Dissipative Solitons in Physical Systems

Transcription

1 Dissipative Solitons in Physical Systems Talk given by H.-G. Purwins Institute of Applied Physics University of Münster, Germany

2 Chapter 1 Introduction

3 Cosmology and Pattern Formation 1-1

4 Complex Behaviour of Pattern Forming Nonlinear Dissipative Systems Patterns in space and time Attractors Complex dependence on initial conditions boundary conditions parameters Dependence on initial conditions multistability Dependence on parameters and boundary conditions bifurcation Lack of reproducibility in the presence of noise Chaos Understanding of self-organized patterns is one of the most important problems of modern science 1-2

5 Chapter 2 Gas-Discharge Systems

6 Lichtenberg Pattern I: Experimental Set-Up elektrophorus Lichtenberg ( ) 2-1

7 Lichtenberg ( ) 2-2 Lichtenberg Pattern II: Reproduction of the Original Pattern

8 Current Filaments in a Pulse Driven Planar Gas-Discharge System with Dielectric Barrier photograph of the luminescence radiation from the discharge space measured on one electrode Buss

9 Chapter 3 Experimental Results from Gas-Discharge Systems with High Ohmic Barrier

10 Experimental Set-Up for Measuring Self-Organized Patterns in Planar DC Gas-Discharge Systems 3-1

11 Superposition of 1000 Subsequent Experimental Frames of a Single Propagating Current Filament coordinate y / mm monotonous tails coordinate y / mm oscillatory tails current density j / Am -2 coordinate x / mm parameters: U 0 = 2,74 kv, ρ SC = 4,95 MΩ cm, R 0 = 20 MΩ, Gas: N 2, T=100 K, p = 280 hpa, D=30 mm, d = 250 µm, a SC =1 mm, I = 46 µa, t exp =20 ms current density j / Am -2 coordinate x / mm parameters: U 0 = 3,6 kv,ρ SC = 3,05 MΩ cm, R 0 = 4,4 MΩ, Gas: N 2, T=100 K, p = 279 hpa, D=30 mm, d = 500 µm, a SC =1 mm, I = 200 µa, t exp =20 ms Bödeker et al

12 Typical Dynamic Behaviour of a Single Experimental Current Filament in the Discharge Plane 1 cm real time parameters: U 0 =2,7 kv, ρ SC =4,95 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=280 hpa, D=30 mm, d=250 µm, a SC =1 mm, I=46 µa Bödeker et al

13 Typical Scattering Process for two Interacting Filaments in the Discharge Plane t=0 ms t=120 ms t=200 ms t=248 ms t=300 ms t=360 ms parameters: U 0 =3 kv, ρ SC =1.98 MΩ cm, R 0 =4.4 MΩ, Gas: N 2, T=100 K, p=244 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=140 µa, t exp =20 ms, f rep =50 Hz Bödeker, Purwins

14 Typical Process for Molecule Formation of two Interacting Filaments in the Discharge Plane t=0 ms t=2 s t=5.68 s t=7.9 s t=9.82 s t=10.42 s parameters: U 0 =3.1 kv, ρ SC =4.19 MΩ cm, R 0 =4.4 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I=170 µa, t exp =20 ms, f rep =50 Hz Bödeker, Purwins

15 Typical Process for the Generation of a Current Filament in the Discharge Plane t=0 ms t=127.5 ms t=212.5 ms t=271 ms t=274 ms t=275 ms parameters: U 0 =3,8 kv, ρ SC =4,14 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I= µa, t exp =0,2 ms, f rep =2 khz Bödeker, Purwins

16 Typical Process for the Annihilation of a Current Filament in the Discharge Plane t=0 ms t=93 ms t=211.5 ms t=293 ms t=306 ms t=306.5 ms parameters: U 0 =3,8 kv, ρ SC =4,14 MΩ cm, R 0 =20 MΩ, Gas: N 2, T=100 K, p=290 hpa, D=30 mm, d=500 µm, a SC =1 mm, I= µa, t exp =0,2 ms, f rep =2 khz Bödeker, Purwins

17 Drift Bifurcation of an Experimental Filament in the Discharge Plane Obtained from Stochastic Data Analysis of Trajectories 2 square of velocity v 0 / mm 2 s -2 parameters: U 0 =3,7 kv, R 0 =10 MΩ, Gas: N 2, T=100 K, p=286 hpa, D=30 mm, d=750 µm, a SC =1 mm, I=107 µa, t exp =20 ms, f rep =50 Hz specific resistivity ρ SC / MΩcm square of the intrinsic velocity as a function of the specific resistivity of the semiconductor wafer and typical experimental trajectories Bödeker et al

18 Strümpel, Purwins Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System I

19 Strümpel, Purwins Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System II

20 Strümpel, Purwins Experimentally Observed Hierarchy of Filamentary Patterns in the DC Gas-Discharge System III

21 Chapter 4 Qualitative Model for Planar Gas-Discharge Systems with High Ohmic Barrier

22 Purwins, Klempt, Berkemeier The Local DC Gas-Discharge System as Activator-Inhibitor System

23 The 3-Component Reaction-Diffusion-Equation (3-k-RD-System) κ2 ut = Du u + f(u) κ 3v κ4 w + κ 1 Ω Ω ud, τ v = D v + u v, θ w = D w+ u w, t t v w Ω 3 f(u) λ u u, u = u(x;t), v = v(x;t), w = w(x;t), x Ω, D,D,D, τ, θ, λ, κ, κ, κ 0. u v w R R, R 1 3, Bode et al

24 Numerical Solution of the 3-k-RD-System: Dissipative Soliton as Localized Travelling Solitary Structure in R 3 τ=48.0, θ=0.5, D u =1.5*10-4, D v =1.86*10-4, D w =9.6*10-3, λ=2.0, κ 1 =-6.92, κ 2 =0, κ 3 =8.5, κ 4 =1.0, Ω=[0,0.466]x[0,0.932], x=0.0155, t=0.01. Liehr et al

25 Numerical Solutions of the 3-k-RD-System: Scattering of Interacting Dissipative Solitons in R 2 t=100 t=300 t=600 t=900 t=1200 t=1480 τ=3.35, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0 Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al

26 Numerical Solutions of the 3-k-RD-System: Formation of Rotating Molecules Due to Collision of Dissipative Solitons in R 2 t=100 t=320 t=540 t=760 t=980 t=1200 τ=3.35, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al

27 Numerical Solutions of the 3-k-RD-System: Generation of a Dissipative Soliton Due to Collision in R 2 t=105 t=250 t=265 t=285 t=405 t=530 τ=3.47, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr et al

28 Numerical Solutions of the 3-k-RD-System: Annihilation of a Dissipative Soliton Due to Collision in t=60 t=192 t=324 R 2 t=456 t=588 t=720 τ=3.59, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[0,1]x[0,1], x=5*10-3, t=0.1. Liehr

29 Drift Bifurcation of an Experimental Filament in the Discharge Plane Obtained from Stochastic Data Analysis of Trajectories 2 square of velocity v 0 / mm 2 s -2 parameters: U 0 =3,7 kv, R 0 =10 MΩ, Gas: N 2, T=100 K, p=286 hpa, D=30 mm, d=750 µm, a SC =1 mm, I=107 µa, t exp =20 ms, f rep =50 Hz specific resistivity ρ SC / MΩcm square of the intrinsic velocity as a function of the specific resistivity of the semiconductor wafer and typical experimental trajectories Bödeker et al

30 Relevance of the 3-k-RD-System exemplaric theoretical investigation 3-component nonlinear partial differential equation simple structure solutions reflect a large variety of particle properties strong relation to electrical transport systems theoretical predictions could be manifested experimentally observed phenomena could be found in the solutions of the equations expectation: deep insight into the formation of self-organized patterns deep insight into the mechanisms of pattern formation of nonlinear dissipative systems in general 4-9

31 Chapter 5 The Reduced Equation

32 The 3-Component Reaction-Diffusion-Equation (3-k-RD-System) 2 ut = Du u + f(u) κ3v κ 4w + κ1 ud Ω, Ω Ω τ v = D v + u v, t t κ = + κ κ + κ Ω v θ w = D w + u w, w λ = = = Ω R R 3 f(u) u u, u u(x;t), v v(x;t), w w(x;t), x,,, R D,D,D, τθλκ,,,, κ, κ 0. u v w Bode et al

33 Reduction of the Field Equation to a Dynamical Equation for Dissipative Solitons I ansatz: normalization of the homogeneous stationary background to 0, 2 3 u(x,t 1,T 2,T 3) = u(x p 1) + u(x p 2) +ε ru ε R u, v(x,t,t,t ) = u(x p ) + u(x p ) +εα u(x p ) +εα u(x p ) +ε r +ε R v v time scale separation: j pi = p i(t 1,T 2,T 3), i = 1,2, ru,v = r u,v(x,t 1), Tj =ε t, j = 1,2,3, εα =εα (T,T ), R = R (x). i i 1 2 u,v u,v visualization of and in R for a dissipative Soliton Bode et al α 2 p isoline of the activator u x 2 x isoline of the inhibitor v

34 Reduction of the Field Equation to a Dynamical Equation for Dissipative Solitons II equation of motion for N dissipative solitons p =κ α W (p,...,p ), i = 1,2,...,N i 3 i i 1 N α i = κ 3 τ α i κ 3Q α i α i W i(p 1,...,p N) κ3 interaction function W (p,...,p ) W (p,p ) F p p ( ) N N j i i 1 N = DS i j = j i j = 1 j = 1 pj pi j i j i p p Bode et al

35 Interaction Function of the Equation of Motion for Dissipative Solitons Derived from the Field Equation u(x) = v(x) F(d) = F p p ( 2 1 ) a+c: τ=0, θ=0, D u =0.5*10-4, D v =0, D w =10-3, λ=3.0, κ 1 =-0.1, κ 3 =1.0, κ 4 =1.0, Ω=[0,1.2], x=5*10-3, b+d: D u =1.1*10-4, D w =9.64*10-4, λ=1.71, κ 1 =-0.15, x=2.5*10-3, others as in a+c. Liehr

36 Determination of the Interaction Function from Experimental Trajectories Using New Stochastic Data Analysis acceleration a / mm s -2 distance d / mm parameters: U 0 =4600 V, ρ SC =3,55 MΩ cm, R 0 =4,4 MΩ, Gas: N 2, T=100 K, p=283 hpa, D=30 mm, d=550 µm, a SC =1 mm, I=233µA, t exp =20 ms, f rep =50 Hz Bödeker et al

37 Examples for the Interaction Behaviour of Dissipative Solitons in R 2 Obtained from the Model Equations reduced equation (a), field equation (b)-(d) interaction diagram trajectories I II ( τ, ξ ) = (3.35,0.02) ( τ, ξ ) = (3.35,0.174) B I white area: scattering : rotating molecules I ξ : parallel offset B II : rotating molecules II τ : relaxation time = ( ( )) = d d F(d) e cos d 0.199, Q 1950 θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[-0.5,0.5]x[-0.5,0.5], x=5*10-3, t=0.1. Liehr et al

38 Phase Separation in the Solutions of the Reduced Equation for many Dissipative Solitons Ν=81 marked solitons in the largest cluster at starting time τ=3.34, θ=0, D u =1.1*10-4, D v =0, D w =9.64*10-4, λ=1.01, κ 1 =-0.1, κ 3 =0.3, κ 4 =1.0, Ω=[-2,2]x[-2,2], x=5*10-3, t=0.1; F(d), Q = Röttger

39 Chapter 6 Quantitative Model for Planar Gas-Discharge Systems with High Ohmic Barrier

40 Experimental Set-Up for Measuring Self-Organized Patterns in Planar DC Gas-Discharge Systems 6-1

41 tne + div( Γ e) = S e, n + div( Γ ) = S, t p p p 1 ϕ= e(n e n p ), ε0 Γ e = µ enee De n e, Γ =µ n E D n, p p p p p Se = Sp =ν ne β nen p, E = ϕ, µ =µ (E), ν ν=ν (E), e,p e,p De,p = kt e,p µ e,p (E)/e, j = e( Γ Γ ). g p e Model Equation for the DC Gas-Discharge System I: Gas-Discharge Space (-d < z < 0) n e,n p electron/ion density S e, S p electron/ion source term Γ, Γ electron/ion particle current density µ e, µ p electron/ion mobility E electrical field D e, D p ν β ϕ e T e, T p j g p electron/ion diffusion constant ionisation rate recombination rate electrical potential electron/ion temperature global electrical current density Amiranashvili, Purwins

42 Model Equation for the DC Gas-Discharge System II: Semiconductor Wafer (0 < z < d SC ) jsc =λ E, E = ϕ, div( ε ϕ ) = 0. j SC λ E ϕ ε global electrical current density specific electrical conductivity electrical field electrical potential dielectric constant of the semiconductor Amiranashvili, Purwins

43 Model Equation for the DC Gas-Discharge System III: Boundary Conditions Gas Semiconductor at z=0 1 ε 0 σ t DS σ= e(j z g j SC) z= 0, σ= ( ε ee) (ee), z z =+ 0 z z = 0 1 ( Γ pe) z z = 0 = ( µ pnee p z + np v p ) z = 0, 4 ( Γ ee z) z = 0 = ( µ eneeez 1 + n e v e γγ p e z ) z = 0, 4 v = 8k T / π m. e,p e,p e,p σ surface charge D S e z ε v e,v p γ k diffusion constand of surface charge unity vector in z-direction dielectric constant of the semiconductor thermal electron/ion speed γ -Townsend-coefficient Boltzmann-constant m e,m p electron/ion mass U voltage drop at the component ( ϕ ) ( ϕ ) = U. z = d z = d SC Amiranashvili, Purwins

44 Chapter 7 Dissipative Solitons in Various Systems

45 Experimental Set-Up for Measuring Self-Organized Patterns in Quasi-1-Dimensional DC Gas-Discharge Systems electric circuit Radehaus et al

46 Cascade of Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System I increasing driving voltage U S Radehaus et al. 1987, Willebrand et al

47 Cascade of Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System II decreasing driving voltage U S Radehaus et al. 1987, Willebrand et al

48 Travelling and Interacting Current Filaments in a Quasi-1-Dimensional DC Gas-Discharge System increasing driving voltage U S Willebrand et al

49 Brauer, Purwins Experimental Set-Up for Measuring Self-Organized Patterns in Planar AC Gas-Discharge Systems

50 Brauer, Purwins Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System I

51 Brauer, Purwins Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System II

52 Experimentally Observed Hierarchy of Filamentary Patterns in the DBD System III Brauer, Purwins

53 Current Filaments in n-gaas Plates Measured at 4.2 K by Electron Microscopy decrease of driving voltage from A to F Mayer et al

54 Clusters of Localized States Measured in a Laser Driven Na Vapour Cell Schäpers et al

55 Keynes Nerve Pulse Propagation I: Mechanism for Generation of Electrical Potential Difference

56 Keynes Nerve Pulse Propagation II: Membrane Potential and Ion Conductivity Plotted as a Function of Time