A Theory of Spatiotemporal Chaos: What s it mean, and how close are we?

Transcription

1 A Theory of Spatiotemporal Chaos: What s it mean, and how close are we? Michael Dennin UC Irvine Department of Physics and Astronomy Funded by: NSF DMR Sloan Foundation Research Corporation

2 Outline 1) What do I mean by theory and why do we need one? 2) What is spatiotemporal chaos? 3) What are the possible elements of a theory of spatiotemporal chaos? 4) How does temporal modulation help? 5) Discuss results

3 What is a theory? Principles and /or Laws Conservation of energy Principle of superposition Minimization of free energy Second Law of Thermodynamics Application of symmetry principles

4 Thus, although the motion of systems with a very large number of degrees of freedom obeys the same laws of mechanics as that of systems consisting of a small number of particles, the existence of many degrees of freedom results in laws of a different kind - Landau and Lifshitz, Statistical Physics Entropy => Minimization of free energy

5 Why look for new laws? Hierarchy of length scales => NONLINEAR Hierarchy of external driving => NONEQUILIBRIUM

6 Hierarchy of Length Scales String theory (?) Standard Model Quantum Mechanics Classical Mechanics Stat Mech & Thermodynamics Continuum mechanics (often nonlinear} Pattern formation (definitely nonlinear) Spatiotemporal chaos (?)

7 Hierarchy of External Driving (1) Thermodynamic equilibrium (2) Nonequilibrium statistical mechanics (3) Pattern formation (4) Spatiotemporal chaos (??) (5) Turbulence (?)

8 What is lost? Nonlinear => NO PRINCIPLE OF SUPERPOSITION Nonequilibrium => NO MINIMIZATION PRINCIPLE

9 Why might there be something new? Symmetry and symmetry breaking remains a powerful idea. In certain limits, similar patterns are observed in many systems. Structures, such as solitons, act as fundamental objects. (Recover superposition?) Deterministic chaos might act as thermal noise.

10 What are patterns and spatiotemporal chaos?

11 NOAA Satellite Photograph Jan Top of Devils Postpile

12 Thermal Convection T T + T 1) Fluid heated from below 2) Initially, uniform conduction 3) Above critical T, convection rolls

13 Hu, et al., PRE 1993 Lerma, et al., PRE 1995 Thomas, et al., PRE 1998

14 Bodenschatz, et al., PRL 1991

15 Shaken Granular Materials H. Swinney, et al.

16 Often, patterns display irregular behavior in space and time spatiotemporal chaos

17 From M. Dennin, G. Ahlers, and D.S. Cannell, Science 272, 388 (1996). From J. Liu, K.M.S. Bajaj, and G. Ahlers, unpublished. From Y. Hu, W. Pesch, G. Ahlers, and R.E. Ecke, Phys. Rev. E 58, 5821(1998)

18 Spiral Defect Chaos: Courtesy of Eberhard Bodenschatz ACTUAL TIME: 3 hrs

19 Characteristics of Spatiotemporal Chaos 1) Deterministic dynamics, irregular behavior in space and time 2) Separation of length scales

20 Simulations of Navier-Stokes: Which is theory and which is experiment? Initial state of the system theory or experiment?

21 Theory of fluid dynamics already exists, but many examples of spatiotemporal chaos are not fluid systems. The question is: are there organizing principles that apply for all cases of spatiotemporal chaos?

22 What are the issues? Systems are nonequilibrium: NO MINIMIZATION PRINCIPLE Systems are nonlinear: NO PRINCIPLE OF SUPERPOSITION How do we approach the development of new principles? Go with what we know!

23 (1) Minimization of Free Energies, Stat Mech. (2) Linear Response (3) Linear Stability Theory & Weakly nonlinear analysis: AMPLITUDE EQUATIONS (REGULAR PATTERNS) (4)?????

24 Elements of theories of pattern formation Fundamental Equations (e.g. Navier-Stokes & Heat flow) Weakly nonlinear perturbation analysis No Fundamental Equations (e.g. granular materials) Symmetry arguments AMPLITUDE EQUATIONS (amplitudes are similar to order parameters)

25 What needs to be added? Chaotic behavior chaos theory? Lyaponuv exponents as a measure of the dimension of the system and the source of the chaotic behavior. Separation of length scales averaging and/or renormalization?

26 Closing in on a theory? 1) From Chaos Theory: Lyapunov exponents (Egolf, et al., Nature). (possible experiment) 2) From Statistical Mechanics: coarse graining (Egolf, Science). (toy model system) 3) From Pattern Formation: Amplitude equations (electroconvection experiments)

27 Focus on Amplitude Equations 1) Identify source of irregular behavior 2) Identify appropriate state variables

28 Review of amplitude equations and instabilities Pattern described by: A(x, t) cos(k x). Mode with wavevector k and amplitude A(x, t). Band of stable wavevectors k. Evolution of amplitude described by amplitude equation: ), ( ), ( ), ( ), ( t x A t x A g t x A A t x A t o + = ξ ε τ

29 Küppers Lortz Instability Rolls of all wavevectors unstable to wavevectors at 60 o ε = 0.06; Ω = 8.8 Defects from sidewalls ε = 0.06; Ω = 19.8 Defects from domains [pictures from Hu, Pesch, Ahlers, Ecke, PRE 58 (1998)]

30 Spiral Defect Chaos Band of stable wavenumbers ε = Skewed Varicose ε = 0.04 Eckhaus Morris, et al., PRL 71 (1993)

31 Why electroconvection? Küppers Lortz: in disagreement with amplitude equations SDC: amplitude equations provide only a qualitative description Electroconvection: amplitude equations provide a quantitative description.

32 Electroconvection V < V c 25 µm V director V > V c V

33 FUNDAMENTAL DESCRIPTION: Navier-Stokes + elastic torques + electric forces AMPLITUDE EQUATIONS: (weakly nonlinear expansion) linear part τ A! o Amplitude Equations A = εa1 + ξ 1 nonlinear part b A2 + c A3 A4 ) A1 g ( A + d + 3 other equations for A 2, A 3, and A 4.

34 System Parameters DEFINITION OF TERMS: Applied voltage: V = [V o +V m cos(ω m t)]cos(ω d t) Control parameter: ε = (V o /V c ) 2-1 Modulation strength: b = V m /V c NUMERICAL VALUES: Driving frequency: (ω d /2π) = 25 Hz Modulation frequency: (ω m /2π) = Hz Hopf frequency: (ω h /2π) = Hz Onset voltage (without modulation): V c = 21 V

35 Predictions of Amplitude Equations At onset, NO STABLE BAND of wavenumbers Spatiotemporal chaos at onset Caveat: only two coupled equations are known Treiber and Kramer, PRE 58 (1998) and Riecke and Kramer, 1998

36 Pattern Modes k θ k = q q θ The four modes can be represented as follows: right zig: A 1 (x,t) cos(qx - ω h t) left zig: A 2 (x,t) cos(qx + ω h t) right zag: A 3 (x,t) cos(kx - ω h t) left zag: A 4 (x,t) cos(kx + ω h t)

37 No Modulation Modulation Movies are in real time

38 Why temporal modulation? Provides an unambiguous test of the relevant amplitude equations. 1) Determines regions of standing wave stability 2) Determines type of standing wave 3) Determines nature of standing wave dynamics: irregular versus regular Provides a useful probe of the instabilities and other possible sources of the irregular dynamics.

39 What can we measure and learn? Determine the onset of frequency locking. Determine the impact of frequency locking on the temporal and spatial ordering. Determine the contribution of wavenumber instabilities and defects to the irregular dynamics. Make comparisons with CGL

40 0.06 Onset of Standing waves b No pattern Frequency locked waves STC closed symbols: stepping up open symbols: stepping down f m =2f h ε

41 Onset of frequency locked patterns for ε = Standing squares b 2 (x 10 3 ) Standing rolls [f * -f m /2] (Hz)

42 Elimination of Irregular Dynamics 0.03 amplitude (arb. units) No modulation (amplitudes shifted) Modulation of 2%: Frequency is locked time (min.) ε = 0.01

43 At t = 128 min., jump from b = 0 to b = 0.05, at ε = 0.04 f m = wavenumber (µm -1 ) frequency wavenumber frequency (Hz) time (min.)

44 Spatial Ordering Standing zig/zag on order of correlation length fast Uniform zig/zag on order of system size domain growth obeys power law Disorder returns rapidly defect generation?

45 Summary of results Method of eliminating spatiotemporal chaos Evidence for source of irregular dynamics Demonstrated effectiveness of temporal modulation as a probe of the system Established experimental results for comparison with amplitude equations (phase diagram, domain growth)

46 Emerging Framework? 1) Amplitude equation/ fundamental equations provide information on types of instabilities 2) Instabilities determine chaotic degrees of freedom calculated by Lyapunov exponents 3) Chaotic degrees of freedom provide thermal noise of a stat. mech. description

47 Future directions Detailed study of the defect dynamics, both in the STC state and growing from the regular state Spatial studies. In particular, what happens when two systems at different temperatures are placed in contact. FUNDAMENTALLY NEW PHYSICS!!

48 References and Collaborators Experiments: Carina Kamaga Lynne Purvis Theory: Hermann Riecke Martin Treiber Lorenz Kramer Theory Papers: Riecke, Silber, and Kramer, PRE V49 (1994) Treiber and Kramer, PRE V58 Experiments: Rehberg, et al., PRL V61 (1988) Juarez and Rehberg, PRA V 42 (1990) Dennin, Cannell, Ahlers, PRE V57 (1998)