Transcript Overview
John Doyle Control and Dynamical Systems Caltech Research interests • Complex networks applications – Ubiquitous, pervasive, embedded control, computing, and communication networks – Biological regulatory networks • New mathematics and algorithms – robustness analysis – systematic design – multiscale physics Collaborators and contributors (partial list) Biology: Csete,Yi, Borisuk, Bolouri, Kitano, Kurata, Khammash, ElSamad, … Alliance for Cellular Signaling: Gilman, Simon, Sternberg, Arkin,… HOT: Carlson, Zhou,… Theory: Lall, Parrilo, Paganini, Barahona, D’Andrea, … Web/Internet: Low, Effros, Zhu,Yu, Chandy, Willinger, … Turbulence: Bamieh, Dahleh, Gharib, Marsden, Bobba,… Physics: Mabuchi, Doherty, Marsden, Asimakapoulos,… Engineering CAD: Ortiz, Murray, Schroder, Burdick, Barr, … Disturbance ecology: Moritz, Carlson, Robert, … Power systems: Verghese, Lesieutre,… Finance: Primbs, Yamada, Giannelli,… …and casts of thousands… Background reading online • On website accessible from SFI talk abstract • Papers with minimal math – HOT and power laws – Chemotaxis, Heat shock in E. Coli – Web & Internet traffic, protocols, future issues • Thesis: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization • Recommended books – A course in Robust Control Theory, Dullerud and Paganini, Springer – Essentials of Robust Control, Zhou, Prentice-Hall – Cells, Embryos, and Evolution, Gerhart and Kirschner Biochemical Network: E. Coli Metabolism Mass Transfer in Metabolism* + Regulatory Interactions Complexity Robustness Supplies Materials & Energy Supplies Robustness From Adam Arkin * from: EcoCYC by Peter Karp Robustness Complexity An apparent paradox Component behavior seems to be gratuitously uncertain, yet the systems have robust performance. Mutation Selection Darwinian evolution uses selection on random mutations to create complexity. Transcription/ translation Microtubules Neurogenesis Angiogenesis Immune/pathogen Chemotaxis …. Regulatory feedback control • Such feedback strategies appear throughout biology (and advanced technology). • Gerhart and Kirschner (correctly) emphasis that this “exploratory” behavior is ubiquitous in biology… • …but claim it is rare in our machines. • This is true of primitive, but not advanced, technologies. • Robust control theory provides a clear explanation. Component behavior seems to be gratuitously uncertain, yet the systems have robust performance. Transcription/ translation Microtubules Neurogenesis Angiogenesis Immune/pathogen Chemotaxis …. Regulatory feedback control Overview • Without extensive engineering theory and math, even reverse engineering complex engineering systems would be hopeless. (Let alone actual design.) • Why should biology be much easier? • With respect to robustness and complexity, there is too much theory, not too little. Overview • Two great abstractions of the 20th Century: – Separate systems engineering into control, communications, and computing • Theory • Applications – Separate systems from physical substrate • Facilitated massive, wildly successful, and explosive growth in both mathematical theory and technology… • …but creating a new Tower of Babel where even the experts do not read papers or understand systems outside their subspecialty. “Any sufficiently advanced technology is indistinguishable from magic.” Arthur C. Clarke “Any sufficiently advanced technology is indistinguishable from magic.” Arthur C. Clarke “Those who say do not know, those who know do not say.” Zen saying Today’s goal • Introduce basic ideas about robustness and complexity • Minimal math • Hopefully familiar (but unconventional) example systems • Caveat: the “real thing” is much more complicated • Perhaps any such “story” is necessarily misleading • Hopefully less misleading than existing popular accounts of complexity and robustness Complexity and robustness • Complexity phenotype : robust, yet fragile • Complexity genotype: internally complicated • New theoretical framework: HOT (Highly optimized tolerance, with Jean Carlson, Physics, UCSB) • Applies to biological and technological systems – Pre-technology: simple tools – Primitive technologies use simple strategies to build fragile machines from precision parts. – Advanced technologies use complicated architectures to create robust systems from sloppy components… – … but are also vulnerable to cascading failures… Robust, yet fragile phenotype • Robust to large variations in environment and component parts (reliable, insensitive, resilient, evolvable, simple, scaleable, verifiable, ...) • Fragile, often catastrophically so, to cascading failures events (sensitive, brittle,...) • Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …) • There is a tradeoff between – ideal or nominal performance (no uncertainty) – robust performance (with uncertainty) • Greater “pheno-complexity”= more extreme robust, yet fragile Robust, yet fragile phenotype • Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …) • In many complex systems, the size of cascading failure events are often unrelated to the size of the initiating perturbations • Fragility is interesting when it does not arise because of large perturbations, but catastrophic responses to small variations Complicated genotype • Robustness is achieved by building barriers to cascading failures • This often requires complicated internal structure, hierarchies, self-dissimilarity, layers of feedback, signaling, regulation, computation, protocols, ... • Greater “geno-complexity” = more parts, more structure • Molecular biology is about biological simplicity, what are the parts and how do they interact. • If the complexity phenotypes and genotypes are linked, then robustness is the key to biological complexity. • “Nominal function” may tell little. An apparent paradox Component behavior seems to be gratuitously uncertain, yet the systems have robust performance. Mutation Selection Darwinian evolution uses selection on random mutations to create complexity. Transcription/ translation Microtubules Neurogenesis Angiogenesis Immune/pathogen Chemotaxis …. Regulatory feedback control Loss of Protein Function Cell Network failure Death Unfolded Proteins Folded Proteins Aggregates Temp cell Temp environ Loss of Protein Function Unfolded Proteins Folded Proteins Cell Network failure How does the cell build “barriers” (in state space) to stop this cascading failure event? Death Aggregates Temp cell Temp environ Insulate & Regulate Temp Folded Proteins Temp cell Temp environ olded roteins Thermotax Temp cell Temp environ More robust ( Temp stable) proteins Unfolded Proteins Folded Proteins Aggregates Temp cell Temp environ • Key proteins can have multiple (allelic or paralogous) variants • Allelic variants allow populations to adapt • Regulated multiple gene loci allow individuals to adapt Unfolded Proteins Folded Proteins Aggregates Temp cell Temp environ ve AE RT 37o Log of E. Coli Growth Rate 42o 46o 21o -1/T Heat Shock Response Robustness/performance tradeoff? 37o Log of E. Coli Growth Rate 42o 46o 21o -1/T Refold denatured proteins Unfolded Proteins Folded Proteins Heat shock response involves complex feedback and feedforward control. Temp cell Temp environ Alternative strategies Why does biology (and • Robust proteins – Temperature stability – Allelic variants – Paralogous isozymes advanced technology) overwhelmingly opt for the complex control systems instead of just robust components? • Regulate temperature • Thermotax • Heat shock response – Up regulate chaperones and proteases – Refold or degraded denatured proteins T dependent DnaK : Punfold T dependent E. Coli Heat Shock 3 2 translation rate 032 1 s 0.03 kdis t 32 fr ee Dna K 0 DnaK fr ee Dnak translation & transcription dynamics k 21 (with Kurata, El-Samad, Khammash, Yi) protease k3 FtsH 0 32 d eg rad ation rate 32 : Dn a K rpoH gene Outer Feedback Loop r1 Transcription r2 32 - Heat - hsp1 Local Loop mRNA Feedforward Translation 32 Heat stabilizes hsp2 Transcription & Translation FtsH Lon DnaK GroL GroS 32 : p ro tea se Proteases Chaperones 32 Heat 32 : Dn a K : FtsH Heater Thermostat Tail Added mass Moves the center of pressure aft. Moves the center of mass forward. Thus stabilizing forward flight. At the expense of extra weight and drag. For minimum weight & drag, (and other performance issues) eliminate fuselage and tail. Why do we love building robust systems from highly uncertain and unstable components? d (disturbance) r - P + y P(r ) d Assumptions on components: • Everything just numbers • Uncertainty in P • Higher gain = more uncertain y ( P P)r d P1 P2 P1 P2 P1 P2 d (disturbance) r r - + P d - G K + y P(r ) d y G r GK ( y d ) Negative feedback 1 y GSr Sd 1 S r Sd K S 1 1 GK r d - G + y K Design recipe: • 1 >> K >> 1/G • G >> 1/K >> 1 • G maximally uncertain! • K small, low uncertainty 1 G 1 GK 1 K 1 S 1 y r K Results for y (1/K )r: • high gain • low uncertainty • d attenuated 1 y GSr Sd 1 S r Sd K S = sensitivity function S 1 1 GK r d - G + y K Extensions to: • Dynamics • Multivariable • Nonlinear • Structured uncertainty All cost more computationally. Design recipe: • 1 >> K >> 1/G • G >> 1/K >> 1 • G maximally uncertain! • K small, low uncertainty Results for y (1/K )r: • high gain • low uncertainty • d attenuated r - G y Uncertain high gain K Transcription/translation Microtubule formation Neurogenesis Angiogenesis Antibody production Chemotaxis …. Regulatory feedback control Summary • Primitive technologies build fragile systems from precision components. • Advanced technologies build robust systems from sloppy components. • There are many other examples of regulator strategies deliberately employing uncertain and stochastic components… • …to create robust systems. • High gain negative feedback is the most powerful mechanism, and also the most dangerous. • In addition to the added complexity, what can go wrong? d d (disturbance) y - G y + + F GK F K y F ( y) d 1 y d 1 F 1 d if F F 1 1 y d 1 F If y, d and F are just numbers: y 1 S d 1 F S = sensitivity function d y + F S measures disturbance rejection. It’s convenient to study ln(S). Negative F ( F 0) ln( S ) 0 Disturbance attenuated Positive F ( F 0) ln( S ) 0 Disturbance amplified y 1 S d 1 F ln(S) F>0 ln(S) > 0 amplification F F<0 ln(S) < 0 attenuation ln( |S| ) Negative F ( F 0) ln( S ) 0 Disturbance attenuated Positive F ( F 0) ln( S ) 0 Disturbance amplified y 1 S d 1 F ln(S) F1 ln(S) extreme sensitivity F F ln(S) extreme robustness d + y F 1 S 1 F If these model physical processes, then d and y are signals and F is an operator. We can still define S( = |Y( /D( | where E and D are the Fourier transforms of y and d. ( If F is linear, then S is independent of D.) Under assumptions that are consistent with F and d modeling physical systems (in particular, causality), it is possible to prove that: log S ( ) d 0 ( F 0) ln( S ) 0 attenuate ( F 0) ln( S ) 0 amplify he amplification (F>0) must at least balance the attenuation (F<0). log|S | (Bode, ~1940) ln|S| log|S | F …yet fragile ln|S| log|S | Robust F Robustness of HOT systems Fragile Robust (to known and designed-for uncertainties) Fragile (to unknown or rare perturbations) Robust Uncertainties Feedback and robustness • Negative feedback is both the most powerful and most dangerous mechanism for robustness. • It is everywhere in engineering, but appears hidden as long as it works. • Biology seems to use it even more aggressively, but also uses other familiar engineering strategies: – – – – – Positive feedback to create switches (digital systems) Protocol stacks Feedforward control Randomized strategies Coding Robustness Complexity Current research • So far, this is all undergraduate level material • Current research involves lots of math not traditionally thought of as “applied” • New theoretical connections between robustness, evolvability, and verifiability • Beginnings of a more integrated theory of control, communications and computing • Both biology and the future of ubiquitous, embedded networking will drive the development of new mathematics. Robustness of HOT systems Fragile Robust (to known and designed-for uncertainties) Fragile (to unknown or rare perturbations) Robust Uncertainties Robustness of HOT systems Fragile Humans Archaea Chess Meteors Robust Robustness of HOT systems Fragile Humans Archaea Humans + machines? Chess Meteors Machines Robust Diseases of complexity Fragile Cancer Epidemics Viral infections Auto-immune disease Uncertainty Robust • In a system – Environmental perturbations – Component variations • In a model – – – – Parameter variations Unmodeled dynamics Assumptions Noise Sources of uncertainty Fragile F () Robust F () ? Fragile Sources of uncertainty Robust F () ? Typically NP hard. • If true, there is always a short proof. • Which may be hard to find. , F () ? Typically coNP hard. Fundamental asymmetries* • Between P and NP • Between NP and coNP • More important problem. • Short proofs may not exist. * Unless they’re the same… How do we prove that , F () ? • Standard techniques include relaxations, Grobner bases, resultants, numerical homotopy, etc… • Powerful new method based on real algebraic geometry and semidefinite programming (Parrilo, Shor, …) • Nested series of polynomial time relaxations search for polynomial sized certificates • Exhausts coNP (but no uniform bound) • Relaxations have both computational and physical interpretations • Beats gold standard algorithms (eg MAX CUT) handcrafted for special cases • Completely changes the P/NP/coNP picture Bacterial chemotaxis Random walk Ligand Motion Motor Bacterial chemotaxis (Yi, Huang, Simon, Doyle) Biased random walk gradient Ligand Motion Signal Transduction Motor YehC p High gain (cooperativity) “ultrasensitivity” References: Cluzel, Surette, Leibler Ligand Motion Signal Transduction Motor YehC p ligand binding FAST motor +ATT -ATT flagellar motor R +CH3 SLOW MCPs W P A MCPs -CH3 ATP ATP Pi B A P P Y ~ B CW W Z ADP Y Pi Motor References: Cluzel, Surette, Leibler + Alon, Barkai, Bray, Simon, Spiro, Stock, Berg, … Signal Transduction YehC p ligand binding moto r FAST +ATT -ATT R +CH3 MCPs flagellar motor SLOW W P A MCPs -CH3 ATP ATP Pi B W P A Y ~ B CW P Z ADP Y Pi ligand binding moto r FAST +ATT -ATT flagellar motor MCPs MCPs W CW W A Y ~ A P ATP ATP P Z ADP Y Pi Fast (ligand and phosphorylation) Short time Yp response 1 Ligand 0 0 1 2 3 Che Yp Extend run (more ligand) 4 5 6 Barkai, et al No methylation 0 1 2 3 4 5 Time (seconds) 6 Slow (de-) methylation dynamics R +CH3 MCPs SLOW W P A MCPs -CH3 ATP ATP Pi B A ~ B W P ADP ligand binding moto r FAST +ATT -ATT R +CH3 MCPs flagellar motor SLOW W P A MCPs -CH3 ATP ATP Pi B W P A Y ~ B CW P Z ADP Y Pi Long time Yp response 5 3 1 0 0 1000 2000 3000 4000 5000 6000 7000 5000 6000 7000 No methylation B-L 0 1000 2000 3000 4000 Time (seconds) Tumble (less ligand) Ligand No methylation Extend run (more ligand) Biologists call this “perfect adaptation” • Methylation produces “perfect adaptation” by integral feedback. • Integral feedback is ubiquitous in both engineering systems and biological systems. • Integral feedback is necessary for robust perfect adaptation. Tumbling bias Perfect adaptation is necessary … ligand YehC p YehC p Motor Signal Transduction Perfect adaptation is necessary … Tumbling bias …to keep CheYp in the responsive range of the motor. ligand YehC p Fine tuned or robust ? • Maybe just not the right question. • Fine tuned for robustness… • …with resource costs and new fragilities as the price. Biochemical Network: E. Coli Metabolism Mass Transfer in Metabolism* + Regulatory Interactions Complexity Robustness Supplies Materials & Energy Supplies Robustness From Adam Arkin * from: EcoCYC by Peter Karp What about ? • • • • • • • • • • Information & entropy Fractals & self-similarity Chaos Criticality and power laws Undecidability Fuzzy logic, neural nets, genetic algorithms Emergence Self-organization Complex adaptive systems New science of complexity • Not really about complexity • These concepts themselves are “robust, yet fragile” • Powerful in their niche • Brittle (break easily) when moved or extended • Some are relevant to biology and engineering systems • Comfortably reductionist • Remarkably useful in getting published Criticality and power laws • Tuning 1-2 parameters critical point • In certain model systems (percolation, Ising, …) power laws and universality iff at criticality. • Physics: power laws are suggestive of criticality • Engineers/mathematicians have opposite interpretation: – – – – Power laws arise from tuning and optimization. Criticality is a very rare and extreme special case. What if many parameters are optimized? Are evolution and engineering design different? How? • Which perspective has greater explanatory power for power laws in natural and man-made systems? 6 5 Frequency (Huffman) (Crovella) 4 Cumulative Data compression WWW files Mbytes 3 Forest fires 1000 km2 2 (Malamud) 1 Los Alamos fire 0 -1 -6 -5 Decimated data Log (base 10) -4 -3 -2 -1 0 1 Size of events 2 Size of events x vs. frequency log(probability) dP ( 1) p( x) x dx log(Prob > size) log(rank) Px log(size) 6 Web files 5 Codewords 4 Cumulative Frequency -1 3 Fires 2 -1/2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 Size of events Log (base 10) 2 The HOT view of power laws • Engineers design (and evolution selects) for systems with certain typical properties: • Optimized for average (mean) behavior • Optimizing the mean often (but not always) yields high variance and heavy tails • Power laws arise from heavy tails when there is enough aggregate data • One symptom of “robust, yet fragile” Based on frequencies of source word occurrences, Select code words. To minimize message length. Source coding for data compression Shannon coding Data Compression • Ignore value of information, consider only “surprise” • Compress average codeword length (over stochastic ensembles of source words rather than actual files) • Constraint on codewords of unique decodability • Equivalent to building barriers in a zero dimensional tree • Optimal distribution (exponential) and optimal cost are: length li log( pi ) pi exp(cli ) Avg. length = pl pi log( pi ) i i length li log( pi ) pi exp(cli ) Data 6 5 How well does the model predict the data? DC 4 3 2 1 0 Avg. length = pl pi log( pi ) i i -1 0 1 2 length li log( pi ) pi exp(cli ) Data + Model 6 5 How well does the model predict the data? DC 4 3 Not surprising, because the file was compressed using Shannon theory. 2 1 0 Avg. length = pl pi log( pi ) i i -1 0 1 2 Small discrepancy due to integer lengths. Web layout as generalized “source coding” • Keep parts of Shannon abstraction: – Minimize downloaded file size – Averaged over an ensemble of user access • But add in feedback and topology, which completely breaks standard Shannon theory • Logical and aesthetic structure determines topology • Navigation involves dynamic user feedback • Breaks standard theory, but extensions are possible • Equivalent to building 0-dimensional barriers in a 1- dimensional tree of content document split into N files to minimize download time A toy website model (= 1-d grid HOT design) split into N files to minimize download time # links = # files Forest fires dynamics Weather Spark sources Intensity Frequency Extent Flora and fauna Topography Soil type Climate/season A HOT forest fire abstraction… Fire suppression mechanisms must stop a 1-d front. Burnt regions are 2-d Optimal strategies must tradeoff resources with risk. Generalized “coding” problems • Optimizing d-1 dimensional cuts in d dimensional spaces… • To minimize average size of files or fires, subject to resource constraint. • Models of greatly varying detail all give a consistent story. • Power laws have 1/d. • Completely unlike criticality. Data compression Web Fires Theory d=0 d=1 d=2 data compression web layout forest fires pi li c 1 (1 ) d P( l ) l d 0 1 d 1 d pi exp(cli ) d 0 Data 6 DC 5 WWW 4 3 FF 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Data + Model/Theory 6 DC 5 WWW 4 3 FF 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Forest fires? Fire suppression mechanisms must stop a 1-d front. Burnt regions are 2-d Forest fires? Geography could make d <2. California geography: further irresponsible speculation • Rugged terrain, mountains, deserts • Fractal dimension d 1? • Dry Santa Ana winds drive large ( 1-d) fires Data + HOT Model/Theory 6 5 California brushfires 4 d=1 3 2 1 FF (national) d=2 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Data + HOT+SOC 6 5 4 3 2 d=2 1 .15 0 -1 -6 SOC FF d=1 -5 -4 -3 -2 -1 0 1 2 Critical/SOC exponents are way off Data: > .5 SOC < .15 18 Sep 1998 Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 4 data sets 3 10 HOT FF d=2 2 10 1 10 SOC FF 0 10 -2 10 -1 10 0 10 1 10 2 10 Additional 3 data sets 3 10 4 10 Fires 1930-1990 Fires 1991-1995 SOC and HOT have very different power laws. d d=1 HOT d d=1 SOC d 1 10 1 d • HOT decreases with dimension. • SOC increases with dimension. • HOT yields compact events of nontrivial size. • SOC has infinitesimal, fractal events. HOT SOC infinitesimal size large SOC and HOT are extremely different. SOC HOT Data Max event size Infinitesimal Large Large Large event shape Fractal Compact Compact Slope Small Large Large Dimension d d-1 1/d 1/d HOT SOC SOC and HOT are extremely different. SOC HOT & Data Max event size Infinitesimal Large Large event shape Fractal Compact Slope Small Large Dimension d d-1 1/d HOT SOC Robust Log(freq.) cumulative Gaussian, Exponential Log(event sizes) yet fragile Power laws are inevitable. Gaussian log(prob>size) Improved design, more resources log(size) Power laws summary • Power laws are ubiquitous • HOT may be a unifying perspective for many • Criticality, SOC is an interesting and extreme special case… • … but very rare in the lab, and even much rarer still outside it. • Viewing a complex system as HOT is just the beginning of study. • The real work is in new Internet protocol design, forest fire suppression strategies, etc… Universal network behavior? throughput Congestion induced “phase transition.” Similar for: • Power grid? • Freeway traffic? • Gene regulation? • Ecosystems? • Finance? demand throughput Congestion induced “phase transition.” log(P>) demand Power laws log(file size) Web/Internet? 3 H 2 Networks log(thru-put) Making a “random network:” • Remove protocols – No IP routing – No TCP congestion control • Broadcast everything Many orders of magnitude slower random networks Broadcast Network log(demand) Networks real networks log(thru-put) HOT random networks Broadcast Network log(demand) Turbulence streamlined pipes flow HOT random pipes pressure drop flow HOT turbulence? Robust, yet fragile? streamlined pipes HOT random pipes pressure drop • Through streamlined design • High throughput • Robust to bifurcation transition (Reynolds number) • Yet fragile to small perturbations • Which are irrelevant for more “generic” flows Shear flow turbulence summary • Shear flows are ubiquitous and important • HOT may be a unifying perspective • Chaos is interesting, but may not be very important for many important flows • Viewing a turbulent or transitioning flow as HOT is just the beginning of study The yield/density curve predicted using random ensembles is way off. designed HOT Yield, flow, … random Densities, pressure,… Similar for: • Power grid • Freeway traffic • Gene regulation • Ecosystems • Finance? Turbulence in shear flows Kumar Bobba, Bassam Bamieh wings channels Turbulence is the graveyard of theories. pipes Hans Liepmann Caltech Chaos and turbulence • The orthodox view: • Adjusting 1 parameter (Reynolds number) leads to a bifurcation cascade to chaos • Turbulence transition is a bifurcation • Turbulent flows are chaotic, intrinsically nonlinear • There are certainly many situations where this view is useful. low velocity equilibrium periodic high chaotic pressure drop average flow speed “random” pipe flow (average speed) laminar bifurcation turbulent pressure (drop) Random pipes are like bluff bodies. flow Typical flow pressure Streamline wings channels pipes “theory” log(flow) laminar experiment streamlined pipe turbulent Random pipe log(pressure) log(flow) streamlined pipe Random pipe 2 10 3 10 4 10 5 10 log(Re) This transition is extremely delicate (and controversial). streamlined pipe It can be promoted (or delayed!) with tiny 2 10 perturbations. Random pipe 3 10 4 10 5 10 log(Re) Transition to turbulence is promoted (occurs at lower speeds) by Surface roughness Inlet distortions Vibrations Thermodynamic fluctuations? Non-Newtonian effects? None of which makes much difference for “random” pipes. Random pipe 102 3 10 104 105 Shark skin delays transition to turbulence 80 ppm Guar log(flow) water It can be reduced with small amounts of polymers. log(pressure) flow HOT turbulence? Robust, yet fragile? streamlined pipes HOT random pipes pressure drop • Through streamlined design • High throughput • Robust to bifurcation transition (Reynolds number) • Yet fragile to small perturbations • Which are irrelevant for more “generic” flows streamwise Couette flow upflow low speed streaks downflow high-speed region From Kline Spanwise periodic Streamwise constant perturbation Spanwise periodic Streamwise constant perturbation y flow x position z y position flow v flow u velocity z w x v flow velocity w u u 0 u u u p 1 u t R u v w 0 x y z ux vx t w x y position u (u, v, w) uy vy wy u uz / x 2 2 2 1 vz 2 2 2 v / y p R x y z / z wz w 0 x flow x v velocity z flow u w u 0 u u u p 1 u t R u v w 0 x y z ux vx t w x y position u (u, v, w) uy vy wy u uz / x 2 2 2 1 vz 2 2 2 v / y p R x y z / z wz w 0 x flow x v velocity z flow u w u 0 u u u p 1 u t R u v w 0 x y z ux vx t w x u (u, v, w) uy vy wy ( y, x, t ) v ,w z y 2d NS u uz / x 2 2 2 1 vz 2 2 2 v / y p R x y z / z wz w u u u 1 u t z y y z R 1 2 t z y y z R 0 x y v flow position x velocity z 2 dimensions ( y, x, t ) v ,w z y 2d-3c model flow u w 3 components u u u 1 u t z y y z R 1 2 t z y y z R 0 x These equations are globally stable! Laminar flow is global attractor. ( y, x, t ) v ,w z y 2d-3c model u u u 1 u t z y y z R 1 2 t z y y z R R 2 R Total energy R energy 3 (Bamieh and Dahleh) t 10 energyN=10R=1000t=1000alpha=2 5 Total energy 0 energy 10 10 10 vortices -5 -10 0 200 400 600 t 800 1000 u ( z , y, t ) ( z , y, t ) u ( z , y, t ) ( z , y, t ) What you’ll see next. Log-log plot of time response. Random initial conditions on ( z, y, t 0) concentrated at lower boundar u ( z , y, t ) Streamwise streaks. ( z , y, t ) u ( z , y, t ) ( z , y, t ) Long range correlation. Exponential decay. flow HOT turbulence? Robust, yet fragile? streamlined pipes HOT random pipes pressure drop • Through streamlined design • High throughput • Robust to bifurcation transition (Reynolds number) • Yet fragile to small perturbations • Which are irrelevant for more “generic” flows Complexity, chaos and criticality • The orthodox view: – Power laws suggest criticality – Turbulence is chaos • HOT view: – Robust design often leads to power laws – Just one symptom of “robust, yet fragile” – Shear flow turbulence is noise amplification • Other orthodoxies: – Dissipation, time irreversibility, ergodicity and mixing – Quantum to classical transitions – Quantum measurement and decoherence Epilogue • HOT may make little difference for explaining much of traditional physics lab experiments, • So if you’re happy with orthodox treatments of power laws, turbulence, dissipation, quantum measurement, etc then you can ignore HOT. • Otherwise, the differences between the orthodox and HOT views are large and profound, particularly for… • Forward or reverse (eg biology) engineering complex, highly designed or evolved systems, • But perhaps also, surprisingly, for some foundational problems in physics