Toward a Coupled Oscillator Model of the Mechanisms of Universal Evolution and Development Georgi Georgiev Worcester Polytechnic Institute, Assumption College and Tufts University Work with Thanh Vu, from Assumption College and Atanu Chatterjee and Germano Iannacchione from WPI
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential sinusoidal model 7. External noise – stochastic 8. Examples: Cities, Economy, techno, metabolic cycle, photosynthesis 9. Conclusions: A, f, H all increase exponentially
Cosmic Calendar By Carl Sagan
Accelerating rate of self-organization By Ray Kurzweil
Cosmic Evolution By Eric Chaisson
FERD as measure for complexity By Eric Chaisson
By Ray Kurzweil
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential sinusoidal model 7. External noise – stochastic 8. Examples: Cities, Economy, techno, metabolic cycle, photosynthesis 9. Conclusions: A, f, H all increase exponentially
A first principle • The Least Action Principle for a system states: all processes in nature occur with the least expenditure of action, which is the product of time and energy for them. t 2 ∑ ∑ ∫ δ = δ = 0 I ij L ij dt ij ij t 1
Quantity of organization α = hnm nm ∑ I ij ij • Organization, α , is inversely proportional to the average number of quanta of action per one element and one edge crossing of a network. • n is the number of elements in the system and m is the number of edge crossings per unit time.
Total flow and number of quanta • Recognize that nm, the total number of edge crossings, is the flow, ϕ , of elements per unit time in the network: ϕ =nm. nm ∑ I ij • Recognize that is the total number of quanta of action in the ij Q = h system in certain interval of time. • Therefore:
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential sinusoidal model 7. External noise – stochastic 8. Examples: Cities, Economy, techno, metabolic cycle, photosynthesis 9. Conclusions: A, f, H all increase exponentially
Positive feedback model – exponential solutions ɺ Quantity α α φ Quality = a + a Q + a + 11 12 13 α Q ɺ α φ Q= a + a Q+ a 21 22 23 ɺ + � + φ α φ = a + a Q+ a 31 32 33 FERD 10 100 10 80 12 10 Log( φ ) 10 60 11 10 10 40 10 10 10 20 9 10 Flow 8 10 200 400 600 800 1000 time 7 10 Φ 6 10 5 10 γ τ t 4 α α α = ηφ = e 10 0 2 4 6 8 10 12 14 0 Time [sec] If α or φ stops increasing, 8 x 10 β t Q=Q e φ a Q + a µ ɺ φ = χ Q ɺ 0 or decreases, the other α 12 13 φ = δ t φ φ = e changes in the same α a + a Q 0 31 32 direction.
Exponential growth of α and Q in time -24 36 10 10 -25 10 35 10 Total number of quanta -26 10 Alpha 34 10 -27 10 33 10 -28 10 -29 32 10 10 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Time [sec] 8 Time [sec] 8 x 10 x 10 Data for CPUs since 1971 (closed circles) and an exponential fit (solid line). The transition from single to multicore processors around time 10 9 [sec], does not affect the trend. α and Q do not increase smoothly but in steps. An edge in this system is defined to be one computation. To calculate α , the potential energy of the electrons was taken to be constant. The Lagrangian was then calculated using the kinetic energy. The data for Million Instructions Per Second (MIPS) for each processor was divided by the thermal design power and multiplied by the table value of the Planck’s constant, to solve for α .
α and Q in a positive feedback -24 10 -25 10 -26 10 Alpha -27 10 -28 10 -29 10 32 33 34 35 36 10 10 10 10 10 Total number of quanta Data for α and Q (closed circles) and an power law fit (solid line) with variations around the average.
Confirming Chaisson’s data for CPUs � (FERD) as a function of t (time). The data are from 1982 starting with Intel 286, to 2012.
Power law relations between α , Q and � . Data are filled circles and solid line is the fit. The data are from 1982 starting with Intel 286, to 2012, ending with Intel Core i7 3770k. There is a good agreement between the data and a power law fit.
Expanding to more mutually dependent functions – interfucntions Quality Quantity + α Q 10 100 10 80 Log(N) + + 10 60 � N 10 40 + 10 20 Number Flow 200 400 600 800 1000 10 10 time 9 10 8 10 τ γ t α = η α α = e N Transistor count 7 10 0 λ φ = ω N β t Q=Q e 6 10 0 ψ = σ N Q 5 δ t 10 φ φ = e 0 4 10 ε t N N = e 0 3 10 0 2 4 6 8 10 12 14 Time [sec] 8 x 10
Positive feedback model solutions
The cone of development • This cone has for levels all of the major stages in levels of organization that we know of. • From here we get a sense that there is discreteness, in progress and self- organization in nature. Time Future Civilization Organisms Exponential growth Molecules Atoms Particles Ln(Interfunction)
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential sinusoidal model 7. External noise – stochastic 8. Examples: Cities, Economy, techno, metabolic cycle, photosynthesis 9. Conclusions: A, f, H all increase exponentially
Negative Feedback Proposed Mechanism of development A system of coupled oscillators f j = − − F k f ( f ) h j j H , Which has a solution of the form: = ω y A cos( t ) fn f 1 - F h - F h …….. Δ H f j,H + F h + F h f 2 f 3 Disease – elastic limit Death – plastic limit
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential-sinusoidal model 7. External noise – stochastic 8. Examples: Cities, Economy, techno, metabolic cycle, photosynthesis 9. Conclusions: A, f, H all increase exponentially
Positive and Negative feedback loops + - - α α Q Q H H + • The solution of the system of coupled oscillators: = − − F k f ( f ) h j j H , where ct = f f e j H , j ,0 On a first approximation, best fit is wi th: c t c t c t = + ω f f e ( A Be cos( t e )) 1 2 3 j j ,0
Alpha vs Time -54 Δ H -56 -58 Ln(Alpha) -60 A On a first approximation, best fit is with: -62 c t c t c t = + ω f f e ( A Be cos( te )) 1 2 3 j j ,0 T -64 A, f, H all increase exponentially H -66 8 8 8 8 9 9 9 0.0 2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1.2x10 1.4x10 time (sec) Logarized fit: y = A + B*t + ln(C*exp(M*t) + D*cos(G*t*exp(H*t) + K))
Homeostatic (stability) limits H A T Δ H By Kurzweil Amplitude (A) is increasing and frequency (f=1/T) is increasing. The max deviation from the dynamic equilibrium exponential line (Homeostasis) – is the limit of elasticity of those Interfunctions, i.e. the homeostatic Limits. If the interfunctions deviate more from their Homeostatic values, the system destabilizes and falls apart.
Outline 1. The big questions: Sagan, Chaisson, Kurzweil 2. The search for universality across different systems 3. The Principle of Least Action as a driver for self-organization 4. Positive feedback model: exponential acceleration 5. Negative feedback model: sinusoidal oscillations 6. Combine the two: exponential sinusoidal model 7. External noise – stochastic 8. Examples: Economy, techno, cultural, biological, neural, Benard Cells, Computers. 9. Conclusions: A, f, H all increase exponentially
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