Quantum Mechanical Foundations of Causal Entropic Forces

Transcription

1 Quanum Mechancal Foundaons of ausal Enropc Forces wapnl hah Deparmen of Elecrcal and ompuer Engneerng Norh arolna ae Unversy, U

2 s he number of expers ncrease, each specaly becomes all he more self-susanng and self-conaned. uch balanzaon carres scenfc hough farher away from naural phlosophy whch, nellecually, s he meanng and goal of scence - Isdor Isaac Rab

3 Movaon and Goals Esablsh concree mahemacal relaon beween maxmzng pah dversy and noons of raonaly radonally assocaed wh nellgence Undersand he mcroscopc orgns of forces ha maxmze dversy of fuure sysem pahs Undersand role of crcal dynamcs and long range correlaons n human bran organzaon and behavor Undersand he mechansms behnd self-organzaon of complex srucures capable of sophscaed behavor such as cognon n sysems governed solely by he laws of physcs Use he nowledge of hese mechansms o enhance he capables of he conemporary cognve archecures

4 Oulne ausal Enropc Forces Open Quanum ysems Marovan Maser Equaon Projecve vs. General Ineracons Pah Dversy and Expeced Uly Enropy Balance Relaon and Maxmum Pah Dversy Hamlonan Theory of Dynamc Economcs Economc Uly and Enropy urren Wor nalyss of Brownan Parcle n a Box Dsplaced Oscllaor nsaz Emergence of O (elf Organzed rcaly) n proposed framewor onclusons

5 ausal Enropc Forces 1 Based on he dea of maxmzng enropy producon over fne duraon pahs Enropy over pahs hrough confguraon space: Pah based Enropc Force: X, Prx( ) x() ln Prx( ) x() dx( ) Envronmen as a hea bah a emperaure T r coupled only o a few degrees of freedom of he sysem Formulaon of Pah based Enropc Force under Marovan Langevn dynamcs: F j sympoc Equparon Propery for horzon τ F B X, T X X, X 2T j dx( ) T X, f Prx( ) x() ln Prx( ) x() R T nh X F X, j f j, T R Prx( ) x() dx( ) n 1 ouresy: ausal Enropc Forces,.D. Gross &.E. Freer

6 Open Quanum ysems Projecve Ineracons Densy Marx Evoluon Von Neumann-Louvlle Equaon Lndblad Maser Equaon Quanum Marov Maser Equaon: Expecaon Value of Force: ommon Dsspaon rae μ and Projecve Valued Measure (PVM) Lndbladans: Resemblance wh he pah based Enropc Force, H L H H 2 2 1, F F P F Tr x P x y P y y x F 2 2 1

7 Open Quanum ysems General Ineracons Von-Neumann Enropy always ncreases for Projecve Lndbladans POVM (Posve Operaor Valued Measure) Lndblad operaors for general neracons: F I ysem Enropy can decrease a he cos of Enropy of he bah Enropy balance relaon (econd law for open sysems) H F ssumpons: d d J, ysem evolves as a non-equlbrum dsspave process The POVM Lndbladan for he sysem-bah neracon decreases he Von-Neumann Enropy of he sysem o allow a saonary goal sae o be approached verage nec energy per molecule of he ysem s nally hgher han reservor emperaure

8 Maxmzng Pah Dversy Fuure pah dversy as maxmum enropy ha can be produced from he presen sae o a fuure me horzon: From he Enropy Balance relaon: Maxmum pah dversy: D B Tr ln The sysem s dsspang hea o he reservor and Γ s he maxmum aanable fuure sysem enropy I can easly be nferred ha hs maxmum value of dversy s approached when he enropy producon from he nal o he presen sae (Δσ) s mnmum. H T D H T

9 Hamlonan Theory of Dynamc Economcs 2,3 For he problem of consumpon-opmal growh wh posve rae of me dscoun α >, he equaons of moon are: H s he sysem Hamlonan represenng he producon echnology Q H Q, Opmal pah o he seady sae s gven by he one ha maxmzes followng dscouned expeced uly over (possbly) nfne sequence of nervenons: For large α, he opmal polcy s Marovan n naure and s characerzed by a conrol regon, a complemenary connuaon regon and a se of opmal acons ha can be aen n conrol regon For he dscree me Marovan polcy, reduces o he well nown Bellman equaon: V Q Q Q H, max E 1 U c e d 2 ouresy: Hamlonan pproach o Dynamc Economcs, D. ass and K. hell 3 ouresy: Opmal onsumpon of a generalzed Geomerc Brownan Moon wh Fxed and Varable Inervenon oss,. Baccarn * c max Prc U c Prc V c c * 1 e 1 d

10 Economc Uly and Enropy Repeaed Ineracon model cons n he conrol regon, Measuremens n he connuaon regon (hah 213) Lndblad Dynamcs yeld he followng n he energy egenbass: The frs order lnear DE has he soluon: hange n Von Neumann Enropy n an acon neracon aes form of Bellman Equaon: Under he assumpons made, he maxmum of he Enropc Uly above s obaned when enropy producon per neracon (Δσ) s mnmum. Thus maxmzng pah dversy corresponds drecly o maxmzng he Enropc Uly K f ln ln f,,ln 2 1 ln K, K K e d f e d ln d d d K K K B e d f e e 1

11 urren Wor Quanum Brownan Parcle n a Box 4 Recen sudes on phase dagram of parcle confned o a fne bndng chan coupled o ohmc dsspave bah aldera Legge model for general dsspave dynamcs of a quanum parcle neracng wh a hea bah Hard wall boundary condons affec he phase dagram of he confned parcle The phase localzaon s sharpes when he sysem-bah couplng s crcal (he phase ranson pon) Resuls agree wh behavor of confned brownan parcle under causal enropc forces 4 ouresy: Phase Dagram of he Dsspave Quanum Parcle n a Box, J. abo, L.Borda, F. Gunea and F. ols

12 urren Wor Emergence of rcaly Varous sudes n neuro-magng sugges ln beween effecve bran dynamcs and crcal behavor of physcal sysems (Kzbchler e al. 29) ysem Bah couplng μ deermnes sysem evoluon when all oher parameers held consan, smlar o he rao (T /T R ) n causal enropc forces Opmzng Enropc Uly n conjuncon wh Frs law yelds: du d dv T, du Td PdV d ondon of Thermodynamc sably: df dt PdV Usng Toal dfferenal of Enropy a consan Pressure P and number of parcles N wh he above condons yelds: V T V T, V T T V, K T 1 V V P T, P V T, The sable pon s ndeed he crcal pon of phase ranson as he laen hea of ranson (Λ V ) a he pon. more concree and rgorous analyss usng crcal exponens furnshed by numercal renormalzaon group echnques s under way

13 onclusons Orgn of pah based enropc forces explaned for Marovan projecve neracons n nonequlbrum dsspave processes Esablshed dualy beween maxmzng expeced uly (Enropc Uly) and maxmzng pah dversy Imporance of sysem-bah couplng srengh n deermnng sysem behavor and evdence of crcal sysem dynamcs a he opmal couplng. No need o explcly specfy ules of acons whch he sysem res o maxmze due o he very naure of he dsspave process a opmal couplng emps under way o undersand emergence of O (elf organzed crcaly) n complex sysems n he curren framewor of dsspave open quanum sysems

14 Than You!