Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution

Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution

Complex nanosystem dynamics is analysed by the unreduced solution of arbitrary many-body interaction problem, leading to the fundamental dynamic multivaluedness and universal definition of dynamic complexity in terms of the number of system realisations. As shown, the genuine quantum and classical c...

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Дата:2016
Автор: Kirilyuk, A.P.
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Опубліковано: Інститут металофізики ім. Г.В. Курдюмова НАН України 2016
Назва видання:Наносистеми, наноматеріали, нанотехнології
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Цитувати:Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution / A.P. Kirilyuk // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2016. — Т. 14, № 1. — С. 1-26. — Бібліогр.: 19 назв. — англ.

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spelling irk-123456789-1175892017-05-26T03:03:15Z Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution Kirilyuk, A.P. Complex nanosystem dynamics is analysed by the unreduced solution of arbitrary many-body interaction problem, leading to the fundamental dynamic multivaluedness and universal definition of dynamic complexity in terms of the number of system realisations. As shown, the genuine quantum and classical chaos can only be strong for a free-interaction nanoscale system providing exponentially huge, ‘magic’ efficiency of such unreduced interaction dynamics, which underlies the properties of life, intelligence, and consciousness. Various more or less chaotic regimes of irreducibly complex nanosystem dynamics as well as the rigorously specified transitions between them are reviewed. The obtained unified formalism for description of the unreduced complex nanosystem dynamics is based on the universal symmetry (conservation and transformation) of complexity unifying the extended versions of all usual laws and principles. The main principles of thus obtained new, complex-dynamical nanobiotechnology paradigm are summarised, and as shown, it is the only viable way of further sustainable nanotechnology and society development in the spirit of coevolution of the natural and artificial system complexity. Складну динаміку наносистем проаналізовано за допомогою нередукованого розв'язку задачі довільної взаємодії багатьох тіл, яка приводить до фундаментальної динамічної багатозначности й універсального визначення динамічної складности у термінах кількости реалізацій системи. Показано, що в системі наномасштабних розмірів справжній квантовий і класичний хаос може бути тільки сильним, зумовлюючи експоненційно величезну, «магічну» ефективність динаміки такої нередукованої взаємодії, яка лежіть в основі властивостей життя, інтелекту та свідомости. Розглянуто різні, більш або менш хаотичні режими нередукованої складної динаміки наносистем, а також строго визначені переходи між ними. Одержано об'єднаний формалізм опису нередукованої складної динаміки наносистем, що ґрунтується на універсальній симетрії (збереженні та перетворенні) складності, яка поєднує розширені версії усіх звичайних законів і принципів. Оглянуто основні засади одержаної таким чином нової парадигми складно-динамічної нанобіотехнології та показано, що це єдиний шлях наступного сталого розвитку нанотехнології і суспільства у дусі коеволюції складности природніх і штучних систем. Сложная динамика наносистем проанализирована с помощью нередуцированного решения задачи произвольного взаимодействия многих тел, которое приводит к фундаментальной динамической многозначности и универсальному определению динамической сложности в терминах числа реализаций системы. Показано, что в системе наномасштабных размеров истинный квантовый и классический хаос может быть лишь сильным, давая экспоненциально огромную, «магическую» эффективность динамики такого нередуцированного взаимодействия, которая лежит в основе свойств жизни, интеллекта и сознания. Рассмотрены различные, более или менее хаотичные режимы нередуцированной сложной динамики наносистем, а также строго определённые переходы между ними. Полученный объединённый формализм описания нередуцированной сложной динамики наносистем основан на универсальной симметрии (сохранении и превращении) сложности, которая объединяет расширенные версии всех обычных законов и принципов. Рассмотрены основные принципы полученной таким образом новой парадигмы сложно-динамической нанобиотехнологии и показано, что это единственный путь последующего устойчивого развития нанотехнологии и общества в духе коэволюции сложности естественных и искусственных систем. 2016 Article Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution / A.P. Kirilyuk // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2016. — Т. 14, № 1. — С. 1-26. — Бібліогр.: 19 назв. — англ. 1816-5230 PACS: 03.67.-a, 05.45.-a,05.65.+b,45.50.Jf,87.85.Qr,89.75.-k, 98.80.Bp http://dspace.nbuv.gov.ua/handle/123456789/117589 en Наносистеми, наноматеріали, нанотехнології Інститут металофізики ім. Г.В. Курдюмова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Complex nanosystem dynamics is analysed by the unreduced solution of arbitrary many-body interaction problem, leading to the fundamental dynamic multivaluedness and universal definition of dynamic complexity in terms of the number of system realisations. As shown, the genuine quantum and classical chaos can only be strong for a free-interaction nanoscale system providing exponentially huge, ‘magic’ efficiency of such unreduced interaction dynamics, which underlies the properties of life, intelligence, and consciousness. Various more or less chaotic regimes of irreducibly complex nanosystem dynamics as well as the rigorously specified transitions between them are reviewed. The obtained unified formalism for description of the unreduced complex nanosystem dynamics is based on the universal symmetry (conservation and transformation) of complexity unifying the extended versions of all usual laws and principles. The main principles of thus obtained new, complex-dynamical nanobiotechnology paradigm are summarised, and as shown, it is the only viable way of further sustainable nanotechnology and society development in the spirit of coevolution of the natural and artificial system complexity.
format Article
author Kirilyuk, A.P.
spellingShingle Kirilyuk, A.P.
Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
Наносистеми, наноматеріали, нанотехнології
author_facet Kirilyuk, A.P.
author_sort Kirilyuk, A.P.
title Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
title_short Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
title_full Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
title_fullStr Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
title_full_unstemmed Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution
title_sort complex-dynamical nanobiotechnology paradigm and intrinsically creative evolution
publisher Інститут металофізики ім. Г.В. Курдюмова НАН України
publishDate 2016
url http://dspace.nbuv.gov.ua/handle/123456789/117589
citation_txt Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution / A.P. Kirilyuk // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2016. — Т. 14, № 1. — С. 1-26. — Бібліогр.: 19 назв. — англ.
series Наносистеми, наноматеріали, нанотехнології
work_keys_str_mv AT kirilyukap complexdynamicalnanobiotechnologyparadigmandintrinsicallycreativeevolution
first_indexed 2025-07-08T12:30:15Z
last_indexed 2025-07-08T12:30:15Z
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fulltext 1 PACS numbers: 03.67.-a, 05.45.-a,05.65.+b,45.50.Jf,87.85.Qr,89.75.-k, 98.80.Bp Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution A. P. Kirilyuk G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine, 36 Academician Vernadsky Blvd., UA-03680 Kyyiv-142, Ukraine Complex nanosystem dynamics is analysed by the unreduced solution of arbitrary many-body interaction problem, leading to the fundamental dy- namic multivaluedness and universal definition of dynamic complexity in terms of the number of system realisations. As shown, the genuine quan- tum and classical chaos can only be strong for a free-interaction nanoscale system providing exponentially huge, ‘magic’ efficiency of such unre- duced interaction dynamics, which underlies the properties of life, intelli- gence, and consciousness. Various more or less chaotic regimes of irreduc- ibly complex nanosystem dynamics as well as the rigorously specified transitions between them are reviewed. The obtained unified formalism for description of the unreduced complex nanosystem dynamics is based on the universal symmetry (conservation and transformation) of complexi- ty unifying the extended versions of all usual laws and principles. The main principles of thus obtained new, complex-dynamical nanobiotechnol- ogy paradigm are summarised, and as shown, it is the only viable way of further sustainable nanotechnology and society development in the spirit of coevolution of the natural and artificial system complexity. Складну динаміку наносистем проаналізовано за допомогою нередуко- ваного розв'язку задачі довільної взаємодії багатьох тіл, яка приводить до фундаментальної динамічної багатозначности й універсального ви- значення динамічної складности у термінах кількости реалізацій сис- теми. Показано, що в системі наномасштабних розмірів справжній квантовий і класичний хаос може бути тільки сильним, зумовлюючи експоненційно величезну, «магічну» ефективність динаміки такої не- редукованої взаємодії, яка лежіть в основі властивостей життя, інтеле- кту та свідомости. Розглянуто різні, більш або менш хаотичні режими нередукованої складної динаміки наносистем, а також строго визначені переходи між ними. Одержано об'єднаний формалізм опису нередуко- ваної складної динаміки наносистем, що ґрунтується на універсальній симетрії (збереженні та перетворенні) складності, яка поєднує розши- Наносистеми, наноматеріали, нанотехнології Nanosistemi, Nanomateriali, Nanotehnologii 2016, т. 14, № 1, сс. 1—26 © 2016 ІМФ (Інститут металофізики ім. Г. В. Курдюмова НАН України) Надруковано в Україні. Фотокопіювання дозволено тільки відповідно до ліцензії 2 A. P. KIRILYUK рені версії усіх звичайних законів і принципів. Оглянуто основні заса- ди одержаної таким чином нової парадигми складно-динамічної нано- біотехнології та показано, що це єдиний шлях наступного сталого роз- витку нанотехнології і суспільства у дусі коеволюції складности приро- дніх і штучних систем. Сложная динамика наносистем проанализирована с помощью нереду- цированного решения задачи произвольного взаимодействия многих тел, которое приводит к фундаментальной динамической многозначно- сти и универсальному определению динамической сложности в терми- нах числа реализаций системы. Показано, что в системе наномасштаб- ных размеров истинный квантовый и классический хаос может быть лишь сильным, давая экспоненциально огромную, «магическую» эф- фективность динамики такого нередуцированного взаимодействия, ко- торая лежит в основе свойств жизни, интеллекта и сознания. Рассмот- рены различные, более или менее хаотичные режимы нередуцирован- ной сложной динамики наносистем, а также строго определённые пере- ходы между ними. Полученный объединённый формализм описания нередуцированной сложной динамики наносистем основан на универ- сальной симметрии (сохранении и превращении) сложности, которая объединяет расширенные версии всех обычных законов и принципов. Рассмотрены основные принципы полученной таким образом новой па- радигмы сложно-динамической нанобиотехнологии и показано, что это единственный путь последующего устойчивого развития нанотехноло- гии и общества в духе коэволюции сложности естественных и искус- ственных систем. Key words: dynamic multivaluedness, complexity, chaos, self-organisation, fractal, many-body problem, quantum mechanics, nanobiotechnology, quantum computers. Ключові слова: динамічні багатозначності, складність, хаос, самоорга- нізація, фрактал, проблема багатьох тіл, квантова механіка, нанотех- нології, квантові комп’ютери. Ключевые слова: динамические многозначности, сложность, хаос, са- моорганизация, фрактал, проблема многих тел, квантовая механика, нанотехнологии, квантовые компьютеры. (Received February 22, 2016) 1. INTRODUCTION While the huge and always growing scale of various nanotechnology applications becomes a major driving force of modern world devel- opment, the underlying fundamental science paradigm is mainly limited to traditional (quasi-)regular functioning of artificial nanostructures and nanomachines. However, as shown in previous COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 3 works [1, 2], the true nanoscale range of interacting system ele- ments inevitably leads to strongly irregular dynamic regime of uni- form, or global, chaos, as confirmed, in particular, by chaotic struc- ture and dynamics of natural nanomachines in living organisms. It is important that the difference between the two kinds of dy- namics involves the original, qualitatively deep phenomenon of dy- namic multivaluedness of any real interaction process [1—17], with its regular, dynamically single-valued projection being but a strong- ly simplified model of reality that becomes particularly inadequate in the case of ultimately small, nanoscale structures. The visible regularity of usual artificial nanostructures implies either their ac- tually greater, rather microscale characteristic sizes or a very crude approximation in their theoretical description that can lead to es- sential mistakes and vain hopes. This is the case, for example, of the extremely popular projects of unitary quantum computation, which cannot be realised as such because of the inevitable and pure- ly dynamic chaoticity [2]. On the other hand, the qualitatively dif- ferent kind of real, chaotic (multivalued) nanosystem dynamics opens other perspectives of surprisingly high, ‘magic’ operation ef- ficiency actually realised in natural nanostructures [1, 2, 7, 10—13]. In this paper, we provide a broad original review of this real, in- trinsically complex nanosystem dynamics, including its rigorous mathematical basis, key features, huge efficiency and applied as- pects (section 2). We further argue that our entire universe is es- sentially based on this kind of complex nanoscale dynamics, which provides the unified solution to various stagnating problems of fundamental physics, cosmology, biology and intelligence— consciousness theory, with numerous important applications (sec- tion 3). It follows that we should actively take this way also in arti- ficial nanostructure realisation, using the mentioned huge efficien- cy of their unreduced complex-dynamical version for creation of qualitatively new kind of machinery with really ‘magic’ properties (that were often announced, but never realised, with unitary nanostructures). We draw a number of particular perspectives for such complex-dynamical nanotechnology, emphasizing its links to the omnipresent natural universe structure as additional support for the unique efficiency of this way of further nanotechnology de- velopment by the only possible harmonious coevolution of natural and artificial structures (sections 3 and 4). 2. UNREDUCED CHAOTIC DYNAMICS OF REAL NANOSYSTEMS AND ITS KEY FEATURES We start our universal nanostructure description with a general formulation of a problem of arbitrary interaction of many elements, 4 A. P. KIRILYUK such as atoms, molecules and elementary particles, in the nanoscale size range. Our system existence equation is but a unified Hamilto- nian formulation of many-body interaction problem self-consistently confirmed by further analysis, in addition to such popular cases as Schrödinger equation for quantum dynamics and Hamilton—Jacobi equation for classical systems [2, 3, 7—15]: ( ) ( ) 0 , ( ) ( ) N N k k kl k l k l k h q V q q Q E Q = > ⎧ ⎫⎡ ⎤+ Ψ = Ψ⎨ ⎬⎢ ⎥ ⎣ ⎦⎩ ⎭ ∑ ∑ , (1) where ( )k kh q is the generalised Hamiltonian (expressing a dynamic complexity measure, see below) of the k-th system component with the degrees of freedom kq , ( ),kl k lV q q is the (arbitrary) potential of interaction between the k-th and l-th components, ( )QΨ is the sys- tem state-function describing exhaustively its configuration, { }0 1, ,..., NQ q q q≡ , E is the generalised Hamiltonian eigenvalue, and summations are performed over all (N) system components. This more general timeless form of interaction description and its fur- ther analysis covers also the special case of time-dependent interac- tion (for open systems), where the eigenvalue E on the right is re- placed by the partial time derivative. A more relevant form of existence equation (1) is obtained by separating common system variable(s), 0 q ≡ ξ (describing, e.g., sys- tem element position or interaction time dependence): ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 , , , , N N k l k k k k k kl k lh h q V q V q q Q E Q = > ⎧ ⎫⎡ ⎤ξ + + ξ + Ψ ξ = Ψ ξ⎨ ⎬⎢ ⎥ ⎣ ⎦⎩ ⎭ ∑ ∑ , (2) where now { }1,..., NQ q q≡ , and , 1k l ≥ here and below. If we express the problem in terms of eigensolutions for system components, then the existence equation (2) is transformed, in a standard way, to an equivalent system of equations for the respec- tive state-function components ( )nψ ξ [2, 3, 7—15]: ( ) ( ) ( ) ( ) ( ) ( ) 0 nn n nn n n n n n h V V ′ ′ ′≠ ⎡ ⎤ξ + ξ ψ ξ + ξ ψ ξ = η ψ ξ⎣ ⎦ ∑ , (3) where n nEη ≡ − ε , kn n k ε ≡ ε∑ , ( ) ( )0 nn nn nn k kl k l k V V V′ ′ ′ > ⎡ ⎤ξ = ξ +⎢ ⎥ ⎣ ⎦ ∑ ∑ , (4) ( ) ( ) 0 0( ) , ( ) Q nn k n k k nV dQ Q V q Q′ ∗ ′ Ω ξ = Φ ξ Φ∫ , (5) ( ) ( ) , ( ) Q nn kl n kl k l nV dQ Q V q q Q′ ∗ ′ Ω = Φ Φ∫ , (6) COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 5 ( ) ( ) ( ) 1 21 1 2 2( ) ... Nn n n Nn NQ q q qΦ ≡ ϕ ϕ ϕ , and ( ){ } kkn kqϕ , { } knε are eigenfunctions and eigenvalues of nonin- teracting system components: ( ) ( ) ( ) k k kk k kn k n kn kh q q qϕ = ε ϕ , (7) ( ) ( ) ( )1 2, ,..., , ( ) N n n n n n n Q Q ≡ Ψ ξ = ψ ξ Φ∑ , (8) with ( )1 2, ,..., Nn n n n≡ running through all eigenstate combinations (starting at 0n = for the ground state). As we do not want to use any usual approximation for problem solution (killing the essential features of unreduced interaction dy- namics), we try to solve the unreduced system of equations (3) by expressing the excited state-function component ( )nψ ξ ( 0n > ) through the ground-state component ( )0ψ ξ with the help of stand- ard Green’s function technique and substituting the result into the equation for ( )0ψ ξ , which leads to the effective existence equation for ( )0ψ ξ [2, 3, 7—16]: ( ) ( ) ( ) ( ) 0 0 0;effh V⎡ ⎤ξ + ξ η ψ ξ = ηψ ξ⎣ ⎦ , (9) where the effective (interaction) potential (EP) ( );effV ξ η is defined as ( ) ( ) ( ) ( ) 0 000;effV Vξ η ψ ξ = ξ ψ ξ + ( ) ( ) ( ) ( ) ( ) ξ ∗ Ω ′ ′ ′ ′ξ ψ ξ ξ ψ ξ ξ ψ ξ∫ + η − η − ε∑ 0 0 0 , 0 0 0 0 n ni ni n n i ni n V d V (10) and includes the eigensolutions ( ){ }0 niψ ξ ,{ }0 niη of a truncated system of equations (containing no equations and contributions for ( )0ψ ξ in (3)): ( ) ( ) ( ) ( ) ( ) ( ) 0 nn n nn n n n n n h V V ′ ′ ′≠ ⎡ ⎤ξ + ξ ψ ξ + ξ ψ ξ = η ψ ξ⎣ ⎦ ∑ , (11) with , 0n n′ ≠ here and below, 0 0Eη ≡ η = − ε , and 0 0n nε ≡ ε − ε . The eigensolutions, ( ){ }0 ,i iψ ξ η , of the effective problem formu- lation (9), (10) are used to obtain other state-function components ( )ψ ξni and then the total system state function (4) (the general problem solution): 6 A. P. KIRILYUK ( ) ( ) ( ) 0 0 0 , ( ) ( )i i ni n i n Q c Q Q > ⎡ ⎤Ψ ξ = ψ ξ Φ + ψ ξ Φ⎢ ⎥ ⎣ ⎦ ∑ ∑ , (12) where coefficients ci are determined by state-function matching at the boundary/configuration where interaction vanishes. The key point in the effective description of the unreduced many- body interaction, crucially important for real nanosystem dynamics, is the explicitly nonlinear structure of the effective dynamic equa- tion (9), (10), which contains the eigensolutions ( ){ }η ψ ξ( , )ni to be found. It is not difficult to see that it leads to essential growth of the number of its equally valid solutions, due to the respective growth of the highest power of its characteristic equation for ei- genvalues [1—17]. As these solutions, called system realisations, are all equally real and mutually incompatible (due to their physical completeness), they are forced to permanently replace each other in dynamically random, or chaotic, order thus defined. Any measured quantity, suitably represented by system density ( ),Qρ ξ (given by the squared modulus of the state-function for ‘wave-like’ levels or the state-function itself for ‘particle-like’ levels), is obtained as a specific, dynamically probabilistic sum of respective quantities for all realisations: ( ) ( ) ( ) ( ) 2 2 1 1 , , , ,r r r r N N Q Q Q Q = = ℜ ℜ⊕ ⊕ρ ξ ≡ Ψ ξ = ρ ξ = Ψ ξ∑ ∑ , (13) where Nℜ is the number of realisations (determined by the number of system eigenmode combinations), ( ) ( ) 2 , ,r rQ Qρ ξ ≡ Ψ ξ is the r-th realisation density, and the dynamically probabilistic sum, desig- nated by ⊕, describes the mentioned dynamically random change of system realisations. In accord with (12), the r-th realisation state- function ( ),r QΨ ξ is obtained from the effective problem solution as ( ) ( ) ( ) 0 0, r r r i i i Q c Q⎡Ψ ξ = Φ ψ ξ +⎣∑ ( ) ( ) ( ) ( ) ( ) , 0 0 0 0 0 0 * r n ni ni n i rn i i ni n Q d V′ ′ ′ ′ ξΩ ⎤′ ′ ′ ′Φ ψ ξ ξ ψ ξ ξ ψ ξ∫ ⎥+ ⎥ η − η − ε ⎥⎦ ∑ , (14) where 0n ≠ , r ic are matching coefficients giving the generalised Born's rule for realisation probabilities [1] (see below), and ( ){ }0 ,r r i iψ ξ η are the r-th realisations’ eigensolutions of effective ex- istence equation (9), (10). COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 7 Thus discovered plurality of mutually incompatible, but equally real solutions of the unreduced interaction problem is called dynam- ic multivaluedness, or redundance, the property leading to the uni- versally defined, omnipresent dynamic randomness, or chaos, and dynamic complexity [1—17] (see below). The complementary and equally universal feature of the unreduced interaction process is due to the dynamic entanglement of interacting system components (or degrees of freedom) described by sums of products of eigenfunc- tions of different components depending on their respective degrees of freedom in the general solution of unreduced problem (12)—(14). Similar to dynamic multivaluedness, it is absent in usual, dynami- cally single-valued (perturbative or ‘model’) problem solution and further amplified by ultimately detailed elaboration of the unre- duced solution, where the auxiliary, truncated system equations (11) participating in the formulation of the first-level solution (9)- (14) is progressively solved by the same EP method applied to a (fi- nite) series of ever more truncated higher-level auxiliary systems of equations. This leads to the ultimately complete expression of the general solution of the unreduced interaction problem in the form of probabilistic (multivalued) dynamical fractal whose hierarchical- ly connected structural levels are made of incompatible, chaotically changing realisations (as opposed to usual, dynamically single- valued and dynamically trivial fractals) [2, 6, 7, 10, 13, 15, 18, 19]: ( ) ( ) ... ... , , , ,rr r r r r N Q Q′ ′′ ′ ′′ ℜ ⊕ρ ξ = ρ ξ∑ , (15) where the dynamically probabilistic sum is taken over all realisa- tions of all levels of the dynamical fractal. The fractally structured dynamic entanglement of interacting de- grees of freedom within the chaotically changing system realisa- tions determine the tangible material quality of resulting system structure, which is absent in any usual, dynamically single-valued model providing only a schematic, purely abstract, ‘immaterial’ pro- jection of real system structure and dynamics. This conventional dynamically single-valued projection is also called unitary solution and description, as it provides a qualitatively smooth system struc- ture and only formally introduced evolution, without real change and causal time flow (see also below). The property of dynamic multivaluedness and related randomness naturally includes also the universal dynamic origin and definition of probability, actually referring to (also universally defined) events of system realisation emergence. Since all elementary realisations are equally probable by origin, the a priori dynamic probability αr of each r-th realisation emergence is given by 1r Nℜα = , but as in 8 A. P. KIRILYUK the general case these elementary realisations are grouped into compound realisations containing dense groups of similar, experi- mentally unresolved elementary realisations (cf. multivalued self- organisation below), the general expression of dynamic probability (of realisation emergence) is given by 1,..., ; , 1r r r r r r r N N N N N N ℜ ℜ ℜ ⎛ ⎞α = = = α =⎜ ⎟ ⎝ ⎠ ∑ ∑ , (16) where 1 rN Nℜ≤ ≤ is the number of elementary realisations within the r-th observed compound realisation. This expression is directly generalised to multilevel fractal structure of the unreduced solution (15), so that the averaged, expectation value of the observed system density, ( ),ex Qρ ξ , (for long enough observation times) is given by ( ) ( )... ... , , ... , ,ex rr r rr r r r r N Q Q′ ′′ ′ ′′ ′ ′′ ℜ ρ ξ = α ρ ξ∑ . (17) Note, however, that the general, dynamically probabilistic sum of system realisations (15) represents the general solution of unre- duced interaction problem also for the case of any small number of realisation emergence events and remains valid even before any event occurs at all. It describes the ‘living’ structure of dynamical fractal permanently probabilistically moving in real time, the prop- erty absent in any unitary theory and crucially important for real nanosystem dynamics and efficiency (see below). The detailed algebraic and geometric analysis of effective problem formulation (9)—(10) revealing the dynamic multivaluedness phe- nomenon shows [1—16] that in addition to the complete set of Nℜ system realisations determining its structure, the unreduced inter- action problem solution contains one special realisation with a strongly reduced number of contributing elementary eigenvalues, which describes a transient system state during its transition be- tween those ordinary, or regular, structure-forming realisations. We call that transitional state the intermediate, or main, system realisation and show that it provides the universal and causally complete extension of usual quantum mechanical wave function to any system dynamics [2, 3, 10—15]. This generalised wave function, or distribution function, describes the state of transiently quasi-free system components that dynamically disentangle in the realisation change process, which explains the reduced number of eigenvalues and vanishing effective interaction magnitude for that intermediate realisation of generalised wave function. At the lowest, quantum levels of complexity, it provides the realistic extension of usual, ab- stract wave function, and at higher, classical complexity levels it COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 9 gives the physically real version of distribution function. This causally complete version of generalised wave function in our unreduced interaction description naturally includes the im- portant link to the above concept of dynamic realisation probability (16), in the form of generalised Born’s rule stating that the wave function magnitude at a given local configuration determines the probability rα of realisation emergence just around that configura- tion: ( ) 2 r rxα = Ψ , (18) where ( )xΨ is the generalised wave function, and xr–the r-th reali- sation configuration (one may also have here ( )r rxα = Ψ for ‘parti- cle-like’ complexity levels). This physically transparent relation can be rigorously derived from the state-function matching conditions (determining the coefficients r ic in (16)) and is practically useful for system description due to the universal dynamical equation for ( )xΨ (the generalised Schrödinger equation, see below) [2, 3, 10— 15]. The proposed causal interpretation of the generalised wave func- tion (or distribution function) of any real system and interaction process as the physically real ‘realisation probability field’ implies also the generalised and causal interpretation of the (originally) quantum-mechanical, probabilistically interpreted ‘linear combina- tion’ of eigenstates. It becomes clear now, within the dynamic re- dundance paradigm [2, 3, 10—15], that the unreduced interaction dynamics consists in permanent system transitions between those state-realisations, which are all really taken by the system probabil- istically (at random, cf. (15)), with probabilities determined dynam- ically by respective numbers of elementary realisations, in accord with the dynamic probability definition (16) and the above causal Born's rule interpretation (18). We can finally provide the universal and rigorous definition of dynamic complexity C of any real system as a growing function of the number of system realisations or their change rate, equal to ze- ro for the (unrealistic) case of only one realisation [1—16]: ( ) ( ) , 0, 1 0C C N dC dN Cℜ ℜ= > = , (19) with, for example, ( ) ( )0 lnC N C Nℜ ℜ= or ( ) ( )0 1C N C Nℜ ℜ= − . The case of 1Nℜ = , ( ) 0C Nℜ = exclusively considered in usual, dynami- cally single-valued (unitary) theory, including its numerous, inevi- tably non-universal and contradictory imitations of complexity, cor- responds only to effectively zero-dimensional, point-like projection of unreduced, multivalued system dynamics, which destroys all its 10 A. P. KIRILYUK essential features (see below). The latter are particularly important, as we shall see, for the unreduced nanosystem dynamics, where fre- quent transitions between various regimes of genuine complex dy- namics are inevitable and omnipresent. Specific features of real nanosystem dynamics are revealed by the detailed analysis of the EP formalism Eqs. (10), (14) taking into ac- count all the unreduced interaction effects. We note, first of all, that contrary to usual, unitary nanoscience models, any real nanosystem dynamics is a dynamically multivalued, chaotic one, with permanent change of system realisations in causally random order. Moreover, we show that, just for the case of nanoscale dy- namics, one is usually left with the particularly irregular regime of uniform, or global, chaos, while the opposite case of relatively regu- lar, though always multivalued, self-organisation is a much less ev- ident situation, which can actually emerge with growing character- istic sizes and transition from quantum to classical behaviour [1, 2]. This limiting regime of externally regular multivalued dynamics is obtained in the case of essentially differing characteristic system frequencies, or eigenvalue separations, such as the frequencies of internal element dynamics, ω = Δε 0q n A , and inter-element dynam- ics, ξω = Δη 0i A , where nΔε is the separation of eigenvalues 0εn in (10), (14), iΔη the separation of eigenvalues 0 niη , and 0A is the characteristic action value. If, for example, qξω ω (or i nΔη Δε ), then one can neglect the 0 niη dependence on i in (10), (14), which leads to the local limit for the EP and state-function due to the completeness of eigenfunction set ( ){ }0 niψ ξ : ( ) ( ) ( ) 2 0 00 0 0 ; n eff ni nn V V V ξ ξ η = ξ + η − η − ε∑ , (20) ( ) ( ) ( ) ( ) ( ) 0 0 00 0 , n nr r r i ir i n i ni n Q V Q c Q ′ ⎡ ⎤Φ ξ Ψ ξ = Φ + ψ ξ⎢ ⎥η − η − ε⎣ ⎦ ∑ ∑ , (21) where the respective averaged values of 0 niη , 0 ni′η are implied, and the usual case of Hermitian interaction potential is assumed for brevity. It is easy to see that in this case the system performs very frequent transitions between its very similar realisations thus giv- ing the impression of externally regular shapes and trajectories. We therefore call this limiting case multivalued self-organised criticali- ty (SOC), taking into account also the multilevel (fractal) realisa- COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 11 tion hierarchy mentioned above that forms in this limit characteris- tic (but now intrinsically chaotic) SOC patterns of ‘sustainable fluc- tuations’ around the average shape [1—13, 18, 19]. However, this relatively regular regime of ‘ordered chaos’ can hardly be realised at the smallest scale of genuine nanosystem be- cause in this case the eigenvalue separation and frequencies, related to respective spatial sizes, cannot vary and differ essentially. The genuine nanoscale level dynamics corresponds instead to the oppo- site limiting case of comparable level separations or frequencies (the situation of resonance), i nΔη Δε (or qξω ω ). In that case, differ- ent realisation eigenvalues intermingle randomly, and we obtain es- sentially different realisations replacing each other chaotically with relatively low frequencies comparable to those of their internal mo- tions, which give the evident situation of explicitly irregular, strong chaoticity forming the regime of uniform, or global, chaos. Simultaneously, we obtain the true meaning of the phenomenon of resonance practically inevitable for nanoscale systems and giving rise to the onset of global chaoticity, whose criterion acquires the rigorously derived exact expression [1—4, 10—13, 18, 19]: 1i n q ξωΔη κ ≡ = Δε ω , (22) where κ is the introduced parameter of chaoticity. While κ grows from 0 to 1, system behaviour changes from the global regularity of SOC (at 1κ ) to the global chaoticity (at 1κ ), where chaoticity increases by a more abrupt step each time κ passes by a higher res- onance ( m nκ = , with moderate integer 1n m> ≥ ). The situation of unreduced many-body nanosystem dynamics corresponds thus to strong chaoticity of 1κ ∼ , which can pass to a more regular SOC regime of 1κ (or 1κ , for a ‘complementary’, usually less use- ful system configuration) only with growing characteristic sizes (e.g., in ‘measurement’ parts). This dominating regime of strong chaoticity and irregular pro- cess configuration for genuine nanosystem dynamics is related also to its essential quantum features persisting due to ultimately small, atomic-scale sizes of interacting element configurations. There is the well-known paradox of quantum chaos in usual (dynamically single-valued) chaos theory, where truly random dynamics is impos- sible in Hamiltonian quantum systems due to the absence of ‘expo- nentially diverging trajectories’ describing the origin of classical chaoticity. This fundamental limitation creates the unpleasant vio- lation of the even more fundamental correspondence principle im- posing a noncontradictory, direct transition between respective quantum and classical features in the classical limit of relatively 12 A. P. KIRILYUK small Planck's constant, 0→ . One could also conclude, based on this conventional quantum chaos theory with absent genuine dy- namic randomness on quantum scale, that essentially quantum nanosystems (including those used for quantum computation) are basically regular, which would justify their unitary description, in- cluding (generally accepted) feasibility of unitary quantum comput- ers. While conventional theory tries to replace the absent genuine quantum chaos with ‘quantum signatures of (classical) chaos’, we show in our unreduced, dynamically multivalued description of (ar- bitrary) Hamiltonian quantum system dynamics [2, 3, 16] that the problem does not even appear in the unreduced theory, since the genuine quantum chaos with truly random system realisation change is obtained in the same way as in (any) classical dynamics, including the universal criterion (22) of global (genuine) chaos on- set. The standard correspondence principle is re-established for any real, truly chaotic dynamics, including the transition (22) between the quasi-regular (self-organised) and essentially chaotic dynamical regimes [2, 3, 16]. The obtained result of genuine dynamic randomness in essentially quantum dynamics changes dramatically the general picture of nanosystem dynamics, especially in combination with the dominat- ing regime of strong chaoticity, as explained above (Eq. (22)). As opposed to basically regular, both quantum and classical dynamics in usual picture, we obtain in reality the highly chaotic, truly ran- dom quantum and classical dynamics of any unreduced many-body nanosystem, with multiple occasional transitions between dynamical regimes of various regularity, from SOC to global chaos, as de- scribed above (in particular, unitary quantum computation in essen- tially quantum regimes is impossible in principle, even in a totally noiseless system [2]). This messy, truly complex and permanently changing configura- tion of real nanoworld structure is further complicated by purely dynamic transitions between essentially quantum and classical (lo- calised) kinds of behaviour readily occurring, according to our theo- ry, just at those ultimately small (atomic) scales. Here too, we ob- tain the crucial difference with respect to standard, unitary theory picture, which relates classical behaviour emergence to sufficiently big, practically macroscopic system size, in particular due to the growing ‘decoherence’ effects in such greater systems (in the most popular interpretation). While this conventional picture directly contradicts multiple observation results, for both small-scale and larger structures (see [2, 3, 10, 14, 15] for details), and shows mul- tiple conceptual difficulties, there is no other choice in the artifi- cially limited, dynamically single-valued space of usual theory. COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 13 If we avoid those artificial restrictions in our unreduced, dynam- ically multivalued description of an isolated, purely quantum (nano)system dynamics, then we naturally obtain the purely dynam- ic, intrinsic emergence of classical, permanently localised behaviour in elementary bound, also nanoscopic systems, such as atoms, even when they are totally isolated from any external interactions [2, 3, 10, 14, 15]. The classical localisation effect for an elementary bound system is due to the truly (spatially) chaotic dynamics of the ‘free quantum walk’ of its components (starting from the so called quantum beat process for massive elementary particles [2, 3, 10, 14, 15]), so that in their bound state they cannot continue to walk randomly and coherently (in the same direction), performing in- stead a chaotic localised ‘dance’ around each other, with the proba- bility of greater coherent walk exponentially decreasing with dis- tance. This purely dynamic origin of classical behaviour explains also occasional revivals of quantum behaviour of those bound, nor- mally localised systems due to their special interactions with other systems (e.g., during quantum interference or Bose condensation process). This result means that the diversity of variously (but usually es- sentially) chaotic regimes of real many-body nanosystems is addi- tionally extended by those occasionally emerging (bound) classical states and interaction-driven revivals of quantum behaviour, reveal- ing the huge genuine complexity of unreduced nanosystem dynam- ics [1, 2]. This leads to further qualitatively important feature of unreduced nanoworld dynamics, the (exponentially) huge efficiency of unreduced complex dynamics, explaining the observed ‘magic’ properties of life, intelligence and consciousness (which remain oth- erwise always ‘mysterious’ in the framework of the unitary theory) [1, 2, 7, 10—13, 18, 19]. We first recall the rigorously derived universal structure of the complete interaction problem solution in the form of dynamically probabilistic (multivalued) fractal (see Eq. (15)), which efficiently adapts its permanently chaotically changing branches to the envi- ronment and its own emerging structure. We obtain thus the im- portant universal property of dynamic adaptability of natural structures, essentially based on the dynamically multivalued search process automatically performed by its chaotic realisation change and explaining the optimal performance of natural structure growth and dynamics (for a given level of complexity). The latter can be characterised quantitatively by the mentioned effect of exponentially huge power of unreduced many-body interac- tion dynamics as compared to its usual, dynamically single-valued models [1, 2, 7, 10—13, 18, 19]. The maximum operation power, P, of a real complex system is proportional to the total number Nℜ of 14 A. P. KIRILYUK its realisations given by the number of combinations of its unit linkN N n= (essential) interaction links (where unitN is the number of interaction elements and linkn –the average number of links per element): ! 2 ( )N NP N N N N e Nℜ∝ π ∼ . (23) Since for many real systems with macroscopic total size and nano- scopic operative structure N is already a large number (e.g., 1210N ≥ for brain or genome interactions [7, 10, 18]), we obtain ‘exponentially huge’, practically infinite P values just due to arbi- trary interaction link combinations in the dynamically multivalued interaction search process. By contrast, any usual, dynamically sin- gle-valued (basically regular) operation mode or model for the same system size can only produce the power 0P growing as Nβ ( )1β ∼ , so that 0 N NP P N N−β∼ ∼ → ∞ , which provides the rigorously sub- stantiated quantitative explanation for the underlying ‘magic’ effi- ciency and properties of life, intelligence and consciousness (as well as for the persisting inefficiency of their usual, unitary modelling) [7, 10, 18]. We conclude this section with applied aspects of the revealed complex (multivalued) and essentially chaotic dynamics of real nanosystems. Note, first of all, that one may see certain contradic- tion between thus revealed character of unreduced interaction dy- namics and the dominating nanotechnology paradigm and models based on essentially regular behaviour (e.g., nanoscopic elements of computer chips and many other artificial structures). This contra- diction is easily resolved by noting that this quasi-regular behaviour is possible due to the realised regime of SOC, far from the global chaos onset condition (22), i.e., at 1κ , according to the above unified classification of dynamic regimes. This situation can be re- alised especially for larger structures of the nano-microscale, show- ing predominantly classic behaviour, but also in a specially fixed, effectively one-dimensional (sequential) configuration, which is very far from arbitrary, unreduced nanosystem structure (e.g., in natural nanobiosystems). However, this ‘easy regularity’ of usual nanotechnology (which can never be absolute!) is obtained at the expense of losing the huge efficiency (23) of the unreduced, essentially chaotic dynamics, to- gether with the ensuing ‘magic’ properties of (genuine) life, intelli- gence and consciousness, becoming increasingly important for prac- tical applications. On the other hand, various attempts to realise unreduced nanomachines at ultimately small scales and in essential- ly quantum regimes within the same, unitary nanotechnology para- digm easily lead to fundamentally wrong directions and vain illu- COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 15 sions of ‘magic’ efficiency. This is the case of unitary ‘quantum computation’, which has grown to a vast field of theoretical re- search, but in fact cannot be realised in principle, due to the inevi- table quantum chaos condition (22), in accord with our genuine quantum chaos concept (absent in usual, basically regular quantum dynamics) [2]. One can realise instead the highly complicated, mixed, but indeed hugely efficient multivalued dynamics of unre- duced many-body nanosystems described above. This extended, complex-dynamical nanotechnology paradigm has deep technological and social implications related to the necessary real, unreduced sus- tainability of further civilisation development (see section 4 below). Detailed realisation of complex-dynamical nanotechnology concept should rely on the universal laws of complex dynamics of real struc- tures reviewed in the next section. 3. REAL NANOSYSTEM COMPLEXITY AS A BASIS FOR THE UNIVERSE STRUCTURE DEVELOPMENT AND LAWS We have seen in the previous section that the unreduced many-body nanoscale system dynamics exceeds essentially any its usual, dy- namically single-valued (or unitary) version by the richness of the complete, dynamically multivalued behaviour with a variety of dy- namic regimes of permanently chaotically changing system realisa- tions. In order to properly understand the emerging multilevel, fractally structured system dynamics one must rely on the funda- mental laws of the unreduced universal complexity (19) of this mul- tivalued interaction dynamics. In accordance with the universal nature of dynamic complexity introduced above, there is the single, unified law defining its dy- namics and evolution, the universal law of conservation and trans- formation, or symmetry, of complexity, which also generalises and extends various particular dynamical laws and principles [2—13, 18, 19]. It starts from the fact that universal dynamic complexity ( )C Nℜ is determined by the number of system realisations Nℜ (Eq. (19)) depending only on the number of element eigenmode combina- tions, which does not change during system evolution and interac- tion development. However, if the total system complexity C does not change in its structure formation process, it does change its form, passing from the latent, potential form of dynamic infor- mation, I, at the start of interaction process development (initial system configuration giving rise to the existence Eq. (1)) to the ex- plicit, unfolded complexity form of dynamic entropy, S, so that their sum, total complexity C, remains unchanged: const, 0, 0C I S C I S S I= + = Δ = Δ + Δ = Δ = −Δ > , (24) 16 A. P. KIRILYUK where the last inequality reflects the unstoppable interaction devel- opment process (irreversible realisation change), giving the general- ised second law (entropy growth) [2—13, 18, 19]. Both complexity forms, dynamic information I and entropy S, as well as the total complexity C I S= + , are determined by the same universal equa- tion (19), but whereas system realisations are only ‘planned’ in the initial state I C= , 0S = , they really emerge and replace each other in further system structure evolution I C< , 0 S C< < , until its final state of fully developed realisations (in the form of dynamical- ly probabilistic fractal, Eq. (15)) 0I = , S C= . In order to transform the basic expression (24a) of the symmetry of complexity to a dynamic equation, we need to introduce the no- tions of emerging real space and time at each complexity level (where the most fundamental levels of ‘physical’ space and time are obtained in this way at the lowest complexity levels of elementary particles, see below). The element of dynamically discrete, or quan- tised, space, xΔ , is determined by the explicitly found eigenvalue separation of the unreduced EP Eq. (9), (10), r ixΔ = Δη , where the elementary length λ is given by neighbouring realisation separation, r r r ixλ = Δ = Δ η , and the elementary size (of real space ‘point’) 0r emerges as eigenvalue separation within the same realisation, 0 r i i ir x= Δ = Δ η (index r stands for realisation number). Emerging space quantisation is thus due to the fundamental dynamic dis- creteness of the underlying realisation formation and change pro- cess, while the tangible, physically real character of this emerging space is determined by the fractal dynamic entanglement of the in- teracting degrees of freedom (section 2). The elementary time interval, tΔ , of a given complexity level emerges as intensity, actually measured by frequency, ν, of rigor- ously (and universally) defined events of incompatible realisation formation and change, 1tΔ = τ = ν . The irreversible, unstoppable time flow is thus due to the same fundamental phenomenon of dy- namic multivaluedness and randomness of any real interaction pro- cess and resulting system structure, which is inevitably absent in any usual, dynamically single-valued theory or model, leading to the famous, unsolvable (we now know why) ‘problem of time’ or ra- ther absence of real, irreversible time flow in unitary science framework [2—13, 18, 19]. The concrete value of real time interval tΔ = τ can be obtained also from the above quantised space interval rx λ = Δ (of the same complexity level) and the velocity 0v of signal propagation in the material of interaction components (necessarily known from this lower complexity level dynamics), 0τ λ= v . This purely dynamic link between equally real space and time is not re- duced, however, to any their real ‘mixture’ within a ‘space-time’ entity (introduced in unitary theory): while space is a tangible ‘ma- COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 17 terial’ entity dynamically ‘woven’ from interacting components, equally real time is not a tangible entity and describes only perma- nent chaotic (and therefore irreversible) change of material space structure elements. Since universal dynamic complexity ( )C C Nℜ= is determined by the number Nℜ of permanently changing system realisations, Eq. (19), while the emerging space and time elements specify the same process development, it becomes evident that the basic, universal, integral measure of complexity is provided by the simplest linear combination of (independent) space and time elements, known as action A (now extended to any system dynamics and its intrinsic nonlinearity): p x E tΔ = Δ − ΔA , (25) where all dynamic increments are determined by real system leaps between realisations, coefficients p and E are recognised as momen- tum and energy, now generalised as universal differential measures of complexity, 0 consttp x = Δ= Δ λ AA , (26) 0 constxE t = Δ= − Δ τ AA , (27) and 0A is the characteristic action magnitude at the complexity level in question (x and p should be understood as vectors if neces- sary). With irreversible time flow ( 0tΔ > ) and positive energy ( 0E > ) action ( , )x tA is a decreasing function of time, 0Δ <A (see Eq. (27)). This permanently decreasing integral form of complexity is therefore directly associated with dynamic information I introduced above, after which the main law of the universal symmetry of com- plexity, Eq. (24), takes a more dynamically sensible form: const, 0, 0C S C S S= + = Δ = Δ + Δ = Δ = −Δ >A A A . (28) In addition to the absolute universality of this law eventually unify- ing the extended versions of all usual dynamical laws and ‘princi- ples’ (now properly understood, see below), we note the natural uni- ty within one law of energy/complexity conservation and transfor- mation laws (the extended first and second laws of thermodynamics respectively), so that the balance, or symmetry, of total complexity (or energy) conservation is maintained by its ‘suitable’ transfor- mation from dynamic action-complexity (information) to dynamic 18 A. P. KIRILYUK entropy-complexity, where ‘symmetry’ is not different from ‘con- servation’ (as opposed to unitary theory) [2—13, 18, 19]. The differential, dynamically important form of the universal symmetry of complexity is obtained by division of the last equality (28) by real time increment constxt = Δ : const const, , 0x tH x t t x= = Δ Δ⎛ ⎞+ =⎜ ⎟Δ Δ⎝ ⎠ A A , = > 0H E , (29) where the generalised Hamiltonian, ( , , )H H x p t= , is a differential form of dynamic entropy-complexity, const( ) xH S t == Δ Δ , in accord with Eqs. (26), (27). We obtain thus the generalised, universally valid and rigorously derived version of the Hamilton—Jacobi equa- tion plus the generalised differential version of the second law (first and second parts of Eq. (29), respectively). In the basic case of a closed system, the Hamiltonian does not depend on time explicitly and we obtain a simpler version of the generalised Hamilton—Jacobi equation: const, tH x E x = Δ⎛ ⎞ =⎜ ⎟Δ⎝ ⎠ A . (30) The ultimately extended version of the ‘second law’ or the ‘arrow of time’ in (29), 0H E= > , is obtained by introduction of the gen- eralised Lagrangian, L, defined as the total (discrete) time deriva- tive of action-complexity A : const constx t x L p E p H t t x t = = Δ Δ Δ Δ= = + = − = − Δ Δ Δ Δ A A A v v , (31) where x t= Δ Δv is the (global) system motion speed, and the obvi- ous vector version of these relations is implied if necessary. Intrin- sic randomness of plural realisation choice gives entropy-complexity growth or the equivalent action-complexity decrease (see (28)), i.e. < > ≥0, , 0L H E pv , (32) which is reduced to a simpler version, , 0H E > , of the same gener- alised second law (time arrow in the direction of entropy-complexity growth) for 0=v . Returning now to the generalised wave function ( , )x tΨ of the intermediate system realisation (introduced in section 2, see Eq. (18)), we can complete the unified Hamilton—Jacobi expression of the symmetry of complexity (29), (30) for regular, localised realisa- tions by the corresponding equation for the distributed realisation COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 19 of the system wave function. This is achieved with the help of the generalised, causal (dynamic) quantisation rule, extending usual postulated Dirac quantisation in quantum mechanics and expressing complexity conservation in one cycle of transition between two reg- ular system realisations through the main realisation of the wave function [2, 3, 10—15, 18, 19]: ( ) 0Δ Ψ =A , 0 ΔΨΔ = − Ψ A A , (33) where 0A is a characteristic action value that may include a numer- ical constant depending on the detailed nature of the complexity level considered (quantum, classical, wave-like, corpuscular, etc.). Using unified quantisation (33) in the universal Hamilton—Jacobi equation (29), (30), we obtain the generalised Schrödinger equation for the generalised wave function (or distribution function) [2, 3, 10—15, 18, 19]: ( ) const const0 ˆ , , ,x tH x t x t t x= = ΔΨ Δ⎛ ⎞= Ψ⎜ ⎟Δ Δ⎝ ⎠ A , (34) const ˆ , ( ) ( )tH x x E x x = Δ⎛ ⎞Ψ = Ψ⎜ ⎟Δ⎝ ⎠ , (35) where the Hamiltonian operator, ˆ ˆ( , , )H x p t , is obtained from the Hamiltonian function ( , , )H x p t of Eq. (29) by replacement of the momentum variable ( ) const/ |tp x = = Δ ΔA with the corresponding momentum operator, 0 constˆ ( / ) |tp x = = Δ ΔA . One should add here the generalised Born’s rule (18) relating system wave function (or dis- tribution function) with realisation probability distribution. Note that this complex-dynamic origin of the generalised Schrödinger equation provides also the causally complete interpretation of usual quantum-mechanical Schrödinger equation at the corresponding lowest levels of complexity, with 0 i=A [2, 3, 14, 15] (just rele- vant for real nanosystem dynamics, see also below). We thus finally obtain the unified Hamilton—Schrödinger formal- ism of arbitrary many-body system dynamics, Eqs. (29)—(35), (18), expressing the universal symmetry of complexity at its any given level. It is supposed that both the generalised Hamilton—Jacobi equation for regular realisations and Schrödinger equation for the generalised wave function are further analysed by the same unre- duced EP method (section 2) giving plural realisations of emerging new complexity (sub)levels. This is especially important for real nanosystem dynamics represented, as we have seen above (see the 20 A. P. KIRILYUK end of section 2), by a dense, permanently changing and highly cha- otic mixture of lowest quantum and classical complexity levels. At the same time, emergence of greater complexity steps in system structure development towards superior levels of complexity is bet- ter described by explicitly discrete form of the symmetry of com- plexity (28). Note finally that the universal symmetry of complexity and the ensuing Hamilton—Schrödinger formalism (28)—(35) underlie and unify the extended versions of all known (correct) laws, equations and principles, including the Hamiltonian form of our starting ex- istence equation (1), thus self-consistently confirmed. One can spec- ify this unified origin of various dynamic equations by inserting the Hamiltonian expansion in powers of p into our universal Hamil- ton—Jacobi and Schrödinger equations: ( ) ( ) 0 , , , n n n H x p t h x t p ∞ = = ∑ , (36) ( ) const const 0 , 0 n x n t n h x t t x ∞ = = = Δ Δ⎛ ⎞+ =⎜ ⎟Δ Δ⎝ ⎠∑A A , (37) ( ) ( ) const const 0 0 , , n x n t n h x t x t t x ∞ = = = ΔΨ Δ⎛ ⎞= Ψ⎜ ⎟Δ Δ⎝ ⎠∑A , (38) where ( , )nh x t can be arbitrary functions. In the limit of usual de- rivatives, the last two equations become as follow: ( ) 0 , 0 n n n h x t t x ∞ = ∂ ∂⎛ ⎞+ =⎜ ⎟∂ ∂⎝ ⎠∑A A , (39) ( ) 0 0 , n n n n h x t t x ∞ = ∂Ψ ∂ Ψ= ∂ ∂∑A , (40) thus, including, in a truncated form, many ‘model’ equations. Their scope grows even more, if we take into account various models for nonlinear dependencies in the unreduced EP formalism (9), (10) used for solution of universal equations (29), (30), (34), (35). The universal symmetry of complexity (28), following rigorously from the dynamic multivaluedness of any real interaction process, leads to the intrinsically unified, causally complete description of entire world structure and dynamics in the form of multilevel dy- namically probabilistic fractal, Eq. (15) (section 2) [2, 6, 7, 10, 13, 15, 18, 19]. Progressive levels of this unified complex-dynamical world structure are properly specified by the detailed analysis of respective interaction processes, starting from the lowest levels of elementary particles and fields, together with physically real fun- COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 21 damental space and time (see above in this section), their intrinsic properties, dynamically unified fundamental interactions, quantum, emerging classical and relativistic behaviour [2, 3, 13—16]. These fundamental and now causally complete results are particularly im- portant for proper understanding and design of real nanosystem dynamics, already because it belongs just to these lowest, dynami- cally entangled complexity levels [1, 2]. While the natural world evolution and dynamics can be causally understood within the dy- namic redundance paradigm [1—19], the emerging new stage of its artificial, man-made coevolution necessitates practical application of the same causally complete understanding, if we want to obtain the intrinsically creative, sustainable world development. By contrast, usual, unitary theory models containing essential ‘mysteries’, ‘un- solvable’ problems, omnipresent ruptures, contradictions, and many ‘invisible’ or ‘dark’ entities (matter, energy, dimensions, elemen- tary species) are not suited for the full-scale nano interaction de- scription, including the particularly popular case of (unitary) ‘quan- tum computers’ [2]. Our intrinsically unified, complete and realistic world structure starts from the unreduced (attractive) interaction in the simplest possible configuration of two initially homogeneous, physically real media, the electromagnetic protofield, giving rise to directly ob- served particles, and the gravitational protofield, eventually identi- fied as directly unobservable quark condensate [2, 3, 14, 15]. Due to the intrinsic dynamic instability, this initially homogeneous con- figuration gives rise to local emergence of essentially nonlinear lo- cal self-oscillation process called quantum beat (periodic local proto- field squeeze and extension) and constituting the essential, causal dynamic structure of a simplest (massive) elementary particle, such as the electron. The dynamically multivalued quantum beat charac- ter accounts for non-zero dynamic complexity of even such simplest objects and leads to spatially chaotic permanent wandering of the squeezed (corpuscular) state of virtual soliton, which provides the universal dynamical definition of inertial (and eventually gravita- tional), relativistic mass-energy, without any additional entity, such as the Higgs field in usual, dynamically single-valued theory [2, 3, 14, 15]. While the number of forming elementary particles grow, the protofield tension increases and finally no new quantum beat process can emerge (which is the self-tuned universe feature and re- spective unitary problem solution). After that, only massless pho- tons with quasi-regular dynamics can appear and disappear because of protofield and massive particle interaction. One obtains also equally transparent dynamic origin of other in- trinsic properties and their observed features, including electric charge and spin, and of the number and character of fundamental 22 A. P. KIRILYUK interactions transmitted by the two protofields and now intrinsical- ly, dynamically unified from the start. Elementary particle interac- tions then lead to emergence of higher complexity sublevels, in the form of compound elementary particles (represented by hadrons) and elementary bound states, such as atoms. The latter represent also the elementary classical, permanently localised states, due to internal chaotic dynamics of their components (obtained without evoking any other ‘decoherence’, see section 2) [2, 3, 10, 14, 15]. Genuine quantum chaos (dynamic randomness) and quantum meas- urement are other characteristic cases of elementary quantum ob- ject interaction (in the Hamiltonian and weakly dissipative system configurations respectively), naturally emerging within the same, unreduced analysis of dynamically multivalued interaction process- es, thus overcoming the ‘hard’ limitations of usual theory [2, 3, 16]. As mentioned above, the unreduced natural or artificial nano(bio)system dynamics is a highly entangled, irregular and permanently changing mixture of these characteristic kinds of low- er-level complex dynamics (quantum and classical chaos, emerging classicality and quanticity revivals, quantum measurement, collapse and extension). Therefore, it can be properly understood and man- aged in applications only within the described causally complete, dynamically multivalued picture (including the important property of exponentially huge power of unreduced interaction dynamics, Eq. (23)). In a more general vision, the entire Universe dynamics and struc- ture is largely determined by the same unreduced interaction pro- cesses that locally give rise to the full-scale, ‘living’ nanobiosystem dynamics, and therefore the former can be seen now as macroscopic, astrophysical manifestations of living complex-dynamical (multi- valued) structure development, generally governed by various mani- festations of the universal symmetry of complexity (28)—(40). Therefore, proper understanding of the self-tuned Universe dynam- ics, without ‘mysteries’, ‘dark’ entities, ‘hidden’ dimensions and ‘invisible’ particle species (the main content of usual science pic- ture), is obtained within the same analysis and dynamical picture as the causally complete description of Universe’s local nanostruc- tures, in their extended, complex-dynamical, or ‘living’, version (with the mathematically exact definition of the properties of life and related intelligence and consciousness [2, 3, 10, 14, 15]). In- deed, we show that macroscopic living, intelligent and conscious systems are obtained as superior levels of unreduced dynamic com- plexity (with exponentially huge interaction power, Eq. (23)), where essential components are provided already at nanoscale complexity sublevels. This is of direct importance for respective applications of artificial intelligence, machine consciousness, advanced robotics, COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 23 intelligent computer systems and communication networks, reliable genomics, and integral medicine [2, 3, 6—13, 18, 19]. 4. COMPLEX-DYNAMICAL NANOBIOSCIENCE PARADIGM AND UNIFIED PRINCIPLES OF NANOSYSTEM DYNAMICS The rigorously specified laws of unreduced complex dynamics and their outlined applications to the unreduced nano(bio)system dy- namics (section 3) lead to the ultimately extended paradigm of com- plex, multivalued nanosystem dynamics, governed by the universal symmetry (conservation and transformation) of complexity, Eqs. (28)—(40). While the detailed system dynamics and evolution is gov- erned by these equations, their applications provide also certain general principles of real, complex-dynamical interaction process development, which can be useful as such for the unreduced nanosystem design and management (while being consequences of the same universal symmetry of complexity) [2, 3, 11—13, 18, 19]. One may start with the basic complexity development principle as the guiding line for complex system design and dynamics. This is simply the applied version of the symmetry of complexity (24), (28) stating that any real system dynamics is governed by the permanent transformation of its initial dynamic information I (expressed by action-complexity A ) to the equal amount of dynamic entropy- complexity S, so that their sum of total complexity C S= +A re- mains unchanged (with any complexity form being determined by the system realisation number, Eq. (19)). The process details are irregular, highly unpredictable and generally extremely numerous (due to the exponentially huge efficiency, Eq. (23)) and therefore, contrary to usual design and programming, cannot be followed and designed exactly, one by one. Instead of detailed step-by-step pro- gramming, one should use here suitable arrangement and adjust- ment of potential action-complexity through system configuration, while the detailed ways of complexity development are found by the unreduced complex dynamics itself, spontaneously and with the ex- ponentially huge efficiency (23). One can call this approach complex- dynamical programming. Close to it, it is the complexity correspondence principle, accord- ing to which most efficient interaction with ‘interesting’ (creative) results takes place for interacting components of comparable unre- duced complexity. In particular, higher-complexity participants tend to control and determine the behaviour of lower-complexity units (generalised self-organisation principle), while lower- complexity component or system cannot efficiently control or simu- late higher-complexity behaviour. It is especially this latter mani- festation of the complexity correspondence principle that has im- 24 A. P. KIRILYUK portant practical applications in various fields of usual, unitary sci- ence and technology, which tends to neglect the unreduced complex- ity effects. First of all, it clearly expresses the basic general defi- ciency of usual, dynamically single-valued (zero-complexity) science paradigm being applied to understanding of real, dynamically mul- tivalued (complex) system and interaction dynamics (especially at higher complexity levels). A particular application becoming ex- tremely popular in the last decades, unitary quantum computation, appears now impossible as such, irrespective of technical details, because here the lowest, essentially quantum complexity levels are supposed to efficiently solve, reproduce or simulate much higher, classical complexity processes and tasks (even apart from the real quantum chaos obstacle, section 2) [2]. By contrast, it is the unre- duced, complex-dynamical (and thus inevitably chaotic) nanosystem operation that can indeed show ‘magic’, exponentially huge effi- ciency (see Eq. (23)) characteristic of living and intelligent behav- iour. The complex-dynamical control principle implies optimal system complexity development (from dynamic information to dynamic en- tropy) as the unified real control purpose, as opposed to usual con- trol purpose of making system behaviour ‘more regular’ (in the predetermined shape or direction), including various pseudo- complexity features of ‘synchronisation’ etc. As the unreduced in- teraction analysis explicitly shows, real, dynamically multivalued system behaviour, including thus any ‘control’ scheme, can never be really, internally regular (section 2), and therefore relying on usual ‘total control’ approach in the unitary control paradigm can actual- ly produce higher, catastrophic deviations from the desired stabil- ity, than lower-control influences. What can be truly reliable in re- al, highly chaotic nanosystem design is the creative, rather than re- strictive, control principle, using proper system development be- tween dynamically multivalued, internally chaotic SOC regimes (see section 2). One can generally speak here about sustainable control (and design) principle, where global stability is obtained with the help of local creativity. Suitable transitions between now clearly de- fined regimes of strong chaos and self-organisation (section 2, Eq. (22)) can be efficiently used here for system configuration and dy- namics management. Finally, the unreduced (free) interaction principle separately em- phasizes the huge power of unreduced, essentially chaotic interac- tion dynamics (Eq. (23)), variously used also in the other principles. Only the general purpose of optimal entropy-complexity growth can be fixed, while the detailed nanosystem evolution remains highly chaotic and largely unknown, where this intrinsic chaoticity is a normal and useful system operation regime (see also the above com- COMPLEX-DYNAMICAL NANOBIOTECHNOLOGY PARADIGM AND EVOLUTION 25 plexity development principle). The magic, exponentially huge pow- er (23) of unreduced complexity can also be described as the genu- ine, interactive parallelism of multivalued fractal dynamics. The entire free-interaction nano- and biosystem dynamics can be de- scribed thus as constructive alternation of strong-chaos and multi- valued SOC regimes realising the optimal combination of efficient chaotic search and ordered creation respectively. In summary, the rigorously substantiated picture of the unre- duced nanoscale interaction dynamics, sections 2—4, opens new per- spectives for the full-scale, free-interaction nanobiosystem design and control, actually realising the coevolution of natural and artifi- cial complex-dynamical, living and intelligent systems, organisms and communities. One obtains thus the new paradigm of unreduced complex-dynamical nanobioscience and technology, necessitating the corresponding complexity transition (or revolution) with respect to the dominating dynamically single-valued, unitary models and ap- proaches [1—3]. 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