UNIVERSAL SCIENCE UNIVERSAL SCIENCE OF COMPLEXITY OF COMPLEXITY - PowerPoint PPT Presentation

Andrei P. Kirilyuk Andrei P. Kirilyuk Institute of Metal Physics, Kiev, Ukraine Institute of Metal Physics, Kiev, Ukraine http://myprofile.cos.com/mammoth http://myprofile.cos.com/mammoth UNIVERSAL SCIENCE UNIVERSAL SCIENCE OF COMPLEXITY OF COMPLEXITY Consistent Understanding of Consistent Understanding of Ecological, Living Living and Intelligent and Intelligent Ecological, System Dynamics System Dynamics

Escaping Complexity Escaping Complexity • Can Can real real , higher , higher- -complexity system dynamics complexity system dynamics • be understood at the level of rigour level of rigour of of be understood at the Newtonian science? Newtonian science? • The problem of The problem of unreduced unreduced many many- -body body • interaction ( ( unsolved unsolved in Newtonian science) in Newtonian science) interaction • The unique possibility for a new progress of The unique possibility for a new progress of • fundamental science (otherwise (otherwise “ “ending ending” ”) ) fundamental science • Universal Science of Complexity Universal Science of Complexity • http://books.google.com/books?ie=UTF- -8&hl=en&vid=ISBN9660001169&id=V1cmKSRM3EIC 8&hl=en&vid=ISBN9660001169&id=V1cmKSRM3EIC http://books.google.com/books?ie=UTF

Universal Science of Complexity Unreduced Interaction Unreduced Interaction Cosmos ICT Dynamic Complexity Dynamic Complexity Nano Brain Bio http://arxiv.org/find/quant- http://arxiv.org/find/quant -ph,gr ph,gr- -qc,physics/1/au:+Kirilyuk/0/1/0/all/0/1 qc,physics/1/au:+Kirilyuk/0/1/0/all/0/1

Multivalued Dynamics of Unreduced Interaction Multivalued Dynamics of Unreduced Interaction Arbitrary many- -body interaction process: body interaction process: Arbitrary many   N N ( ) ( ) ( ) ( )   ( ) + = =   h q V q q , Ψ Q E Ψ Q Q q q , ,..., q   , ∑ ∑ 1 2 N  k k kl k l    = >   k 0 l k       or   N N ( ) ( ) ( ) ( ) ( ) ( )   ξ + + ξ ξ = ξ ξ ≡   + h h q V , q V q q , , Q E Ψ , Q q   , Ψ 0 ∑ ∑ 0  k k 0 k k kl k l    = >   k 1 l k       The unreduced (nonperturbative) general solution is always probabilistic The unreduced (nonperturbative) general solution is always probabilistic (phenomenon of dynamic multivaluedness (phenomenon of dynamic multivaluedness = = i intrinsic chaoticity ntrinsic chaoticity ): ): N ℜ ⊕ ( ) ( ) ρ ξ = ∑ ρ ξ , Q , Q r = r 1 Dynamically determined probability Dynamically determined probability N α α = , ∑ r = 1 r r N ℜ r

Unreduced Interaction Dynamics Unreduced Interaction Dynamics Arbitrary interaction process in terms of (free) component eigenvalues: Arbitrary interaction process in terms of (free) component eigen values: ( ) ( ) ( ) ( ) ( ) ξ ψ ξ + ξ ψ ξ = η ψ ξ h V ′ ′ ∑ 0 n nn n n n ′ n where the total system state- -function is obtained as function is obtained as where the total system state ( ) ( ) ( ) ( ) ( ) ( ) ( ) ξ = ψ ϕ ϕ ϕ ≡ ψ ξ , Q q q q ... q Q ∑ ∑ Ψ n 0 n n n n Φ n 1 1 2 2 N N 1 2 N ( ) ≡ n n n , ,..., n n 1 2 N Usual perturbative (mean perturbative (mean- -field) field) approximations: approximations: Usual ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ξ + ξ + ξ ψ ξ = η ψ ξ ξ < ξ < ξ ɶ ɶ h V V , V V V   ′ ∑ 0 nn n n n n 0 n nn   ′ n

Unreduced general solution of the same problem: of the same problem: Unreduced general solution N ℜ ⊕ ( ) ( ) ( ) ( ) ( ) 2 2 ρ ξ ≡ ξ = ρ ξ ρ ξ = ξ , Q , Q , Q , , Q , Q ∑ Ψ r r Ψ r = r 1 ( ) ( ) ( ) ( ) ( )   Φ ψ ξ ξ ψ ′ ξ ′ ξ ψ ′ ξ ′ * r 0 0 Q d V ′ ′ n ni ∫ ni n i 0 0     Ω ( ) ( ) ( ) ξ ξ = Φ ψ ξ + r r , Q c Q ∑ ∑   Ψ r i i 0 0 η − η − ε r 0   ′ i ni n 0   ′  n i ,    i ( ) ψ ξ η r r { , } where i i are solutions solutions of the of the effective potential (EP) effective potential (EP) equation equation where 0 are ( ) ( ) ( ) ( ) ( ) ξ ψ ξ + ξ η ψ ξ = ηψ ξ h V ; eff 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ′ ′ ′ ′ ξ ψ ξ ξ ψ ξ ξ ψ ξ * r 0 0 V d V ′ ′ ∫ n ni ni n i 0 0 0 ( ) ( ) ( ) ( ) Ω ξ ξ η ψ ξ = ξ ψ ξ + r r r V ; V ∑ eff i i i 0 00 0 η − η − ε r 0 ′ i ni n 0 ′ n i , Δ = λ = Δ η , Δ = Δ , Δ = Δ r Elementary length x time t x action A V t v i eff 0

Unreduced Interaction: Dynamic Multivaluedness (Chaos) Unreduced Interaction: Dynamic Multivaluedness (Chaos) Dynamically redundant interaction result: Second First incompatible object object system realisations N points N points N points N points ( modes ) ( modes ( modes ) ) ( modes ) First Second Third INTERACTION INTERACTION a 1 a3 a1 a2 b 2 b2 b3 b1 c 3 c3 c2 c1 (N N) combinations combinations × of mode entanglement of mode entanglement (a ( a1 1, ,a a2 2, ,a a3 3, ,b b1 1, ,b b2 2,etc.) ,etc.) Permanent realisation change realisation change Permanent ⇓ ⇓ in causally (dynamically) random causally (dynamically) random order order in N- -fold redundance fold redundance N

Unreduced Interaction Complexity Unreduced Interaction Complexity UNIVERSAL DEFINITION OF DYNAMIC (INTERACTION) COMPLEXITY: UNIVERSAL DEFINITION OF DYNAMIC (INTERACTION) COMPLEXITY: ( ) ( ) dC = > = C C N , 0, C 1 0 ℜ dN ℜ N ℜ where is the (dynamically derived) system realisation number where is the (dynamically derived) system realisation number ( ) ( ) = = − C C ln N , C C N 1 , etc. ℜ ℜ for example: for example: 0 0 Universal dynamic complexity includes includes intrinsic intrinsic chaoticity chaoticity Universal dynamic complexity due to the dynamically probabilistic due to the dynamically probabilistic problem solution: problem solution: N ℜ ( ) ⊕ ( ) ρ ξ = ∑ ρ ξ , Q , Q r = r 1 with the dynamically determined probability dynamically determined probability with the N α α = , ∑ r = 1 r r N ℜ r

Unreduced Complexity Measures Unreduced Complexity Measures Two universal, emerging emerging forms of complexity, forms of complexity, space space and and time time Two universal, Δ = λ = Δ η Δ = Δ r x , t x v r i 0 A Generalised action action is the simplest combination of space & time: Generalised is the simplest combination of space & time: Δ = Δ − Δ p x E t A = ∝ pdx N ℜ A ∫ is a universal integral integral complexity measure complexity measure is a universal Differential complexity measures: complexity measures: Differential ∂ Δ A A = = p = momentum momentum ∂ Δ t const x x ( spatial spatial rate of realisation emergence) rate of realisation emergence) ( ∂ Δ A A = = − = − 2 E m v = energy / / mass mass x const energy ∂ Δ 0 t t ( temporal temporal rate of realisation emergence) rate of realisation emergence) ( Δ x v = = p E m , = v v Δ Dispersion relation : : � causal relativity causal relativity Dispersion relation � t c 2

Generalised wavefunction (distribution function) Generalised wavefunction (distribution function) Total number of unreduced EP eigenvalues: Total number of unreduced EP eigenvalues: 2 N ) 2 ξ ( ξ N ξ ) ξ � ℜ = ξ “ N max = N N ξ ( N N ξ N q + 1) = ( N N ξ N q + N N ξ � N N ℜ = N N ξ “regular regular” ” realisations realisations N max = q + 1) = ( q + N ξ ξ N N ξ ξ eigen of N of N q q eigen eigen- -solutions each + solutions each + “ “incomplete incomplete” ” set of set of N eigen- -solutions = solutions = transitional realisation , , generalised wavefunction generalised wavefunction , or , or distribution function distribution function transitional realisation ⇓ ⇓ Ψ ( mechanical wavefunction Ψ causal extension of usual quantum of usual quantum- -mechanical wavefunction ( x x ) ) causal extension transiently weak EP, disentangled components, system restructuring ng transiently weak EP, disentangled components, system restructuri � � Ψ ( ⏐ Ψ : α α r 2 = ⏐ ) ⏐ ⏐ 2 Causally generalised Born probability rule Born probability rule : ( X X r Causally generalised r = r ) � � A ( complexity A Generalised Hamilton- Generalised Hamilton -Jacobi equation Jacobi equation for action for action- -complexity ( x x ) ) Δ A Δ Ψ Ψ / Ψ , Δ 0 Δ / Ψ A = = ̶̶ A A 0 + causal quantisation rule causal quantisation rule , + � � universal Schrö ödinger equation dinger equation universal Schr Δ Ψ Δ ( )   Ψ ˆ A = H x , , t x t , = = 0 x const  t const  Δ Δ t x  