Outline Chaotic maps Deterministic diffusion End From normal to anomalous deterministic diffusion Part 1: Normal deterministic diffusion Rainer Klages Queen Mary University of London, School of Mathematical Sciences Sperlonga, 20-24 September 2010 From normal to anomalous diffusion 1 Rainer Klages 1
Outline Chaotic maps Deterministic diffusion End Setting the scene nonequilibrium conditions nonequilibrium nonequilibrium equilibrium steady states non-steady states ergodic microscopic chaos dynamical systems microscopic strong weak hypothesis complexity Gibbs statistical mechanics fractal SRB measures infinite measures ensembles deterministic transport thermodynamic macroscopic thermodynamics normal anomalous properties theory of nonequilibrium statistical physics starting from microscopic chaos? approach should be particularly useful for small nonlinear systems From normal to anomalous diffusion 1 Rainer Klages 2
Outline Chaotic maps Deterministic diffusion End Outline three parts: Normal deterministic diffusion: 1 some basics of dynamical systems theory for maps and escape rate theory of deterministic diffusion From normal to anomalous diffusion 1 Rainer Klages 3
Outline Chaotic maps Deterministic diffusion End Outline three parts: Normal deterministic diffusion: 1 some basics of dynamical systems theory for maps and escape rate theory of deterministic diffusion From normal to anomalous deterministic diffusion: 2 normal diffusion in particle billiards and anomalous diffusion in intermittent maps From normal to anomalous diffusion 1 Rainer Klages 3
Outline Chaotic maps Deterministic diffusion End Outline three parts: Normal deterministic diffusion: 1 some basics of dynamical systems theory for maps and escape rate theory of deterministic diffusion From normal to anomalous deterministic diffusion: 2 normal diffusion in particle billiards and anomalous diffusion in intermittent maps Anomalous (deterministic) diffusion: 3 generalized diffusion and Langevin equations, fluctuation relations and biological cell migration From normal to anomalous diffusion 1 Rainer Klages 3
Outline Chaotic maps Deterministic diffusion End 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 • single steps uncorrelated : Markov position process (coin tossing) • define diffusion coefficient as 5 1 2 n < ( x n − x 0 ) 2 > D := lim 10 n →∞ with discrete time step n ∈ N and average over the initial density 15 � < . . . > := dx ̺ ( x ) . . . of positions x = x 0 , x ∈ R 20 • for sailor: D = s 2 / 2 time steps From normal to anomalous diffusion 1 Rainer Klages 4
Outline Chaotic maps Deterministic diffusion End Bernoulli shift and dynamical instability idea: study diffusion on the basis of deterministic chaos Bernoulli shift M ( x ) = 2 x mod 1 with x n + 1 = M ( x n ) : 1 M 0.5 0 0 0.5 1 x 0 x 0 + ∆ x x apply small perturbation ∆ x 0 := ˜ x 0 − x 0 ≪ 1 and iterate: ∆ x n = 2 ∆ x n − 1 = 2 n ∆ x 0 = e n ln 2 ∆ x 0 ⇒ exponential dynamical instability with Ljapunov exponent λ := ln 2 > 0: Ljapunov chaos From normal to anomalous diffusion 1 Rainer Klages 5
Outline Chaotic maps Deterministic diffusion End Ljapunov exponent local definition for one-dimensional maps via time average : n − 1 1 � � M ′ ( x i ) � , x = x 0 � � λ ( x ) := lim ln n n →∞ i = 0 if map is ergodic: time average = ensemble average, λ = � ln | M ′ ( x ) |� Birkhoff’s theorem with average over an invariant probability density ̺ ( x ) that is � x related to the map’s SRB measure via µ ( x ) = 0 dy ̺ ( y ) Bernoulli shift is expanding : ∀ x | M ′ ( x ) | > 1, hence ‘hyperbolic’ normalizable pdf exists, here simply ̺ ( x ) = 1 ⇒ λ = ln 2 From normal to anomalous diffusion 1 Rainer Klages 6
Outline Chaotic maps Deterministic diffusion End Kolmogorov-Sinai entropy define a partition { W n i } of the phase space and refine it by iterating the critical 1 point n times backwards M 0.5 0 0 0.5 1 x From normal to anomalous diffusion 1 Rainer Klages 7
Outline Chaotic maps Deterministic diffusion End Kolmogorov-Sinai entropy define a partition { W n i } of the phase space and refine it by iterating the critical 1 point n times backwards let µ ( w ) be the SRB measure of a partition element w ∈ { W n i } M 0.5 0 0 0.5 1 x From normal to anomalous diffusion 1 Rainer Klages 7
Outline Chaotic maps Deterministic diffusion End Kolmogorov-Sinai entropy define a partition { W n i } of the phase space and refine it by iterating the critical 1 point n times backwards let µ ( w ) be the SRB measure of a partition element w ∈ { W n i } M 0.5 � define H n := − µ ( w ) ln µ ( w ) , w ∈{ W n i } where n denotes the level of refinement 0 0 0.5 1 x From normal to anomalous diffusion 1 Rainer Klages 7
Outline Chaotic maps Deterministic diffusion End Kolmogorov-Sinai entropy define a partition { W n i } of the phase space and refine it by iterating the critical 1 point n times backwards let µ ( w ) be the SRB measure of a partition element w ∈ { W n i } M 0.5 � define H n := − µ ( w ) ln µ ( w ) , w ∈{ W n i } where n denotes the level of refinement 0 0 0.5 1 1 x the limit h ks := lim nH n n →∞ defines the Kolmogorov-Sinai (metric) entropy (if the partition is generating) for Bernoulli shift with uniform measure refinement yields H n = n ln 2, hence h ks = ln 2 > 0: measure-theoretic chaos From normal to anomalous diffusion 1 Rainer Klages 7
Outline Chaotic maps Deterministic diffusion End Pesin theorem note: for Bernoulli shift λ = ln 2 and h ks = ln 2 Theorem For closed C 2 Anosov systems the KS-entropy is equal to the sum of positive Lyapunov exponents. Pesin (1976), Ledrappier, Young (1984) believed to hold for a wider class of systems for one-dimensional hyperbolic maps: h ks = λ From normal to anomalous diffusion 1 Rainer Klages 8
Outline Chaotic maps Deterministic diffusion End Escape from a fractal repeller piecewise linear map, slope a = 3, with escape : escape 1/3 1/3 1 take a uniform ensemble of N 0 points; calculate the number N n of points that survive after n iterations: N n = ( 2 / 3 ) N n − 1 = N 0 e − n ln ( 3 / 2 ) =: N 0 e − γ n M 0.5 0 0 0.5 1 x escape From normal to anomalous diffusion 1 Rainer Klages 9
Outline Chaotic maps Deterministic diffusion End Escape from a fractal repeller piecewise linear map, slope a = 3, with escape : escape 1/3 1/3 1 take a uniform ensemble of N 0 points; calculate the number N n of points that survive after n iterations: N n = ( 2 / 3 ) N n − 1 = N 0 e − n ln ( 3 / 2 ) =: N 0 e − γ n M 0.5 for hyperbolic maps N n decreases exponentially with escape rate γ ; 0 0 0.5 1 repeller forms a fractal Cantor set x escape From normal to anomalous diffusion 1 Rainer Klages 9
Outline Chaotic maps Deterministic diffusion End Escape rate formula note: for open systems λ , h ks must be computed with respect to the invariant measure on the fractal repeller R for our example: λ ( R ) = ln 3 , h ks ( R ) = ln 2 (as before) , γ = ln ( 3 / 2 ) ⇒ γ = λ ( R ) − h ks ( R ) no coincidence: this is the escape rate formula of Kantz, Grassberger (1985) From normal to anomalous diffusion 1 Rainer Klages 10
Outline Chaotic maps Deterministic diffusion End Escape rate formula note: for open systems λ , h ks must be computed with respect to the invariant measure on the fractal repeller R for our example: λ ( R ) = ln 3 , h ks ( R ) = ln 2 (as before) , γ = ln ( 3 / 2 ) ⇒ γ = λ ( R ) − h ks ( R ) no coincidence: this is the escape rate formula of Kantz, Grassberger (1985) • proven for Anosov diffeomorphisms with ‘holes’ by Chernov, Markarian (1997) • ∃ position dependence of escape rates, cf. Bunimovich, Yurchenko (2008) and ff From normal to anomalous diffusion 1 Rainer Klages 10
Outline Chaotic maps Deterministic diffusion End A simple deterministic diffusive map continue the previous map on the unit interval by a lift of degree one , M a ( x + 1 ) = M a ( x ) + 1, where a denotes the slope: Grossmann/Geisel/Kapral et al. (1982) M (x) a three questions: 3 Does this map exhibit 2 diffusion? a If so, can one calculate the 1 diffusion coefficient? x And if so, is there any relation 0 3 1 2 between this coefficient and dynamical systems quantities? From normal to anomalous diffusion 1 Rainer Klages 11
Outline Chaotic maps Deterministic diffusion End Escape rate formalism, Step 1: diffusion equation solve the ordinary one-dimensional diffusion equation ∂ t = D ∂ 2 n ∂ n ∂ x 2 with n = n ( x , t ) distribution function at point x and time t ; D defines the diffusion coefficient solution for absorbing boundaries, n ( 0 , t ) = n ( L , t ) = 0: ∞ � � π m � 2 � a m sin ( π m � n ( x , t ) = exp − Dt L x ) L m = 1 with a m determined by the initial density n ( x , 0 ) Q: do we get the same for our deterministic chaotic model? From normal to anomalous diffusion 1 Rainer Klages 12
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