to deterministic random walks simple maps with non-trivial diffusion coefficients 2 diffusion in chaotic particle billiards density-dependent diffusion in the periodic Lorentz gas from theory towards applications Deterministic chaos and diffusion Rainer Klages 2

3 The drunken sailor at a lamppost random walk in one dimension (K. Pearson, 1905): steps of length s with probability p(±s) = 1/2 to the left/right time steps position single steps uncorrelated: Markov process define diffusion coefficient as 1 D := lim n 2n < (x n x 0 ) 2 > with discrete time step n N and average over the initial density <... >:= dx (x)... of positions x = x 0, x R for sailor: D = s 2 /2 Deterministic chaos and diffusion Rainer Klages 3

4 Dynamics of a chaotic map goal: study diffusion on the basis of deterministic chaos key idea: use chaos instead of stochasticity for drunken sailor why? determinism preserves all dynamical correlations model a single step x 0 x 1 by a deterministic map: 1 y=m(x) y x 1 =M(x 0 ) 0 x 0 x 1 1 x steps are iterated in discrete time n according to the equation of motion with x n+1 = M(x n ) M(x) = 2x mod 1 Bernoulli shift Lyapunov exponent: λ = ln 2 > 0 paradigm of a chaotic map Deterministic chaos and diffusion Rainer Klages 4

5 A deterministic random walk 2 study diffusion in the piecewise linear deterministic map { M h (x) = 2x + h 0 x < 1 2 2x 1 h 1 2 x < 1 lifted onto the real line by M h (x + 1) = M h (x) + 1 with symmetric shift h 0 as a control parameter (Gaspard, RK, 1998) deterministic random walk generated by x n+1 = M h (x n ) problem: calculate D(h) y -1 1 h 2 3 x Geisel/Grossmann/Kapral (1982) Deterministic chaos and diffusion Rainer Klages 5

6 The Takagi function method start from the Einstein formula 1 D = lim n 2n < (x n x 0 ) 2 >, x = x 0, with <... >= 1 0 dx h(x)... over the invariant density h(x) of m h (x) := M h (x) mod 1; it is h h(x) = 1! define integer jumps j k = x k+1 x k at discrete time k and rewrite D(h) via telescopic summation to D(h) = 1 2 j0 2 + j 0 j k k=1 Taylor-Green-Kubo formula structure of formula: first term: random walk solution other terms: higher-order dynamical correlations Deterministic chaos and diffusion Rainer Klages 6

7 Generalized Takagi/de Rham functions problem: calculate j 0 k=0 j 1 k = 0 dx j 0 k=0 j k defining Th n(x) = x 0 dy n k=0 j k(y) yields the de Rham-type equation Th n (x) = t(x) T n 1 h (m h (x)) with dt(x)/dx := j 0 (x). example: for h = { 1 we get the famous Takagi function (1903) T1 n(x) = x T n 1 1 (2x) 0 x < x T n (2x 1) 2 x < 1 Deterministic chaos and diffusion Rainer Klages 7

8 Solving the functional recursion relation can be solved to Th n (x) = t(x) T n 1 h (m h (x)) T n h (x) = n k=0 1 2 k t(mk h (x)) For 0 h and T h (x) = lim n Th n (x) this leads to ( ) ) D(h) = h ĥ 2 (1 2 h ) + T h (ĥ Knight, RK, Nonlinearity (2011) (with ĥ := h mod 1 (h / N), ĥ := 1 (h N), ĥ := 0 (h = 0)) Deterministic chaos and diffusion Rainer Klages 8

9 Diffusion coefficient for the lifted Bernoulli shift On large scales we recover the drunken sailor s result, D(h) h 2 /2 (h 1). On small scales, D(h) is partially a fractal function. Local maxima are the solutions to mh n (1/2) = 1/2: topological instability under parameter variation. Deterministic chaos and diffusion Rainer Klages 9

10 Why the plateau regions? For 0.5 h 1 ergodicity is broken and topology conserved: The phase space is split up into two invariant sets, see the mod 1 map: For a uniform initial density, the diffusion coefficient is calculated to D(h) = D(h) + D(h) = (1 h) + (h 1 2 ) = 1 2. Deterministic chaos and diffusion Rainer Klages 10

11 More maps: the lifted negative Bernoulli shift Same Takagi function method: 0 h 0.5: again ergodicity breaking which suggests: 0.5 h 1: topological instability topological instability fractal diffusion coefficient non-ergodicity linear diffusion coefficient Deterministic chaos and diffusion Rainer Klages 11

12 The lifted V-map This map suggests topological instability under parameter variation and no ergodicity breaking......but Takagi function method yields a piecewise linear D(h): explanation via dominating branch Deterministic chaos and diffusion Rainer Klages 12

13 The lifted tent map Finally, the lifted tent map T(x)......via T(x) = V( x):...is topologically conjugate to the V-map V(x)... Fortunately, D(h) is invariant under topological conjugacy (Korabel, RK, 2004): Takagi function solution for this map! Deterministic chaos and diffusion Rainer Klages 13

14 Summary: Linear and fractal diffusion coefficients Deterministic chaos and diffusion Rainer Klages 14

15 The periodic Lorentz gas w Lorentz (1905) moving point particle of unit mass with unit velocity scatters elastically with hard disks of unit radius on a triangular lattice only nontrivial control parameter: gap size w Deterministic chaos and diffusion Rainer Klages 15

16 two limiting cases for diffusion under parameter variation: w w w 0 = 0, D(w 0 ) = 0 trivial localization w , D(w ) ballistic flights paradigmatic example of a chaotic Hamiltonian particle billiard: positive Ljapunov exponent; D(w), w 0 < w < w Bunimovich, Sinai (1980) How does the diffusion coefficient D(w) look like? Deterministic chaos and diffusion Rainer Klages 16

17 Diffusion coefficient for the periodic Lorentz gas <(x(t) x(0)) diffusion coefficient D(w) = lim 2 > t 4t simulations (RK, Dellago, 2000): from MD 0.2 D(w) w Deterministic chaos and diffusion Rainer Klages 17

18 Diffusion coefficient for the periodic Lorentz gas <(x(t) x(0)) diffusion coefficient D(w) = lim 2 > t 4t simulations (RK, Dellago, 2000): from MD 0.2 D(w) 0.1 residua(w) irregularities on fine scales w Can one understand these results on an analytical basis? w Deterministic chaos and diffusion Rainer Klages 18

19 Taylor-Green-Kubo formula for billiards map diffusion onto correlated random walk on hexagonal lattice: lr rl (1) w ll (3) zz lz rz rr l (2) r zr zl z rewrite diffusion coefficient as Taylor-Green-Kubo formula: D(w) = 1 j 2 (x 0 ) + 1 j(x 0 ) j(x n ) 4τ 2τ n=1 τ: rate for a particle leaving a trap; j(x n ): inter-cell jumps over distance l at the nth time step τ in terms of lattice vectors l αβγ... RK, Korabel (2002) Deterministic chaos and diffusion Rainer Klages 19

20 TGK formula can be evaluated to D n (w) = l2 4τ + 1 n p(αβγ...)l l(αβγ...) 2τ αβγ... p(αβγ...) : probability for lattice jumps with this symbol sequence first term: random walk solution for diffusion on a two-dimensional lattice, calculated to (Machta, Zwanzig, 1983) w(2 + w) 2 D 0 (w) = π[ 3(2 + w) 2 2π] other terms: higher-order dynamical correlations; for time step 2τ: D 1 (w) = D 0 (w) + D 0 (w)[1 3p(z)] 3τ: D 2 (w) = D 1 (w) + D 0 (w)[2p(zz) + 4p(lr) 2p(ll) 4p(lz)] Deterministic chaos and diffusion Rainer Klages 20

21 open problem: conditional probabilities p(αβγ...) analytically? here results obtained from simulations: 0.2 D(w) 0.1 simulation results for D(w) random walk approximation 1st order approximation 2nd order approximation 3rd order approximation 1st order and coll.less flights w variation of convergence as a function of w indicates presence of memory due to dynamical correlations Deterministic chaos and diffusion Rainer Klages 21

22 Diffusion in the flower-shaped billiard hard disks replaced by flower-shaped scatterers with petals of curvature κ: 3 simulation results for the diffusion coefficient and analysis as before: 0.2 y x D 0.1 num. exact results Machta-Zwanzig 1st order 2nd order 3rd order 4th order 5th order κ Harayama, RK, Gaspard (2002) irregular diffusion coefficient due to dynamical correlations Deterministic chaos and diffusion Rainer Klages 22

23 Outlook: molecular diffusion in zeolites zeolites: nanoporous crystalline solids serving as molecular sieves, adsorbants; used in detergents, catalysts for oil cracking example: unit cell of Linde type A zeolite; periodic structure built by silica and oxygen forming a cage Schüring et al. (2002): MD simulations with ethane yield non-monotonic temperature dependence of diffusion coefficient < [x(t) x(0)] 2 > D(T) = lim t 6t in Arrhenius plot; explanation similar to previous TGK expansion Deterministic chaos and diffusion Rainer Klages 23

24 Summary fractal transport coefficients in simple maps irregular transport coefficients of the same origin in particle billiards I expect this phenomenon to be typical for classical transport in low-dimensional, deterministic, spatially periodic chaotic systems. Deterministic chaos and diffusion Rainer Klages 24

25 Advertisement similar results for: current and diffusion in the piecewise linear map M a,b (x) = ax + b, x [0, 1) diffusion in the climbing sine map C a (x) = x + k sin(2πx) diffusion in a dissipative particle billiard with oscillatory driving furthermore: fractal parameter dependencies for mobility, chemical reaction rate, anomalous diffusion in simple maps stability of such curves with respect to random perturbations some indication of such phenomena in experiments: antidot lattices, granular vibratory conveyors, surface diffusion, Josephson junctions, zeolites Deterministic chaos and diffusion Rainer Klages 25

26 Acknowledgements and literature work performed in collaboration with: G.Knight (Bologna) Chr.Dellago (Vienna) J.R.Dorfman (College Park/USA) P.Gaspard (Brussels) T.Harayama (Kyoto) N.Korabel (Manchester) Part 1 Deterministic chaos and diffusion Rainer Klages 26