Outline Particle billiards Weakly chaotic map CTRW theory End From normal to anomalous deterministic diffusion Part 2: From normal to anomalous Rainer Klages Queen Mary University of London, School of Mathematical Sciences Sperlonga, 20-24 September 2010 From normal to anomalous diffusion 2 Rainer Klages 1
Outline Particle billiards Weakly chaotic map CTRW theory End Outline yesterday: Normal deterministic diffusion: 1 some basics of dynamical systems theory for maps and escape rate theory of deterministic diffusion reference: R.Klages, From Deterministic Chaos to Anomalous Diffusion book chapter in: Reviews of Nonlinear Dynamics and Complexity , Vol. 3 H.G.Schuster (Ed.), Wiley-VCH, Weinheim, 2010 http://www.maths.qmul.ac.uk/˜klages From normal to anomalous diffusion 2 Rainer Klages 2
Outline Particle billiards Weakly chaotic map CTRW theory End Outline yesterday: Normal deterministic diffusion: 1 some basics of dynamical systems theory for maps and escape rate theory of deterministic diffusion reference: R.Klages, From Deterministic Chaos to Anomalous Diffusion book chapter in: Reviews of Nonlinear Dynamics and Complexity , Vol. 3 H.G.Schuster (Ed.), Wiley-VCH, Weinheim, 2010 http://www.maths.qmul.ac.uk/˜klages today: From normal to anomalous deterministic diffusion: 2 normal diffusion in particle billiards and anomalous diffusion in intermittent maps From normal to anomalous diffusion 2 Rainer Klages 2
Outline Particle billiards Weakly chaotic map CTRW theory End The periodic Lorentz gas idea: study more physically realistic models of deterministic diffusion 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 w Lorentz (1905) From normal to anomalous diffusion 2 Rainer Klages 3
Outline Particle billiards Weakly chaotic map CTRW theory End The periodic Lorentz gas idea: study more physically realistic models of deterministic diffusion 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 paradigmatic example of a chaotic w Hamiltonian particle billiard: ∃ positive Ljapunov exponent; ∃ diffusion in certain range of w Lorentz (1905) (Bunimovich, Sinai, 1980) How does the diffusion coefficient D ( w ) look like? From normal to anomalous diffusion 2 Rainer Klages 3
Outline Particle billiards Weakly chaotic map CTRW theory End Diffusion coefficient for the periodic Lorentz gas < ( x ( t ) − x ( 0 )) 2 > diffusion coefficient D ( w ) = lim from MD 4 t t →∞ simulations: 0.2 D(w) 0.1 0 0 0.1 0.2 0.3 w From normal to anomalous diffusion 2 Rainer Klages 4
Outline Particle billiards Weakly chaotic map CTRW theory End Diffusion coefficient for the periodic Lorentz gas < ( x ( t ) − x ( 0 )) 2 > diffusion coefficient D ( w ) = lim from MD 4 t t →∞ simulations: 0.2 D(w) 0.0008 0.1 residua(w) 0.0004 0 -0.0004 0.24 0.26 0.28 0.3 w 0 0 0.1 0.2 0.3 w ∃ irregularities on fine scales (R.K., Dellago, 2000) Can one understand these results on an analytical basis? From normal to anomalous diffusion 2 Rainer Klages 5
Outline Particle billiards Weakly chaotic map CTRW theory End Taylor-Green-Kubo formula for billiards map diffusion onto correlated random walk on hexagonal lattice: lr rl (1) (3) zz w ll rz rr lz l r zr zl (2) z From normal to anomalous diffusion 2 Rainer Klages 6
Outline Particle billiards Weakly chaotic map CTRW theory End Taylor-Green-Kubo formula for billiards map diffusion onto correlated random walk on hexagonal lattice: lr rl (1) (3) zz w ll rz rr lz l r zr zl (2) z rewrite diffusion coefficient as Taylor-Green-Kubo formula : ∞ D ( w ) = 1 + 1 � � � j 2 ( x 0 ) � j ( x 0 ) · j ( x n ) � 4 τ 2 τ n = 1 τ : rate for a particle leaving a trap; j ( x n ) : inter-cell jumps over distance ℓ at the n th time step τ in terms of lattice vectors ℓ αβγ... R.K., Korabel (2002) From normal to anomalous diffusion 2 Rainer Klages 6
Outline Particle billiards Weakly chaotic map CTRW theory End TGK formula can be evaluated to n D n ( w ) = ℓ 2 4 τ + 1 � p ( αβγ . . . ) ℓ · ℓ ( αβγ . . . ) 2 τ αβγ... p ( αβγ . . . ) : probability for lattice jumps with this symbol sequence From normal to anomalous diffusion 2 Rainer Klages 7
Outline Particle billiards Weakly chaotic map CTRW theory End TGK formula can be evaluated to n D n ( w ) = ℓ 2 4 τ + 1 � p ( αβγ . . . ) ℓ · ℓ ( αβγ . . . ) 2 τ 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 ) = √ 3 ( 2 + w ) 2 − 2 π ] π [ From normal to anomalous diffusion 2 Rainer Klages 7
Outline Particle billiards Weakly chaotic map CTRW theory End TGK formula can be evaluated to n D n ( w ) = ℓ 2 4 τ + 1 � p ( αβγ . . . ) ℓ · ℓ ( αβγ . . . ) 2 τ 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 ) = √ 3 ( 2 + w ) 2 − 2 π ] π [ other terms: higher-order dynamical correlations; for time step 2 τ : D 1 ( w ) = D 0 ( w ) + D 0 ( w ) [ 1 − 3 p ( z )] 3 τ : D 2 ( w ) = D 1 ( w ) + D 0 ( w ) [ 2 p ( zz ) + 4 p ( lr ) − 2 p ( ll ) − 4 p ( lz )] From normal to anomalous diffusion 2 Rainer Klages 7
Outline Particle billiards Weakly chaotic map CTRW theory End open problem: conditional probabilities p ( αβγ . . . ) analytically? Here results obtained from simulations: 0.2 D(w) simulation results for D(w) 0.1 random walk approximation 1st order approximation 2nd order approximation 3rd order approximation 1st order and coll.less flights 0 0 0.1 0.2 0.3 w variation of convergence as a function of w indicates presence of memory due to dynamical correlations • approach was incorrectly criticized by Gilbert, Sanders (2009) • theory can be worked out exactly for one-dimensional maps From normal to anomalous diffusion 2 Rainer Klages 8
Outline Particle billiards Weakly chaotic map CTRW theory End Diffusion in the flower-shaped billiard hard disks replaced by flower-shaped scatterers with petals of curvature κ : 3 2 1 y 0 -1 -2 -3 -2 -1 0 1 2 3 x From normal to anomalous diffusion 2 Rainer Klages 9
Outline Particle billiards Weakly chaotic map CTRW theory End Diffusion in the flower-shaped billiard simulation results for the hard disks replaced by diffusion coefficient and flower-shaped scatterers analysis as before: with petals of curvature κ : 0.2 0.2 0.2 0.2 0.2 0.2 0.2 3 2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 D D D D D D D 1 y num. exact results Machta-Zwanzig 0 1st order 2nd order 3rd order 4th order -1 5th order 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 -2 κ κ κ κ κ κ κ -3 -2 -1 0 1 2 3 Harayama, R.K., Gaspard (2002) x ∃ irregular diffusion coefficient due to dynamical correlations From normal to anomalous diffusion 2 Rainer Klages 9
Outline Particle billiards Weakly chaotic map CTRW theory End 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” From normal to anomalous diffusion 2 Rainer Klages 10
Outline Particle billiards Weakly chaotic map CTRW theory End 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 S ( T ) = lim 6 t t →∞ in Arrhenius plot; explanation similar to previous TGK expansion From normal to anomalous diffusion 2 Rainer Klages 10
Outline Particle billiards Weakly chaotic map CTRW theory End Polygonal billiard channels instead of convex scatterers, look at polygonal ones: (b) (a) (c) (d) φ ψ • weak chaos: dispersion of nearby trajectories ∆( t ) grows weaker than exponential (Zaslavsky, Usikov, 2001) • pseudochaos: algebraic dispersion ∆ ∼ t ν , 0 < ν (Zaslavsky, Edelman, 2002); above: special case ν = 1 From normal to anomalous diffusion 2 Rainer Klages 11
Recommend
More recommend
