Extended Prigogine Theorem: Method for Universal Characterization of Complex System Evolution

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1 Extended Prgogne Theorem: Method for Unveral Characterzaton of Complex Sytem Evoluton Sergey amenhchkov* Mocow State Unverty of M.V. Lomonoov, Phycal department, Rua, Mocow, Lennke Gory, 1/, Publhed unaltered at Chao and Complexty Letter, Volume 8, Iue 1, 14, pp eyword: elf organzaton, control parameter, lmontovch S theorem, Prgogne theorem. Abtract Evoluton of arbtrary tochatc ytem wa condered n frame of phae tranton decrpton. Concept of Reynold parameter of hydrodynamc moton wa extended to arbtrary complex ytem. Bac phae parameter wa expreed through power of energy, njected nto ytem and power of energy, dpated through nternal nonlnear mechanm. It wa found out that bac phae parameter a control parameter mut be delmted for two type of ytem - accelerator and decelerator. It wa uggeted to elect zero tate entropy on through condton of zero value for entropy producton. Zero tate ntroduce unveral prncple of dorder characterzaton. On ba of elf organzaton S theorem we have derved relaton for entropy producton behavor n the vcnty tatonary tate of ytem. Advantage of thee relaton n comparon to clacal Prgogne theorem veratlty of ther applcaton to arbtrary nonlnear ytem. It wa found out that extended Prgogne theorem ntroduce two relaton for accelerator and decelerator correpondngly, whch remark ther quanttatve dfference. At the ame tme clac Prgogne theorem make poble decrpton of lnear decelerator only. For untable moton t correpond to trange attractor. TABLE OF CONTENTS Introducton. Sergey amenhchkov, Mocow State Unverty of M.V. Lomonoov, Phycal department; 1. Stablty and bac phae parameter. Sergey amenhchkov, Mocow State Unverty of M.V. Lomonoov, Phycal department;. Entropy demarcaton crteron. Sergey amenhchkov, Mocow State Unverty of M.V. Lomonoov, Phycal department; 3. Unveral Prgogne theorem. Sergey amenhchkov, Mocow State Unverty of M.V. Lomonoov, Phycal department. Abbrevaton UPT - Unveral Prgogne Theorem; DDF - Denty dtrbuton functon. * Correpondng author: Rua, Mocow, rmky val, 4, d.1, amenhchkov Sergey A., kamphy@gmal.com, Skype: kamenhchkov_ergey 1

2 Introducton Evoluton of complex ytem,.e. ytem, contanng of tattcally large partcle number, can be decrbed, ung tochatc tate functon. Th approach wa frtly appled to thermodynamc ytem and wa extended nto area of arbtrary complex ytem. It wa convenent to repreent evoluton of tochatc ytem a et of tranton between phae tate. However, mot general characterzaton of complex ytem need ntroducton of unveral meaure of ytem dorder and ytem ntablty. It neceary to undertand connecton between thee two charactertc and to defne ther behavor n vcnty of tatonary, table tate. Veratlty of uggeted decrpton mpoble wthout proper and convenent electon of control parameter whch can be ued for any type of complex ytem. Next three chapter are devoted to oluton of thee problem and goal. 1. Stablty and bac phae parameter Evoluton of tochatc ytem under defned control parameter et bac queton of ynergetc cence. It connected wth problem of tattcal decrpton of elf organzaton,.e. decrpton n term of dtrbuton denty evoluton. Stochatc ytem evoluton may be repreented a conequence of phae tate and phae tranton f ung term of tattcal thermodynamc. Let generalze thee term for an arbtrary ergodc tochatc ytem (ES ytem). If we degnate q + and q - for power nput and output per ytem volume ma, then energy balance condton can be formulated n the followng way: q ( R ( f ( 1 R f 1 (1) q ( Here R( o called bac phae parameter and ( et of control parameter (charactertc vector). Quantte q + and q - correpond to power nput and output per ytem volume ma uch that q( v v q ( q (. We may ue example of hydrodynamc bfurcaton. Then 1 nput/output energy mechanm are provded by flow nertal force and by vcou dpaton correpondngly. Reynold number Re play role of bac phae parameter n th cae and gven below: l u l u dt Re( R( D( D( dt ( Here l patal cale of ytem, u velocty of energy ource (nput flow) whch aumed to be contant n th example. The bac potulate of th chapter can be formulated n the followng way: bfurcaton necearly correpond to the condton: R ( 1, whle phae condton realzed for R 1 (Hypothe I). Scheme of ES ytem frt order phae tranton then may be repreented by et of chan, followng below. q ( R( R( 1 q ( R( R( t1) 1 (3) q ( R( R( 1 q ( R( R( t1) 1 (4) ()

3 q ( R( R( 1 q ( R( R( 1 q ( R( R( 1 q ( R( R( t1) 1 (6) Here and how fnte ncreae and decreae of correpondng parameter for t1 t t. Intal condton of ytem correpond to R ( t ) 1. A t follow from et (3) - (6) potve feedback for nput/output power mechanm compulory condton for phae tranton. Wthout lo of generalty one can be repreented n the followng way: q ( / q ( 1. For tuaton when fxed nput power q + wtched to anther contant (3, 4) bac phae parameter can be repreented a gven expreon (7). q ( R ( f ( (7) q ( t ) q ( q (... 1 Here perturbaton member of denomnator decompoton correpond to bfurcaton and devaton from one table phae tate to another one uch that R( 1 n both cae. Th nontatonary proce can be called phae tranton f we ue termnology of tattc phyc. Let tet Hypothe I, ung auxlary entropy of olmogorov Sna h d [1]: h( x( ). Here averagng n phae pace degnated a and averaged quantty can h d be expreed a um of potve Lyapunov factor pace: Vector x ( h h N ln N 3 (5) h for each dmenon of generalzed phae ( charactertc phae vector of ytem tate. Factor x ( x (8) how dtance growth x ( n drecton for two nfntely cloely located pont n phae pace. Condton of tatonary tate then equal to h or 1 ( N, gven below: q( t 1 v 1 v v ). Relaton for pecfc ytem power q ( q Sytem tablty condton lead to h and 1 f we conder all Lyapunov factor. Gven nequalte lead to expreon (1) for velocty component v 1,. x ( x ( / t v ( ( ( 1 (1) x x / t v Relaton (1) n fact allow recevng component of acceleraton ( : v ( v lm v ( v v( lm (11) t t t t Indeed, conderaton of pecfc power q ( can be reduced to two cae: a) ( and x ( x ( ; b) ( and x ( x (. Sgn of ( and x x v ( x (9) x match - th condton oblgatory for defnton of Lyapunov factor. Then ( doen t depend on ntal gn of coordnate hft. x

4 For cae a) and b) we then receve: a) v and v ; b) v and v. In both cae wth ue of relaton (8) we receve that q (. Accordng to defnton (7) of bac phae parameter th mean that R ( 1 for h. Condton of R f,.e. R ( 1 correpond to phae tate of d ytem. Let how how realzaton of condton R ( 1 nfluence ytem tablty we tet reverblty of tatement, gven above. Relaton (1) how that n th cae q ( q ( and conequently / t. Here pecfc energy energy of ytem ma unt. Wthout lo of generalty th requrement lead to relaton v v 1, two cae: a) v and v ( ; b) v and v (.. Th condton can be reduced to Accordng to expreon (1) and (11) we have followng conequence: a) x x ; b) ( x ( x. Here we agan ue condton of potve tme delay and concdence of ( and x gn. A t wa hown n general cae R-parameter defne neceary but not uffcent requrement of tablty. Thu ue of R ( a control parameter mut be delmted for two type of ytem: a) accelerator - v ( ; b) decelerator - v (. For frt type of ytem moton tablty lo and bfurcaton are realzed for R ( 1; decelerator come to phae tranton only f R ( 1. However for both type of moton bfurcaton necearly correpond to the condton: R ( 1, whle phae condton realzed for R 1. Hypothe I ha been proved. x. Entropy demarcaton crteron Local elf organzaton S theorem, formulated by U. lmontovch n 1983 [] how dependence of Lyapunov functon from arbtrary control parameter a( fluctuaton: ( a S S1 (1) ) Index correpond to normalzaton of econd tate, whch ntroduce artfcal conervaton of Hamlton functon for both tate. Normalzaton procedure wll be partcularly condered n the begnnng of next chapter. In relaton (1) two tate of tochatc ytem are condered State 1, correpondng to control parameter a and State, whch correpond to control parameter a a a. State more dordered tate then State 1. In S theorem entropy ued n fact a dorder meaure. However t can be ued only a relatve charactertc, for n S - theorem entropy of regular, zero tate, not ntroduced,.e. no demarcaton crteron ext. One of poble way for t formulaton baed on bjectve connecton between dynamc entropy and Gbb entropy - (S). Let ue eparate moton of ytem n accelerated and h d decelerated tage. Then bac phae charactertc R ( can be ued a the control parameter for each tage. A t wa mentoned above ntablty take place f R( 1for accelerator or f R ( 1 decelerator. Bjecton h d (S) vald n vcnty of table phae tate when R 1and h. d 4

5 Condton of energy conervaton realzed n th cae for q ( q (. Louvlle theorem allow makng concluon about conervaton of phae pace volume and ngle value of entropy n vcnty of phae ( R 1). Then zero tate entropy S can be defned from mplct condton: lm ( S). If ytem ha contnuou or dcrete et of phae tater, mnmum h d SS value of entropy hould be elected. 3. Unveral Prgogne theorem Ue of demarcaton crteron allow formulaton of two dordered tate S and S 1, condered n S theorem: S S1 S. S theorem realzed n frame of frt bac aumpton that dtrbuton functon of both tate have Boltzmann form. Then ndex, appled to entropy of econd tate S defne normalzaton, expreed by relaton for Hamlton functon (13): Symbol H a a H a a 1 (13a) H f ( T ) cont exp (13b) kt degnate phae pace averagng. In fact expreon (13a) reveal ytem energy vrtual conervaton - normalzaton acheved artfcally by normalzaton of dtrbuton functon f(t): T T where T ytem temperature. If S S1 thent T, f control parameter a ( choen correctly. Vrtual conervaton mean that correct comparon of ytem dorder poble wthn uch prelmnary calbraton of Hamltonan. Indeed for Boltzmann form of denty dtrbuton entropy functon of ytem Hamltonan S ln B H, T. Therefore comparon of two tate wth dfferent dtrbuton and value of entropy can be correct only wthn relaton (13a), repreentng artfcal conervaton of total ytem energy. Accordng to defnton of dynamc entropy [1] tme dervatve of Lyapunov functon can be repreented n the followng form: t S t S1 h h1 Relaton (1) can be modfed for tme dependent control parameter a a t ) a ( t ) a ( t ) t ( : 1 1 h h1 (15) ( a) ( a) at t at Conequence of S theorem repreented by relaton (16). Here lower ndex t correpond to tme dervatve. Let ue R ( a control parameter. h a t (14) (16) 5

6 Then condton (16) can be expreed n the followng way n the vcnty of tatonary condton R ( 1: R h h h (16a) t t R h h h (16b) t t In (16a) and (16b) we have ued replacement lm h/ t h after dvdng both de of t nequalty (16) by t. Tranton 1 to tatonary table tate condered. Accordng to concluon of Chapter 1 relaton (16a) correpond to decelerator type of moton, when tablty lo caued by ncreae of bac phae parameter R ( : R. (16b) decrbe the cae of accelerated moton - v (. Relaton (16a) and (16b) gve content of generalzed Prgogne theorem [5] for R( control ytem. We ued hypothe that value of control parameter for table tate matche control parameter of tatonary moton. I.e. we aume that relaxaton of table ytem to tatonary tate fnally occur f control parameter tay fxed. Unlke Prgogne theorem we have acheved tatement wth general area of applcaton. Indeed, relaton (16a), (16b) are vald for ytem wth arbtrary cla of lnearty. Clac Prgogne theorem defne entropy producton n vcnty of tatonary tate. However, concluon of the theorem baed on lnear approxmaton of ere expanon for control parameter component: F F a l, k... (17) a a Here F thermodynamc free energy and l are contant knetc factor of Onanger relaton, [5]. In frame of lnear expreon relaton (16a) acheved clacal Prgogne theorem decrbe lnear decelerated type of ytem. That why advantage of (16a) and (16b) relaton n comparon to clacal Prgogne theorem veratlty of t applcaton to arbtrary ytem controlled by bac phae parameter R (. Let lt bac aumpton of Nonlnear Prgogne Theorem, whch were ued above: t. Th condton natural for phycal decrpton of ytem evoluton; H Denty dtrbuton functon (DDF) ha Boltzmann form f f B cont exp. kt Here H Hamlton functon of ytem and T t temperature. DDF vald for ytem wth ndependent partcle trajectore n phae pace. Th condton vald for chao tate [4], when h d and phae pace reoluton fnte -. Here element of phae pace; H a a 1 H a a normalzaton of ytem temperature - T. t. Condton of vrtual energy conervaton reached by T 6

7 Concluon Evoluton of arbtrary tochatc ytem wa condered n frame of phae tranton decrpton. Concept of Reynold parameter of hydrodynamc moton wa extended to arbtrary complex ytem and bac phae parameter q ( R ( wa ntroduced. q ( t ) q ( q (... 1 We came to concluon that ue of R ( a control parameter mut be delmted for two type of ytem: a) accelerator - v ( ; b) decelerator - v (. For frt type of ytem moton tablty lo and bfurcaton are realzed for R ( 1; decelerator come to phae tranton only f R ( 1. However for both type of moton bfurcaton necearly correpond to the condton: R ( 1, whle phae condton realzed for R 1. Entropy a dorder meaure commonly appled for characterzaton of complex ytem tate. However arbtrarne of addtve contant lead to abence of unveral approach to dorder characterzaton. In the current work t wa uggeted to defne zero tate entropy S from mplct condton: lm h ( ) d S. For contnuou or dcrete et of phae tater, SS mnmum value of entropy ha to be elected. On ba of elf organzaton S theorem we have derved relaton for entropy producton behavor n the vcnty tatonary tate of ytem. That why advantage of thee relaton n comparon to clacal Prgogne theorem veratlty of ther applcaton to arbtrary ytem controlled by bac phae parameter R (. Clacal Prgogne theorem decrbe only lnear decelerated type of ytem. Extended Prgogne theorem defne two relaton for decelerated moton ( v( ) and accelerated one ( v ( ). Frt type correpond to h ht connecton mnmum entropy producton acheved n tatonary, table phae tate. Second type of moton decrbed by oppote nequalty: h ht. It could be ueful to remark qualtatve and quanttatve dfference n accelerated and decelerated moton. Both of thee type could be untable. But n one cae retorng force ( v( ) correpond to attracton nto certan phae pace. In another cae we have repulon a determnatve mechanm. If we conder untable trajectore frt type of moton correpond to trange attractor and econd one to repeller. Extended Prgogne theorem gve oppote cenaro of evoluton n vcnty of phae tate for thee two type of moton. Reference [1] Zalavky, G.M., Sagdeev, R.Z., Introducton to nonlnear phyc: from the pendulum to turbulence and chao, Nauka: Mocow, 1988, p. 99-1; [] lmontovch, U.L., Turbulent moton and tructure of chao. New approach to tattc theory of open ytem, M.:omnga, 1, p.76-81; [3] Prgogne, I.R., Introducton to thermodynamc of rreverble procee, M.:Nauka, 196; 7

8 [4] amenhchkov, S.A., Unveral Prncple of Perfect Chao, Dcontnuty, Nonlnearty, and Complexty, DNC-D-13-6, under revew, ubmtted on Feb. 1, 13. Orgnal paper n Arxv - [5] Stratonovch, R.L., Nonlnear nonequlbrum thermodynamc, M.:Nauka, 1985, p

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