Isotropy, entropy, and energy scaling

Isotropy , entropy , and energy scaling 1 Isotropy , entropy , and ener gy scaling Robert Shour T oronto , Canada Abstract T wo principles e xplain emergen ce. First, in the Receipt’ s reference f rame, Deg ( S ) = 4 3 Deg ( R ), where Supply S is an isotropic radiati ve ener gy source, Receipt R re- cei ves S ’ s ener gy , and Deg is a system’ s de grees of freedom based on its mean path length. S ’ s 1 3 more degrees of freed om relati ve to R enables R ’ s growth and increasing complexity . S econd, ρ ( R ) = Deg ( R ) × ρ ( r ), where ρ ( R ) represents the collectiv e rate of R and ρ ( r ) represents t he rate of an indi vidual in R : as Deg ( R ) in- creases due to the ﬁrst principle, the multipli er e ﬀ ect of networking in R increases. A univ erse lik e ours with isotropic energy distribution, in which both principles are operati ve, is therefore predisposed to exhibit em ergence, and, for reasons sh own, a ubiquitous role for the natural logarithm. Contents 1 Introduction 1 2 The Network Rate Theorem 8 2.1 Observations leading t o Th e Network Rate Hypothesis . . . . . . . . . 8 2.2 Deriving and modeling η . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 The NRT : Observations, implications, and speculation s . . . . 14 3 The 4 3 Degrees of Fr eedom Theorem 18 3.0.2 The 4 3 DFT : Observations, implications, speculations . . . . . 21 4 Discussion 27 List of T ables 1 Calculations of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 Introd uction This article derives two theorems. One, The 4 3 De grees o f F r eed om Theor em , describes how more degrees of freedom in an energy so urce compared to the system receiving the energy initiates and d rives emergence. The oth er , The Network Rate Theorem , Isotropy , entropy , and energy scaling 2 explains the thermo dynam ic beneﬁt o f networking. T oge ther they p rovide a theo ry o f emergence. R elated id eas possibly explain the natural logarithm and con nect quantum scaling and gravity . Emergence is the name assigned to a process whereby an e nsemble of simple com- ponen ts results in a system or process w ith featur es that th e comp onents d o not have. An emergent phenomen on ca n not be predicted on the basis of the attributes of its fun- damental components. Life, markets, language, and ecosystems are emergent systems. The 4 3 De grees o f F r eedom Theor em is more fun damental than The Network Ra te Theor em for two reasons. First, The 4 3 De grees o f F reedom Theorem suggests ho w a system begins. Second , The 4 3 De grees of F r eedom Theorem can be proved with on ly mathematics. This article derives The Network Rate Theorem ﬁrst because it leads to The 4 3 De grees of F r eedom Theorem . The Network Rate Theorem is revealed by an analytical appr oach applied to language as an emergen t system. An an alytical app roach attempts to account for an observed phenom enon. A synth etic approach instead w ou ld start with com ponen ts a nd deduce an outcome based o n th em. A synthetic a pproac h highligh ts a minimal set of r elev ant conditio ns necessary to dedu ce an outcome, but is hopeless as a star ting poin t for an emergen t outcome because it is imp ossible to know in advance what conditions are m inimally r elev ant. “. . . no collectiv e organizational pheno menon . . . has ever been deduced” (Laughlin, 2005, p. 88). That study o f a co llectiv e phen omeno n like langu age, which emerges due to rela- tionships amon g n etworked mem bers o f a society , could reveal laws of p hysics is not unexpected. The p hysicist David Bo hm wro te: “Ein stein’ s basically n ew step [ in spe- cial r elativity] was in the ad option o f a r elation al a pproac h to ph ysics” ( Bohm, 196 2, p. xv i). “ . . . the phy sical facts con cerning time and space coor dinates co nsist o nly o f r elationship s between observed phenomena and instruments . . . . Like wise . . . the facts concern ing perceptio n in common exp erience show tha t this also is always con cerned with relation ships . . . ” ( Bohm, 1 962, p. 62). The ph ysicist L ee Smolin writes that “ net- works do not exist in space—they simply are. It is their network o f in terconn ections that d eﬁne, in app ropria te cir cumstances, the geo metry o f space . . . ” (Smolin, 1997, p. 285) . Robert Laug hlin, wh o won a No bel Prize in p hysics, writes “The laws of n a- ture that we care ab out . . . emerge throu gh collective self-organization . . . ” (Lau ghlin, 2005, p. xi). The philosopher David Hume wrote: “T is evident, that all the sciences have a re lation, greater or less, to human nature” (Hume, 1739, p. x v). The Network R ate Theor em emerges from conceptual f oraging . Just as social in - sects follow promising pa ths over a physical land scape, hu mans follow p romising path s over a concep tual land scape. Honeybees scout potential hi ve loca tions; their society collectively e valuates alternatives until a con sensus eme rges ( Seeley , 2010). Foraging ants follow other ants’ pheromon e trails, an d reinforce chemical signals. Du e to ap- praisal, cho ice, revie w and reﬁnement, a tested idea emerges from random foraging. In connection with computerized heuristics the mathematician s Zbigniew Mich alewicz and David Fogel note: “The essential idea of evolutionary problem solving is quite sim- ple. A p opulatio n of candidate solution s to the task at h and is evolv ed over successiv e iterations of rand om v ariation and selection. Random variation provides th e mecha- nism f or discovering new solutions. Selection d etermines wh ich solution s to m aintain as a basis for further exploration” (Michalewicz, 2 004, p. 16 1). Isotropy , entropy , and energy scaling 3 Four ideas ar e p re-eminen t in this article. One: the me an path length σ is th e intrinsic scaling factor of a networked system o f size n . T wo: for n = σ η the expo nent η gi ves the system’ s intrin sic degree s of freedo m and its intrinsic entropy . Using a parameter other than σ , as in the u sual d eﬁnition o f en tropy , gives only an indir ect measure of a system’ s intr insic degrees of freedom . Three : in trinsic degrees of freedom multiplies capacity . Four: in the Receipt’ s reference frame, an isotropic energy Supply has 1 3 more in trinsic d egrees of f reedom th an th e system (Receipt) that receives the energy . Some observations about the m ean path len gth, degrees of free dom, intelligence, languag e an d mathematics follow in this Introdu ction. Leaving out the observations about intelligen ce, languag e an d mathe matics would sho rten the ar ticle and av oid the distraction of po ssibly arguable points which d o not a ﬀ ect the deriv ation s. Th e observa- tions are included because they provid e con text f or rea soning leading to the deriv ation s. On Mea n P ath Leng ths. Th e p hysicist Rud olf Clausius found that oxyg en molecule s at the temperature of me lting ice tr av el an a verage 461 meters p er second (Clausius, 1857; Brush I, p. 131 ). The physicist Buijs-Ballot o bjected that if so, “volumes o f gases in con tact would necessarily speedily mix with o ne ano ther”. “How then do es it happen th at tob acco-smo ke, in ro oms, remains so long extended in immovable lay ers?” (Clausius, 1 858; Brush I, p. 136). I n rep ly , Clausius i n troduced the c oncept of the mean path len gth (Brush I, p. 140). A gas molecule does not travel un impeded but collides with other gas molecules. The mean path length is the average distance (some fraction of a meter) a mo lecule moves befo re its cen ter of gravity c omes into contact with th e ‘sphere of action’ of another molecule. The psychologist Stanley Milgram (Milgra m, 1967) asked, what is the length of an acquaintan ce chain connecting any two peop le selected arbitrarily fro m a large pop u- lation. In his termino logy , “ A tar get po int is said to be of th e i th remove if it is of the i th generation and no lower genera tion. ” Milg ram asked peop le to ma il a documen t tow ard s a target in Boston. He measured the lengths of the acqua intance chains, and found a mean of 5.2 lin ks. His exp eriment is the origin of th e expre ssion ‘six degrees of separation’ presumed to separate, on a verage, any two people in the world. Clausius’ s m ean path length can be made equiv alent to Milgram’ s degre es of sep- aration by equ ating acquaintance and collision. In stead o f ﬁnding the mean distance between gas molecules in meters, ﬁnd the mean numb er o f collisions separating gas molecules. Sup pose that both gases an d s oc ieties collecti vely maximize their energy e ﬃ ciency . Then the sam e fun damental eq uation that char acterizes th e e ﬃ cient use of energy should apply to both g ases and so cieties. In this novel way the concept of mean path length connects physical systems and networks; a general principle related to one may apply to the other . In a seminal 199 8 article, W atts an d Strogatz analy zed ‘a small world network’. Earlier research had mostly s tud ied entirely regular or entirely random networks. They examined rand omly rewired networks o f an intermediate character . They d eﬁned the ‘character istic path length’ as the a verage of the least n umber of steps between pairs of vertices in a g raph, a deﬁnition eq uiv alent to degrees of sepa ration. T he clusterin g coe ﬃ cient C , wh ich measures the resu lt of their r ewiring o f a gr aph, is the network av erag e of all C i where C i is th e proportion of one step away node s that are actually Isotropy , entropy , and energy scaling 4 connected to each node a i in a network: C = P n i = 1 C i n . For example, su ppose node a k has 5 neigh bors one step away . If o nly 3 of them actually lin k to a k in one step, then th e propo rtion connecting in one step is C k = 3 5 . On Degr ees of freedom. The motion of a point in a p lane has two degrees of f reedom . The m otions of N points on a p lane h av e 2 N degrees of fre edom. One d egree of f ree- dom confers one ch oice on a given axis or line; a ch oice o f left or right does no t give two d egrees of freedom . Suppo se a node moves at any given time with 3 degrees of freedom . The rate at which co llisions with o ther no des in the same sy stem occur d oes not a ﬀ ect its degrees of freedom of motion at any g iv en time. On the Natu r e of Intelligence. IQ tests are desig ned and administered by psycholo- gists. A T ask Force of the American Psych ological Association in 1996 ch aracterized intelligence as the “ability to u nderstand comp lex ideas, to a dapt e ﬀ ectively to the e n- vironm ent, to learn fro m experience, to engage in various for ms of reasonin g, to over- come obstacles by takin g though t” (Neisser , 1996). In this article, assume in telligence is the rate of pr oblem solv ing and that solved p roblem s can be cou nted. T hat enables mathematically modeling intelligence. Use of the model permits its appraisal. An IQ test indir ectly measur es an in dividual’ s skill, using th eir innate p roblem solving capacity , at solving the proble m of learning from a so ciety’ s store of solved problem s and fr om experience, and at ap plying (perh aps b y joinin g to gether di ﬀ er ent ideas) what that individual has learned. Since members of a society share the sam e store o f k nowledge, av erag e IQ measures, partly an d in directly , the IQ o f tha t society . Think of a verage indi vidu al IQ, like a collec ti ve economic indicator , as th e society’ s av erag e problem solving rate per capita. On Collective Intelligences. By an alogy to econom ics, society collectively allo cates its c ollective resources to solving pro blems in a con ceptual area un til the outcom e is as beneﬁcial as for collectiv e resources spent on solving problem s in some other conceptu al area. “. . . a po tatoe-ﬁeld sho uld pay as well as a clover -ﬁeld , and a clover- ﬁeld as a tu rnip-ﬁeld ” (th e econo mist Jev ons, 1879, p. li v). Conside r problems to be the conceptual eq uiv alents of tu rnips. “Th e product of the ‘ﬁnal unit’ of labor is the same as th at of every u nit, separately considered ” (the econom ist, Joh n B. C lark , 1899, p. viii): on average, solutio ns with the same en ergy cost sho uld b eneﬁt society to the same extent. Just as sectors in a n ec onomy compete f or ﬁn ancial r esources, problem s in a society compe te for its collective problem solv ing resou rces. If a n alter native use of p roblem solving energy gives society a better yield for its solutions, society d iv erts energy to t h at alternative use u ntil the solution yields are about the same. At all scales, a collective intelligence e xcee ds the intelligence of its compon ent in- dividuals. Bees, wasps, an ts and ter mites locate, design and build nests or hi ves with a co llectiv e skill that exceeds the cognitiv e capacities of individual insects. “Individ- ually , no ant knows wh at the colony is sup posed to be doing, but to gether they act like th ey have a m ind” ( Strogatz 2 003, p. 250). Th e Gree k mathematician Pappu s (c . 290 - c. 350 ) attributed a co llectiv e mathematical in sight to b ees: “Bees then, know just this fact which is of service to themselves, that the hexag on is greater than the square and th e triangle and will hold mo re honey for the same expen diture of material used in con structing the di ﬀ erent ﬁgures” (Heath, V ol. 2, Ch. XIX, p . 39 0). Col- lectiv e intelligence even o ccurs in bacteria (Ben-Jacob, 2010). In the ﬁeld of Swarm Isotropy , entropy , and energy scaling 5 Intelligence (SI) (Kennedy; Bonabeau), to solve di ﬃ cult problem s computer scientists and en gineers use networked alg orithms and robots to mimic the emergent co llectiv e intelligence of social insects. Cultures, eco nomies an d m athematics are collective intelligences. “ Althoug h gen- erated b y the collective actions of lots of brain s, c ultures h av e storag e an d processing capabilities not possessed by a sing le h uman” (Montag ue, 20 06, p. 199), the wisdom of crowds (Surowiecki). A society’ s economy has “disper sed b its of inco mplete and frequen tly contradictory knowledge which all separate indi vid uals possess” (Hayek, p. 77) lead ing to a market so lution th at “mig ht have been arr iv ed at by on e sing le mind possessing a ll the inform ation” (Hayek , p . 86). In m athematics “ the inner logic o f its development remind s o ne much mo re of the w ork o f a sing le intellect, d ev elo ping its though t systema tically and consistently using the variety of human individualities on ly as a m eans” (I.R. Shafare vitch, in Da vis, 199 5, p. 5 6). This also ap plies to physics, literature, biology and so on. Adoption by a society of a p ropo sed solution to a pro blem depen ds on society’ s estimate of its l ikelihoo d of success. A proposed site for a new b ee hiv e i s appraised by the old hive. T he success o f a new e lectronic device in a human society is ap praised, throug h the oper ation of a market, b y all p otential buyers. In SI, prog rammer s imitate this appraisal and approval e ﬀ ect, creatin g ‘ant pherom one trail’ software for robots. An ant has ab out o ne million neur ons; a hum an b rain has about 10 0 b illion. Sup- pose that the same physical la ws govern the ne tworking and problem so lving output capacity of n euron s in ants as in a single human brain. Th en the collective behavior of 10 billion n euron s in 1 0 thou sand ants and of 100 b illion neu rons in a hu man brain should have similarities. Compare a soc iety of 10,00 0 ants to a society of 10,00 0 human b eings. If the same physical la ws govern, then the hu man society is as much mor e inte lligent than the av erag e component human as is the ant society than the a verage compon ent ant. A society o f 100 m illion humans—10,0 00 networked soc ieties of 10 ,000 indi vidu als each—is as much more intelligent than an a verage society of 10,000 humans as is a society of 10 ,000 humans than its average compo nent human. Consider that human ity throug h languag e, wr iting and culture can accum ulate a store of solved p roblems f or hundr eds of human generations. The collective cumulative intelligence of all human societies is much greater than that of an individual human. Bert Holldob ler and Edward O. W ilson write (1990, p. 252): For t wo reasons ants can be e xp ected to practice econom y in the e volution of their communic ation systems, that is, to use a small number of relati vely simple sig nals deri ved fro m a limited nu mber of ancestral stru ctures and movements. First, the small brain and short life span of an t workers lim it the amount of informatio n these insects can process and sto re. Secon d, the tendency toward signal e volution thro ugh ritua lization restricts the r ange of potential ev olutio nary pathways. On a di ﬀ eren t scale, the same observation applies to a human society . Con sider the history of mathematics and language. On Mathematics and language. For this article , it is not necessary to ag ree on what im- proves mathematics and lang uage, or how . It is only necessary to assume that language Isotropy , entropy , and energy scaling 6 and ideas improve. Gr ounds for that assumption follow . From the tim e o f the Babylonian s 5,000 years ago until now mathe matics ha s im - proved in the qu ality and e ﬃ ciency of concepts, meth ods and notation (Boyer, 1991 ; Cajori, 19 28; Menninger, 199 2). Like bees ev aluating reports of nest si tes f rom bee scouts, incremen tal imp rovements in p roblem solving by individuals are ev alu ated b y group s of peo ple for e ﬃ cacy and e ﬃ ciency . Sim ilarly , languag e “is a co ntinuou s pro- cess of development” (Aitchison , 198 9, quotin g Wilhelm von Hum boldt, 1836 ). His- torical linguistics (Mc Mahon, 19 94; Camp bell, 19 98) record s a history o f impr ovement in the f ormation of soun ds and words; “a law of economy” (Herde r , 1 772, p. 164). The lingu ist Otto Jespersen ( p. 32 4) ob served that lan guage “dem ands a m aximum o f e ﬃ ciency and a m inimum of e ﬀ or t . . . [this] formula is simp ly one of m odern energet- ics”. Th e linguist April McMahon writes “. . . soun d systems tend to ward ec onomy ” (p. 30). I n general, “. . . saving mental e ﬀ ort may be the most impor tant kind of economy ” (Polya, 1 962, II , p. 9 1). I co njecture that the a verage rate of progress in th e e ﬃ ciency of languag e is, for the econ omic reasons discussed above, the same as th e rate of p rogre ss in mathematics. Both enc ode ideas—data. Data compre ssion software p rovides an analogy to mathematics and languag e. In the 1990 s, as th e amou nt of electr onic d ata transmitted , stored and accessed in- creased, and the pr ocessing power of c omputer s incre ased, e ﬃ cient d ata compression became econ omically impo rtant. D ata comp ression software stead ily improves (Sa- lomon, 2007) . As socie ty’ s knowledge—data —increases, languag e incre ases com - pression of data it encodes t h rough naming ( categorization, or u niﬁcation), contraction (can’t), clipp ing (b ike, bus, condo ) (Campbell, p. 2 78), acronyms (IBM), allusion (his, comp uter, n ext door) , pattern (Salomon , p. 7), an d metapho r (sunny disposition). Grammar—word order and word en dings— “. . . provides relief to memory ” (Dider ot, The En cyclopedie). “Declension s and conju gations ar e mer ely sh ortcuts. . . ” (Herd er , 1772, p. 1 60). For humans “. . . la nguag e ﬁrst of all i s classiﬁcation and arrangement of the stream of sensory experien ce . . . . I n othe r word s, langua ge d oes in a cr uder but also in a broade r an d more versatile way the same thing that science does” (the ling uist Ben- jamin Whorf , 1 956, p . 55). Society tests the u tility , e ﬃ cacy and consistency of words that en code—co mpress—perce ptions and ideas. “ A ll kno wled ge is a structure of ab - stractions, the ultimate test of the validity of which is, howe ver , in the process of com- ing into co ntact with the world that takes place in immediate perception ” (Bohm, 1965, p. 262). T o encode a nd co mpress data into words require s a society to collectively solve problem s (McMahon, p. 1 38) that includ e: ( 1) how to devise and ch oose sound s to be u sed for encoding ; (2 ) ho w to assemble soun ds in to words; (3 ) what percepts and concepts should be enc oded. A language embod ies a set of encoding prob lems collec- ti vely solved by the generatio ns of a society that use it. It is a p roduct of collective intelligence. Society perform s the same function fo r lan guage that software eng ineers (and their users) per form f or data comp ression sof tware. Hierar chies e ﬃ ciently orga- nize categories, structu res, and methods for the assembly of words into larger structu res such as sentences and theories. Analogy eases learnin g, remembering and using a lan- guage; similar endin gs, sound s, sentence stru ctures, r hythms and m usicality provid e a ‘relief to m emory’ . The p roblem solv ing elem ents of la nguag e—identifyin g concepts Isotropy , entropy , and energy scaling 7 and encod ing th em—also apply to mathematics, p hysics, an d ideas gener ally . Peop le who ad opt enco dings with increased compression jug gle mo re info rmation per time unit. Spee dier problem solving is of immense v alue to organisms with ﬁnite lif e spans. Languag e also improves b y adding ne w words: encod ings of new or modiﬁed con- cepts. Co llections of concepts, such as theories, also improve at all scales. Feedback from each u se of a word is a scientiﬁc experim ent. Society has tested more en coded concepts mo re comprehen siv ely , in m ore ways, mor e often , for peri- ods of time f ar lon ger, than they could be tested by any indi vidu al. In dividual intel- ligence relies on an enor mous store of highly tested and reﬁned conceptual problem solving tools created b y thousands of ge nerations of human soc ieties. An individual using someo ne else’ s veriﬁed solution saves energy . Th rough comp ression a langua ge increases in depth, and through the enco ding of ne w con cepts it incre ases in b readth. Diderot r emarked over two centuries ago that “ . . . by merely compar ing the vocab ulary of a nation at di ﬀ erent times, one would get a sense of its progress” (Diderot). Mathematical concepts are more e ﬃ cient, com pressed, deﬁne d, an d ha ve been more widely tested, than concepts encoded into words. Societies test their own lan- guage over many g eneration s, but all societies in all cultur es h av e tested mathematics in m illions of contexts r epeatedly o ver hun dreds of generations. Math ematics can be more precisely tested than words bo th through logical ana lysis and becau se th e accu- racy of mathematics co mpared to physical pheno mena can be measu r ed . “Mathem atics is a p art of physics. Physics is an experimental science, a p art o f n atural science. Math- ematics is the part of physics where experiments are cheap” (Arnold, 1997) . Just as 10,000 ants b uilding a nest exhibit an intelligen ce far grea ter than that of an indi v idual ant, our store of m athematical knowledge created over the past sev eral thousand y ears b y tens of thousands of mathematicians, tested a nd app raised by tens and hund reds of millio ns of people in daily , scientiﬁc and comm ercial c ontexts, exhibits an intelligence beyond human comprehension . Mathema tics as a disembo died n etwork of concepts ‘knows ’ things th at indi vidu als d o n ot. The ‘unreasonab le’ e ﬀ ectiv en ess of mathematics in th e natural sciences (W igner 196 0; Hamming 19 80) is like mag ic. An “education in . . . math ematics is a little like an inductio n into a mystical order” (Smolin, p. 17 8). The di ﬀ erence between the in telligence of m athematics and of an individual human is far gr eater than the di ﬀ eren ce b etween the intelligence of 1 0,000 societies and the intelligence of an individual human. Since mathematical ideas result from collective e ﬀ o rts of millions o f people over hundr eds of generation s, common mathematical ideas such as c ounting n umbers must reﬂect fundamental pr inciples underlying th e natural world. Mathem atical r easoning relies o n a higher, collective, intelligen ce. A mathem atical theo rem th at predicts p he- nomena su bsequently o bserved or that c onnects to o ther mathematical id eas is a fo rm of experimental veriﬁcation, th e outcome of h umanity ’ s collec ti ve scientiﬁc e valuation of mathematical concepts. M athematical deduction can render express what is implicit in collective mathem atical kn owledge. A math ematical concept that appear s to ap- ply to a phen omeno n can be tested by applyin g it in di ﬀ er ent co ntexts, just a s society collectively does. On a q uestion about intellig en ce. A verage IQs increase. Is this due to the improvement of languag e and ideas? This qu estion im pels foragin g over the con ceptual lan dscapes below . Isotropy , entropy , and energy scaling 8 2 The Network Rate Theor em 2.1 Observ ations leading to The Network Rate Hypothesis Consider words as par t o f soc iety’ s accum ulated array of p roblem solvin g tools. If a lexicon improves, so should problem solvin g. Researchers have observed that average IQs hav e increased—imp roved—in the U.S. in the past 6 0 years or so by about 3. 315% per decade; no one kno ws why (Flynn, 2007, p . 1 13, T able 1 at p. 1 80). Are increasing av erag e IQs caused by words improving? Both improving at the same rate w ould be positive evidence. A rate characteristic of language impr ovement is nee ded. Measuring the d epth o f word s is di ﬃ cult. Measuring th e b readth ( number ) of word s is a massi ve undertak ing, but has alread y been accomplishe d b y academic dictio naries. If word counts of a lexicon at two d i ﬀ erent times use the sam e criteria, and if eac h count is large, th en the calculated rate of increase in the lexicon should be a goo d estimate of the rate of collecti ve pro blem solv ing, because le xico ns requir e a large num ber of problem s to be collectively so lved. The English le xico n increased from 200,0 00 words in 1657 (Lancashire, EMEDD) to 6 16,50 0 words in 198 9 (Simp son, OED) , 3 .39% p er deca de. The University of T oron to’ s partly c ompleted Diction ary o f Old Eng lish ( DOE) con tains Old Eng lish words fr om the year 600 to th e y ear 1 150. Eigh t of 22 Old English letter s, up to the letter g , ha d been co mpleted at Dec ember 2 008. Extrapolating from the 12,2 71 words for the 8 completed letters—the dictionary c ounts æ as a sep arate letter —and assuming the sam e av erag e n umber of word s per letter, g iv es 34,020 word s in Old En glish f or the whole Old English alphabet of 2 2 letters. An increase from 3 4,02 0 word s in 1 150 to 616,5 00 words in 1 989 in the OED is an increase of 3.4 5% per decad e. Both English lexical growth rates ar e close to the rate at which av erag e IQs increase. The error arising from using an estimate ( ρ ( Le x )) E st of the actua l rate of English lexical in crease ( ρ ( Le x )) Act is calculab le b y comparin g the di ﬀ ere nce b etween the ac- tual size o f the English lexicon in 1 989, [ N ( t 2 )] Act and an estimate [ N ( t 2 )] E st based on an estimated English lexical growth rate ( ρ ( Le x )) E st applied to an actual in itial English lexicon. Set ∆ N 2 = [ N ( t 2 )] E st − [ N ( t 2 )] Act . Then ( ρ ( Le x )) E st − ( ρ ( Le x )) Act = " ln 1 + ∆ N 2 [ N ( t 2 )] Act !# ÷ ∆ t . (1) The error in the estimate o f ( ρ ( Le x )) Act becomes smaller as the time period ∆ t increases. Do other studies measure the rate of increase in ideas accumulated by soc iety? The e ﬃ ciency of ligh ting in terms of its lab or cost increased fr om 17 50 B.C.E. to 1 992 by 41 . 5 . 000119 = 3 48 , 739 . 5 times accordin g to a stud y by the econo mist W illiam Nordhau s (1997 ). That is 3.41% p er decade, close to th e rates fo r English lexical growth. So- cieties appraise lightin g impr ovements and choo se which ones to adop t. In light of Equation (1), a stud y covering 3,742 years gives a good estimate of the av er age rate at which ideas improve. Do better collec ti ve ideas incr ease lo ngevity and reduce ho micides? Jim Oeppen and James V aupel (2002) found th at, for example, male longevity in Norway increased from 44. 5 year s in 1 841 to 71 .39 years in 196 0; that is 3.9 7% per decade. Manuel Eisner (2003) estimates th e Londo n hom icide rate in 12 78 at about 1 5 per 100,0 00 Isotropy , entropy , and energy scaling 9 inhabitants (p. 84) com pared to th e En glish homicid e rate in 1975 of 1.2 per 100,0 00 inhabitants (p. 99) , a rate o f decrease of 3 .75% per deca de. The rates b ased on these studies ar e o nly in the vicinity of that fo r in creased lig hting e ﬃ ciency , perhaps d ue to the choice of data. Morris Swadesh d evised a meth od to estimate, based on the rate of their divergence, when two d aughter languages had a commo n moth er tongue. His method is called glottochr onolog y . First he compiled a list of 1 00 or 200 word s basic to lang uages (the Basic List). He th en calculate d the rate of cha nge between cog nates (such as moi in Frenc h and me in English) by comparing their use in historical record s. He found ( Swadesh, 1971) an average rate of divergence of two daugh ter lexicons o f about 14% per th ousand years. In 1 966, he used this diver gen ce r ate to estimate that Indo- Europ ean (English’ s ancestral language) existed at least 7,000 yea rs ear lier (p. 84). Gray an d Atkinso n (20 03) dated Ind o-Euro pean to 870 0 year s earlier, using n ewer methods. Upd ating Swadesh’ s calculation using Gray and Atkinson ’ s ﬁnding s, two daugh ter langu ages di verged from each other at ≈ 7000 8700 × 14% = 1 1 . 2% per thousand years; or 5 . 6% per tho usand year s each from a n otionally static mo ther tongu e. Why half th e u pdated Swadesh d iv ergen ce rate 5 . 6% per th ousand years is so much slower than the English le xical growth rate is a new pro blem. Swadesh’ s method of estimating the div ergenc e rate has been severely critiqued on criteria for iden tifying cognates and other ground s (Blust, 2000, p. 2 04; Campbell, 1998). If t h e updated Swadesh divergence ra te estimates th e common o rigin of two dau gh- ter langu ages, does a rate exist wh ich estimates when lang uage itself began? T o ap prox- imate the size of such a r ate, suppose the 616 ,500 words of the 1989 Oxf ord English Dictionary g rew from 100 words in 200,0 00 years. That would b e 4 . 3% p er th ousand years, no t far o ﬀ half the upd ated Swadesh di vergence r ate. Is h alf the upd ated Swadesh div ergen ce ra te a fossil rate ρ ( r ) embedd ed in th e m uch faster En glish lexical gro wth rate? This leads to: The Network Rate Hypothesis : Th ere is a function η (sma ll Greek eta) such that 3 . 39% per dec ade in collecti ve lexical growth = ρ ( R ) = η × ρ ( r ) , where ρ ( r ) is some kind of fossil rate. The signiﬁcance of being able to m easure the ra te of improvement in collective problem solving, via increasing a verage IQs, le x ical growth and improvement in ligh t- ing, is that m easurability conv er ts a qu alitativ e que stion—do concepts improve—into a testable hypoth esis. 2.2 Deriving and mode li ng η T o inv estigate how langu age f acility incre ases for an indi vid ual, ask how a child ac- quires w or ds. A child learns words from two paren ts, who each learn from two par- ents, an d so on. Supp ose there are η antecedent generations, and (as an idealizatio n and simpliﬁcatio n) each genera tion indepen dently increases society ’ s accumula tion of words at the same rate. If p arents were the o nly source for words, th e numb er of word source ge nerations would be lo g 2 (2 η ) = η . But other peo ple can b e word sour ces. Th e scaling factor (th e b ase of th e log function ) is no t 2, but some unk nown a verage value σ . σ m ust be d etermined in order to con vert lo g σ ( n ) into a number . What numb er is σ ? Isotropy , entropy , and energy scaling 10 Is σ an intrinsic average num ber of acquain tances? Primates usually live in ban ds of 50 mem bers; groom ing is part of their social lif e (Dunb ar , 1997, p. 120- 122). Du n- bar sug gests th at a person virtually ‘gro oms’ thr ee times as many p eople using words as is possible grooming man ually . Could σ be 3 or 50? Con sider idea lized sp eakers who seek to transmit info rmation with least e ﬀ o rt, an d idealized listen ers wh o seek to decode inform ation with least e ﬀ ort, as distinct groups (Zipf, 1949 , p. 2 1). Does Dun- bar’ s optimum audience of three balance the competing goals of speakers and hearers? Three o r ﬁfty , these numb ers cannot work. Why? Ap peal to mathematical r eason- ing. If more persons transmit inf ormation (if σ is greater), information received should be g reater . But, on th e co ntrary , as σ increases, log σ ( n ) d ecreases. It is imp ossible, if η multiplies ρ ( r ), that th e Network Rate decreases fo r the individual with more infor- mation sources. The Network Ra te Hypothesis , or an assumptio n, exp licit o r implicit, on which it is based , is wr ong or the fu nction η is n ot logarithm ic. Supp ose The Net- work Rate Hypothesis is valid and that η is a logarithmic function . Then reconsider the assumption that σ is a ﬁxed num ber . What parameter would σ have to be for η to be logarithm ic? 1 σ must cause log σ ( n ) = η to in crease when σ d ecreases. If information tak es less time to reach an individual, then the rate of in crease in the individual’ s store of infor mation should increase. Mor e information can be receiv ed during a lifetime. A faster a verage rate of info rmation transfer implies a shor ter average minimum tran smission ( or relationship ) distance per tim e un it between transmitting and receiving in dividuals. The mean path length co rrespon ds exactly to such a distance. Suppose th en that σ is a network’ s mean p ath length. For simp licity’ s sake, suppose that th e actu al n umber of steps between pairs o f nodes equals the av er age nu mber of steps, σ . Finding η is still n ot co mplete. In an actual network no t all p airs o f n odes σ steps apart are actually conn ected. If eac h node receives an average prop ortion C < 1 of the mu ltiplicativ e e ﬀ ect of log σ ( n ), th e network need s to increase the n umber of nodes to σ η C to have the same value of η as a network with C = 1. Conclude that η = C × lo g σ ( n ) and that in general ρ ( R ) = C log σ ( n ) × ρ ( r ), where ρ ( R ) and ρ ( r ) ar e rates, and C is th e network’ s clustering coe ﬃ cient. On the assumption tha t an average exists. T o apply th e form ula for η to actual networks requires th at av erag e rates prop ortiona l to the mean path length exist. A verag e IQs exist. Econo mists calculate average gross dom estic prod uct per capita. In principle criteria for coun ting di ﬀ erent kinds of problems solved by people can be design ed and the a verage number of problem s solved per t ime period can be calculated. In principle, therefor e, the average rate of prob lem solving p er cap ita is calculable. I f the a verage rate of prob lem solving obeys laws of econom ic e ﬃ c iency , the average r ate of lexical problem solving an d the average rate of solving lighting p roblems in terms of labor cost, can b e u sed as pro xies for the average rate of collective problem solvin g. In this article, only the a verage features of pro blems are of i n terest. Obtaining a count of problem s is not easy , esp ecially counts that are reﬂective o f society’ s collecti ve problem solving (such as word s in a lexicon). All th at is necessary though is to assume counting is possible in p rinciple. If problems can be counted in principle , the a verage collective problem solving rate can be calculated in principle . T esting η . Before spending time and en ergy s co uting for an explan ation fo r the pro- 1 The balanc e of this arti cle sorts out the implicatio ns of the answer to this question. Isotropy , entropy , and energy scaling 11 Networ k Nodes Number of nodes σ C η Notes Actors people 225,226 3.65 0.79 7.52 1 C. ele gans neurons 282 2.65 0.28 1.62 1 Human Brain neurons 10 11 2.49 0.53 14.71 2 1989 English wo rds 616,500 2.67 0.437 5.932 3, 4 1657 English wo rds 200,000 2.67 0.437 5.431 4, 5 1989 populati on people 350,000,000 3.65 0.79 12.0 6, 7 1657 populati on people 5,281 ,347 3.65 0.79 9.445 6, 8 T able 1: Calculations of η Notes to T able 1 1. C and σ are based on valu es in the article by W att s and Strogat z (1998). 2. The number of neuron s: Nicholls, 2001, p. 480. σ and C : Achard, 2006. 3. The number of word s: OE D (Simpson). 4. σ and C : Ferrer , 2001 based on about 3 / 4 of the m illion words appearing in the British National Corpus. Motter (200 2) found σ = 3.16 and C = . 53 based on an English t hesaurus of about 30 ,000 words, a smal ler and less represen tati ve sample. 5. The number of word s: E MEDD (Lancashire ). 6. σ and C : based on the actors study of W atts and Strogatz (1998). 7. The number of people is an estimate of the English spea king societie s in 1989, by adding censuses: 1990 USA, 248.7 million people (Meyer , 2000); 1991 Canada 27,296,859; 1991 England 50,748,000; 1991 Australia , 16,850,540 people. These total 343,595,000 people. 8. The number of peopl e in England: T able 7.8, follo wing p. 207, for the year 1656, Wrigley , 1989. posed η , determine if it works. If it does, then why it does will be the next problem. For data, researchers have m easured the mean path length σ and th e c lustering coe ﬃ cient C for som e networks. Data on a line in T able 1 is used to calculate η ( n ) = C lo g σ ( n ) for the same line. The v alues of σ an d C f or a population of actors (W atts & Strogatz, 199 8) a re applied to human so cieties genera lly; th is is justiﬁed b elow using the Natural Logarithm Theor ems and The 4 3 De grees of F reedom Theor em . If The Network Rate Hypothe sis is valid, the average prob lem solving capacity ( ρ ( R )) av of En glish sp eaking so ciety , not includin g the e ﬀ ect o f u sing lan guage, from 1657 to 1989 is ( η ( po p )) av = 9 . 445 + 12 . 0 2 = 10 . 7 2 times the average indi vidu al rate, ρ ( r ). T reat Eng lish society itself, witho ut the use of lang uage, as a sing le collectiv e brain with innate prob lem so lving capacity ( ρ ( r )) C oll = 10 . 72 × ρ ( r ). Multip ly ( ρ ( r )) C oll by the increase in capacity ( η ( Le x )) av conferr ed on ( ρ ( r )) C oll by the English lexicon. F or 1657 to 19 89, ( η ( Le x )) av = 5 . 431 + 5 . 932 2 = 5 . 68. Now ﬁn d the average individual in nate problem solving capacity ρ ( r ) using society’ s worded problem solving c apacity: ρ ( R ) ≈ 3 . 41% per decade = ( η ( Le x ) ) av × ( ρ ( r )) C oll = ( η ( Le x )) av × ( η ( po p )) av × ρ ( r ). ρ ( r ) = 5 . 6 % per thousand years, exactly half the updated Swadesh di vergence rate. “Such an agreement between results which are obtained from entirely dif- ferent prin ciples c annot be accidental; it rather serves as a powerful con - ﬁrmation of the tw o prin ciples and the ﬁrst subsidiary hypothesis annexed to them” (Clausius, 1850) . In this case, the subsidiary hypothe sis is The Network Rate Hypothesis 2 . 2 From ab out June 2007 to June 2009, I a verag ed a starting indi vidual rate of 0 and ρ ( r ), inste ad of Isotropy , entropy , and energy scaling 12 Using values from about 198 9 i n T able 1, ρ ( R ) = η ( p o p 1989 ) × η ( Le x 1989 ) × ρ ( r ) = 12 × 5 . 932 × ρ ( r ) = 71 × ρ ( r ) . (2) Equation (2) implies that what a 1989 individual experienc ed as a propr ietary ra te of problem solving, 71 × ρ ( r ) , mostly deriv es from η ( po p ) × η ( Le x ). What manner of conce pt is η ? W h y does it work? T o simplify , assume that all binoda l distanc es are σ steps, all n odes hav e equal capacities to rece i ve an d transmit info rmation and all transmissions have an equal amount of information a nd use the same a mount of energy . Assume a network with σ η = n information sou rces. W ith th ese simp lifying assump tions, the fo cus is on net- work level chara cteristics. Like the temperatu re outd oors, component level (molecu - lar) char acteristics and v ariatio ns are ir relev ant. One nu mber su ﬃ ces. If ρ ( r ) = k σ , log σ ( σ η ) = log k σ (( k σ ) η ) = log k σ ( k η σ η ). Instead o f two p arents, f our g randp arents an d so on sup plying words, each ﬁrst g en- eration re ceiv er h as σ 2 second g eneration so urces, σ 3 third g eneration sou rces an d so on up to σ η = n η th generation sources. Ea ch no de receiv es the η beneﬁt of networking, which implies that a ll possible connection s form. Then each node has σ + σ 2 + . . . + σ η sources of information. But since the η th generation alone h as n = σ η nodes as informa- tion so urces, a s well a s in formatio n sources in ge nerations 1 thro ugh η − 1, each nod e would ha ve more information sources than there are nodes. A r elated issue is, suppose the network recei ves n units of energy per time unit for each gener ation of informa- tion exchange inv o lving a particular nod e. T here is not e nough energy ( and th erefore not enoug h time in a round of inform ation transmission) for all possible combin atorial states. Can this coun ting pr oblem be resolved? (This pro blem relate s to the ergodic hypoth esis, discussed below .) I f not, the hypothesis fails. Next is the co mmensur ability pro blem po sed by dimensio nal an alysis (Bridgm an, 1922) : how ca n th e m ean path len gth, a m easure of d istance, scale n , a population size? A scaling su bgrou p for a p opulation should b e a sub -pop ulation, not a d istance. Third is the n − 1 problem: If a giv en node receiv es information from the rest of the network, consistency requires that the argument of the log function s ho uld be n − 1 not n , unless the node tr ansmits, impossibly , ne w information to itself. Fourth, how would such a network be wired? Fifth ( a vexing problem of categorization) : is a m ean path leng th a distance or a scaling factor? If th e ﬁrst fou r prob lems are irr esolvable (the ﬁf th prob lem will be d ealt with sep- arately later), th en η ( n ) = C log σ ( n ) must be false. Y et ρ ( r ) ma tches h alf the upd ated Swadesh di vergence rate too closely to be coin cidence. Since η app lies to transmission of inform ation, ideas from inform ation theory may help. Claude Shann on derived a f ormula (194 8) for the information content η of a string of 0s and 1s, where p i is the prob ability of the i th symbol, η = X p i log 2 1 p i ! . (3) av eraging the η ’ s of the lexi con and populati on and holdi ng ρ ( r ) constant over the rel ev ant time period. This gav e ρ ( r ) = 2.8% per t housand years, a 4 : 1 ratio inst ead of the corre ct 5. 6% per thousand years, a 2 : 1 ratio. I pe rsisted in studying η because of the precision of the 4 : 1 ratio. Some of my ol der preprints on a rXiv ha ve this error . Isotropy , entropy , and energy scaling 13 Shannon used a gra ph to show that η is m aximum for a g iv en numb er of bits when the probab ility of each bit occu rring is th e sam e ( p i = p j , ∀ i , j ), which is also explained (Khinchin , 195 7, p 41) by Jen sen’ s inequality . Sh annon ’ s observation is called the maximum en tropy principle (Jayn es). E quation ( 3) has the same form as that u sed for entropy in thermodyn amics, K × X p i log x 1 p i ! . (4) Assume eq uality of all o f a network’ s nodes, p i = 1 n , ∀ i in Equa tion (4). Substitute σ f or x . Then P 1 n log σ ( 1 1 / n ) = log σ ( n ). η in The Network Rate Hypothesis has the same form as Eq uation (4). K in Equatio n (4) correspond s to the clustering co e ﬃ cient C . “. . . discoveries o f connections between heterogeneous mathematical objects can be compare d with the d iscovery o f the connection between electricity and magnetism . . . ” (V .I. Arno ld); co nnections between di ﬀ er ent m athematical mod els im ply they share a common principle. η ’ s conn ection to entropy connects η to thermody namics. Commensur ability: suppose that the num ber of n odes n and the m ean path length σ are both propo rtional to a com mon me asure o f ener gy . If it takes σ energy u nits to trav el σ steps, then an average of σ peo ple are within σ steps of each of the network’ s nodes. Counting pr oblem: η mathematically requ ires multiple scaling s yet a constant ar - gument n . E nergy must scale in a un iformly nested way . A cluster of nod es scales by σ , not like a pyram id, adding nodes at each next proceeding le vel, b u t internally , by un iformly subd ividing into σ sub clusters. For example , 27 nodes can be in ternally scaled by 3 as follows: [ { aaa } { aaa } { a aa } ] [ { aaa } { aa a } { aaa } ] [ { aaa } { aa a } { aaa } ] . (5) A node when network ed as in (5) has 3 network capacities, depending on in which size cluster , 3 nod es, 9 nodes, or 27- nodes, its capacity is exercised: η 3 (27) = log 3 (27) = 3 . In a next generatio n, the numb er of clusters increases by σ and the number of nodes per cluster d ecreases by 1 /σ . The number of nodes p er generation is co nstant. Th e ﬁrst generation has σ clu sters each with σ η − 1 nodes, the secon d generatio n has σ 2 clusters each with σ η − 2 nodes, an d so on , until th e η th generation of σ η clusters with o ne nod e each. ( k + 1) st generation clusters nest in k th generation clusters. Suppose a network of n equal no des r eceives n energy u nits per time un it. Per time unit, each n ode only has enoug h en ergy to bino dally conn ect to o ne oth er nod e, not to all σ p ossible n odes. What η mu ltiplies is ca pacity ; η m easures R ’ s degrees of fre edom relative to R ’ s mea n path length, σ : η ( n ) = Deg σ ( R ). Sinc e each nod e is in each un iformly scaled nested g eneration , eac h (average) individual has th e same number of degrees of freedo m as the network itself. Th is resolves the wiring prob lem. It also reso lves the n − 1 pr oblem: “The individual agent contributes to the dynam- ics of the who le gro up (society) as w ell as the society contributing to the individual” (Dautenha hn, 1999, p. 103). The intrinsic measure of ρ ( r ) is σ ; ρ ( r ) ∝ σ . Deﬁne the intrinsic entropy o r degrees of freedom of a system X with n nodes as Deg σ ( X ) = lo g σ ( n ). In network theor y , σ equals the av era ge degrees of sepa ration. For a society with σ η = n p eople in it, every Isotropy , entropy , and energy scaling 14 person has the same ‘relationship s trid e’ (acquain tanceship distance) , σ . Treat the start position as the ﬁrst gen eration or stride. Then η strides, or d egrees of f reedom, span s the society . In thermod ynamics, for an id eal g as G , Deg σ ( G ) = log σ ( n ) wh ere σ is the in trinsic mean path length f or collid ing molec ules, m easured in collision steps. In informa tion theory , σ = 2 when inf ormation is represented in bits. I propo se to categorize the hypothesis as The Network Rate The or em ( NRT ): For an isotropic system R with n = σ η nodes and mean path length σ ρ ( R ) = η ( R ) × σ = Deg σ ( R ) × σ (6) and in general for a clustering coe ﬃ cient 0 < C ≤ 1 ρ ( R ) = C lo g σ ( n ) × ρ ( r ) . (7) When C = 1 uniformly scaled nested clusters model R . 2.2.1 The NR T : Obser vations, implications, and speculations On the special r ole of th e mean path len gth. Is there a way , with out using Shann on’ s graph or Jensen’ s inequ ality , to show how the mean path length optimizes η ? Consid er a system of w ater containers. Level 1 has σ water containers, each supported undern eath at level 2 b y σ water contain ers. W ater is supplied at the same r ate to each o f th e ﬁrst lev el w ater contain ers. When a ﬁr st le vel water container is fu ll, water sp ills into its supportin g le vel 2 water containers. If one level 2 containe r is smaller than the rest which are equal in size, it spills water while the other con tainers are still ﬁlling. If one lev el 2 contain er is bigger than the rest which are eq ual in size, the rest spill water while the bigger contain er is still ﬁlling. Analogize water to energy . A networked system utilizing energy supplied at a ﬁxed rate will increa se its rate of outp ut if it uses more (a nd wastes less) of the e nergy supplied p er time u nit. Nested, unifo rmly scaled distribution of ene rgy from a Supply S ind uces a n ested, u niform ly scaled stru cture in a networked system R recei vin g the e nergy , as otherwise energy supplied per time unit by S is no t fully utilized by R . Suppose a centr al energy source radiates energy . E ﬃ cient ﬂow must be uniform in ev ery direction . A wa ve front circular from the source maximizes entropy . What mechanism a llows a n etwork to ﬁnd its a verag e scaling factor? Unite the concept of an ideal network with th e concept of an ideal heat eng ine. T he idealized network discussed above consists o f all p ossible pairs of no des, all σ steps apart and equal in capacity . I n Sadi Carnot’ s (1824 ) ideal heat engine the cylinder con tains a working substance such as air between a ﬁxed p late an d a movable frictionless pis- ton. A furnace transmits heat to the otherwise perf ectly insulated cylinder , causing the working sub stance to expand. The furnace ce ases c ontact with th e cylinder a nd is p ut in touch with a heat sink which removes heat from the working substance causing it to contract. Then the h eat sink is re moved, and the cycle rep eats. Th e piston cycles up and down moving an attached articu lating arm. Carn ot proved that no hea t engine can be mor e e ﬃ cient than an id eal heat en gine. No energy is lost o ther than to movin g the articulating arm. Isotropy , entropy , and energy scaling 15 Consider the p iston’ s initial po sition and the u nique turning po int in the h eat cycle to be two nodes: a heat engine’ s h eat cycle is intrinsically binodal. Treat the furnace as one no de and the h eat sink as the o ther . Remove the a rticulating a rm. Place a fu rnace and a heat sin k at each node, s o that energy can equ ally well move f rom o ne node to the other . A binod al sym metric ideal heat engine can perfectly transfer energy from one node to the other . Suppose that the amount of energy required to transmit information is proportiona l to the amou nt o f informa tion tra nsmitted. By a nalogy , co nstruct a binod al symm etric ideal in formatio n engin e which transmits inform ation from one node to the oth er . All nodes have identical transmission and r eception capacities with n o energy or informa- tion loss. Form an ideal n etwork c onsisting of sym metric ideal information engines. No in formatio n exchange network can be more e ﬃ c ient. Each generation of isotrop ic informa tion e xch ange is eq ually and per fectly e ﬃ cient. I f the physical environment changes, a network wh ose n odes all have equal cap acities in each g eneration o f inf or- mation exchang e will b e the q uickest to cycle throu gh th e gener ations o f info rmation exchange required to reach an optimal ﬁtness landscape. T his (I conjecture) models how n etworks bino dally com municate cha nge to their constituen t comp onents. Opti- mal local binodal exc ha nge leads to global op timality ; social insects are an example of “a decentralized multiagen t system whose control is achieved th rough lo cally sensed informa tion” (Kube & Bo nabeau, 2 000, p. 91), as are lan guage s (speakers an d hearers), markets (buyers and sellers), and genes (two strands of DN A). Comparing entr opy and intrinsic entr op y . In 1848, W illiam Tho mson (L ord K elvin) used Sad i Carnot’ s analysis of an ideal h eat engine an d th e co ntraction of g ases when cooled to ﬁnd absolute ze ro (K elvin, 1848 ). The v olum e o f a n ideal gas contracts in propo rtion to absolute temperature. Clau sius s ou ght and found a n in variant pro perty o f the id eal heat eng ine cycle. He called it entro py (Clausius, 186 5, p. 4 00; in English translation, 18 67, p . 365). I n Clausius’ s deriv ation of entropy (1879 , p . 79), he com- pares the volumes of the working substance at di ﬀ e rent stages of the heat cycle and ﬁnds (p. 83) that ¯ d Q 1 T 1 − ¯ d Q 2 T 2 = 0 , (8) where ¯ d is an inexact di ﬀ erential, ¯ d Q 1 is the heat added to the heat eng ine from the furnace at the absolute temperatu re T 1 , and ¯ d Q 2 is the heat removed fr om the heat engine by the heat sink at the absolute temperature T 2 . ¯ d Q T is the change in entropy . Boltzmann (18 72) remarked that a system can achieve equ ilibrium on a macr o- scopic scale. For example, air has a m easurable tem perature . At a mic roscopic scale, on the oth er hand, there is con stant molecular motio n. He inferred that th e average exchange o f energy o f gas molecu lar co llisions must also be steady . “The determin a- tion of average values is the task of probab ility th eory” (p. 90, English translation) . Boltzmann’ s H Th eory ( − H = entr opy) used probab ility and a log functio n. Build- ing on Boltzmann’ s w or k, the physicist Max Planck derived the form ula f or entropy η = K P p i ln  1 p i  (Planck, 1914) . Clausius’ s deﬁnition of entropy is ba ﬄ ing. ¯ d Q T = d S is the ch ange in entropy , but what doe s it represent? The ratio deﬁnition , based on an id eal heat cycle, relies on Isotropy , entropy , and energy scaling 16 experiment. A degree Kelvin equ als a degree centigrade based on th e freezing and boiling points of water . 0 degrees Kelvin was d etermined by experimen t. The impo rtant practical adv an tage of Cl au sius’ s ratio deﬁnition is its use of temper ature, which can be easily measured. In The Network Rate Theo r em ρ ( R ) = η × σ ⇔ η = ρ ( R ) σ , where σ is the system’ s mean p ath len gth. Changin g a system’ s entropy chang es its degrees o f freed om. For example, d η = log σ ( σ m ) − log σ ( σ n ) = m − n . Assume a system’ s outpu t rate ρ ( R ) equals its energy input rate ρ ( E ). Then, η = ρ ( E ) σ . In Clausius’ s deﬁnition of entropy , ¯ d Q T = d E ǫ , E bein g a to tal amount o f energy and energy ǫ a scaling factor . The numerator o n the left side ¯ d Q T is a change in heat, wh ich is equiv alen t to a change in energy , and the d enomin ator is the absolute temp erature, wh ich is pro portion al to an am ount of energy ǫ . Clausius’ s deﬁnition of entr opy is eq uiv alent to η = ρ ( E ) σ , except th at it uses a scalin g factor T p r oportiona l to σ in the denom inator . Clausius’ s r atio d eﬁnition so indir ectly measures a system’ s intrinsic d egrees of freedo m that it a ltogether obscure s its connection to degrees of freedom. Replace η in The Network Rate Theor em by log σ ( n ), and ρ ( E ) = ρ ( R ) = log σ ( n ) × σ . (9) A mathematical un ion of a ratio deﬁnition o f intrinsic entro py with the statistical deﬁ- nition of intrinsic entropy gi ves The Network Rate Theor em . If two gases at di ﬀ er ent temp eratures mix, they will re ach an equ ilibrium state with a common t em perature. A calculus proof of this uses di ﬀ erential equations. Mor e simply , when two g ases mix, repeated binod al collisions lead to a new mean pa th length σ for the combined system, and hence a common a verage temperatu re ( ∝ σ ). Mechanics studies h ow two par ticles interact. It is not p ossible to con sider e v- ery collision, for exam ple, of 6 .02 times 10 23 oxyge n g as molecules (ab out 3 2 gram s worth). Boltzmann h ad the idea of di vid ing a space up into cells, and calcu lated the exp ected statistical distrib utio n of energies am ong th e d i ﬀ erent cells. Trillions of molecules have a small set of d i ﬀ erent speeds or en ergies, a statistical mech anics. Us- ing the mean path leng th to scale a system reduces the number of parameters f rom a small set to one, a concep tually compressed statistical m echanics. L ike categorizing a country ’ s wealth b y its GDP per capita. On de grees o f fr eedo m and system ca pacity . T he exponent of a system’ s mean path length σ in n = σ η measures its intr insic degrees of freedom. T he collective rate o f a system ρ ( R ) = log σ ( n ) × ρ ( r ), where ρ ( r ) is the rate of an a verage indi vidu al. Sup - pose that ρ ( r ) is a constant, as is the case for average in nate human p roblem solving capacity over the past fe w thou sand years. While av erag e in dividual inn ate capacity is unchan ging, av era ge indi vid ual capacity increases if the indi vid ual’ s innate capacity has more degrees of f reedom to which it can be applied. That occurs when an indi- vidual adds to their store of solved problem s—knowledge. In cellular phone networks, researchers observe that increasing the de g rees of freedom in multiple input m ultiple output (MI M0) antenn a systems lead s to a ‘ gain’ in capacity ( Jafar , 2008; Borade, 2003; Molisch, p. 5 21). M ore antenna s, more degrees of freedom. On glottochr onology: r econ ciling th e diverg en ce rate with the English lexical gr owth rate. 5 . 6% per tho usand y ears, half the upda ted Swadesh di vergence rate, equals the Isotropy , entropy , and energy scaling 17 innate individual average p roblem solving r ate. A po pulation M with a com mon mother tongue divides into daugh ter popu lations D 1 and D 2 isolated f rom ea ch othe r , each initially with the sam e lexicon as M . Assume that D 1, D 2 an d M all ha ve the same size po pulations and same size lexicons: at t 0 , Le x M = Le x D 1 = Le x D 2 , and po p D 1 = po p D 2 = po p M , so η ( po p ) × η ( Le x ) = η for D 1, D 2 and M . T o ﬁnd one h alf the av erag e div ergen ce rate assume Le x M ( t 1 ) = Le x M ( t 0 ), so ( ρ ( r )) M = 0 . Th en compare the rate of change for each daughter language to the rate for a static mother language. Le x D ( t 1 ) Le x M ( t 1 ) = (1 + ( ρ ( r )) D ) × η × Le x D ( t 0 ) (1 + ( ρ ( r )) M ) × η × Le x M ( t 0 ) = (1 + ( ρ ( r )) D ) 1 + 0 = 1 + ρ ( r ) . (10) The η ’ s and the lexical sizes in nume rator and den ominato r of Equation (10) cancel. If the daugh ter to ngues undergo changes independent of each other, the n each Le x D grows at the same a verage rate ρ ( r ) a way from Le x M . Th e av erag e r ate of lexical diver - gence o f the two daug hter langua ges equals 2 ρ ( r ). Swadesh’ s updated d iv ergence rate (remark ably) in directly m easures twice the average in nate ind ividual h uman pro blem solving rate. On mitochond rial Eve. Using matern al m itochon drial DNA, Rebecca Cann , Mark Stoneking and Allan W ilson dated a single woman an cestor to 2 00,00 0 years ago (Cann, 1987) . Nested scalin g implies that d ates a ﬁrst generation, no t an individual (Gould, 2002). On nested scaling and the natural loga rithm. Clusters of size σ scale by σ , so d σ d t = σ . This im plies σ = e . I n T able 1 , the h uman brain, neurons in C. ele gan s, and English words all ha ve a path length close to e ≈ 2 . 7182 8, leading to a conjecture: The Natural Logarithm Theorem 1 The natural loga rithm is a co nsequenc e of uni- formly nested ener gy scaling. The number of generation s is pr oportio nal to time. T he ubiquitous role of the nat- ural log arithm in d ating pr ocesses evidences the uniform nested scaling of isotropic systems. On e conomics. An objection to applying statis tical mech anics to economics is ‘ind i- viduals are no t gas molecules’ (Sinha, 2011, p. 1 47). Th e mean path len gth is a bridge. Let ( ρ ( E c G r ) ) av be a country ’ s average rate of econom ic growth an d ( L P ) its lab or participation rate. Let the econ omic p rodu ctiv e cap acity of the average work ing in di- vidual equa l th eir econ omic problem solving cap acity ( ρ ( r Ec )) av ≈ 3.41 % p er decad e. T o mathematically mo del econ omic g rowth (a problem in econom ics ( Helpman) ) using The Network Rate Theor em , ( ρ ( E c G r )) av = ( η ( po p )) av × ( L P ) av × ( ρ ( r Ec )) av . (11) The a verage individual in a s oc iety has the same number of degrees of freedom in their rate o f economic problem solving ( ρ ( r Ec )) av as the entire society , which at a ny given Isotropy , entropy , and energy scaling 18 time is η ( po p ) × η ( K ), where K re presents society’ s store of kno wled ge. It follows that ( ρ ( E c G r )) av = (( η ( po p )) av ) 2 × ( η ( K )) av × ( L P ) av × ρ ( r ) , (12) where ρ ( r ) is the a verage innate problem solving rate. I f education increases ( ρ ( r Ec )) av in Equation (11), economic growth should increase. Data can test Equation (11). ( LP ) is about 66% fo r the U.S. ( Mosisa). Now estimate the U. S. economic growth rate from 1 880 to 198 0. In 1880 the U.S. had 50,155 ,783 people (18 80 census, T able Ia) and in 1980 , 226 ,545, 805 ( 1980 cen sus, T able 72) . η ( po p 1880 ) = 10 . 8186 and η ( po p 1980 ) = 11 . 738 6, so ( η ( po p )) av = 11 . 4 851. Then ρ ( EcGr ) = 11 . 4 851 × 0 . 66 × 3 . 4 1% per d ecade = 2 . 53% per year . (13) U.S. pro ductivity per h our from 1880 to 1980 increased abou t 2.3% p er year (Romer, 1990) close to the calculated rate in Equation (13). Morality and laws might arise as an emergent set of rules for protecting the factors on the r ight side of E quation (11). Utility theory app lied to econo mic maximization is challenged by p roblems like choo sing whether to hu rt a per son to save se veral people. The dichotom y of utility theory is built into Equation (11). On cosmology . Sup pose th e universe is 13.7 billion years o ld (ab out 4 . 32 × 10 17 sec- onds) with constant scaling factor s prop ortiona l to time. Suppose its entro py is 10 123 (Frampton , 2 008) . For The Netw ork Rate Hypothe sis , let ρ ( S ) be the ag e of the uni- verse. Then ρ ( S ) = 4 . 32 × 10 17 second s = η × s = 1 0 123 × s , (14) which implies that s has a ﬁnite quantum size p ropo rtional to about 10 − 105 of a seco nd, much smaller than one Planck time, about 10 − 43 seconds. Per haps intriguin g. On possible connec tions to q uantum mechanics. η r epetitions of σ is wav e- like. Dis- crete clusters are p article-like. Nestedness of clu ster gener ations resembles superpo si- tion in quantum mechanics. Clusters are co untable like qu anta. In quantum mec hanics, E = h ν , where E is energy , h is Planck’ s constant, and ν (sm all Greek nu) is frequency . Is this an an alog of The Network R ate Theorem ( h being th e ana log of th e scaling fac- tor)? Hugh Everett ( 1957) discussed a ‘many world s’ inter pretation of q uantum mechan- ics. Nested d egrees of freedom can replace ‘many world s’. Robots and algorithms. Nested scaling or degrees of freedom of robots and algorithms should increase the e ﬃ ciency of s uc h systems. On e pidemiology . T ransm ission o f d isease is an alogou s to t ra nsmission of in formation . If ρ ( r ) can be dete rmined, then it may be possible to calculate ρ ( R ) for a population. NRT testing. Scaling occur s in allometry . Does Th e Network R ate Theo r em app ly there? 3 The 4 3 Degr ees of Freedom Theor em Allometry is the study of scaling relationships in organisms. Isotropy , entropy , and energy scaling 19 In the allometry o f metabo lism Y = a M b ; Y is the o rganism’ s metabo lism, a is a constant, and M is the organism’ s b ody mass. In The Network Rate Theorem , the exponent of the scaling factor varies with size, for metabolism the expon ent b is ﬁxed. The Network Rate Theor em must be adapted to apply it to allometry . First, some backg round . In 187 9, Karl Meeh suppo sed that b = 2 3 : an organism’ s surface area dispersing bo dy heat grows by a power of 2 while its m ass supp lying heat grows by a po wer of 3 (Whitﬁeld ). Bu t 2 3 is wron g. Not all en ergy g oes to heat; energy is also used for movement, problem solving , growth an d repro duction. K leiber’ s data ( 1932 ) suppo rts b = 3 4 . W est, B rown an d Enquist (WBE 1 997) c ompare d scaling factors, an idea adapted in the d eriv ation below . T rea ting the circulatory s ystem as a transport system for materials, they foun d b = 3 4 . K ozlowski & K onar zewski ( 2004 and 2005) identiﬁed errors in WBE’ s mathematics; b = 3 4 has not been mathematically proven. The erroneous b ut usefully s imp ler 2 3 scaling hypothesis reveals a way to adapt The Network Rate Theorem to metabolic scaling. Let ρ ( s ) be the average heat supply rate of an organism cell, and ρ ( r ) be the average h eat d ispersing r ate o f a unit area on its surface. Assume heat gener ated is pr oportio nal to a small organism’ s v olu me V 1 (a Supply S ) and all gener ated h eat is uniformly disper sed throug h its sur face ar ea θ 1 (a Receipt R ). Let V 1 ∝ ( ℓ 1 ) 3 , where ℓ 1 is a length. Scale S ’ s length b y s . For a larger organism V 2 ∝ ( ℓ 2 ) 3 = ( s ℓ 1 ) 3 = s 3 ℓ 1 . Surface area θ 2 ∝ ( ℓ 2 ) 2 = σ 2 ℓ 1 where ℓ 1 scales by σ . In general, for S V k + 1 ∝ s 3 k ℓ 1 ρ ( s ) (15) and for R , θ k + 1 ∝ σ 2 k ℓ 1 ρ ( r ) . (16) By The Network Rate Theor em , the ratio of the capacities of Supp ly S to Receipt R is log s ( s 3 k ) log σ ( σ 2 k ) = 3 2 (17) . and of R to S is 2 3 , th e ra tio o f 2 d imensions to 3. I n 3 4 metabolic scalin g, Su pply S is the cir culatory system, Receipt R is th e o rganism, and the ratio of th eir d imensions is 1. Ho w is a 4 3 ratio possible? A co nnection to 4 3 occurs in an interme diate step in the pro of of Stefan’ s Law (Allen & Maxwell, p. 742–74 3; Lon gair, 200 3, p. 256–2 58) concerning isotropic energy radiation. Boltzmann deriv ed Stefan’ s empirically determined law (Boltzmann, 1884). Planck has the intermediate step as (1914, Ch. I I, p. 62) ∂ S ∂ V ! T = 4 u 3 T , (18) where S is entropy , T abso lute temperature, V volume, and u = U V is energy density . U is the total energy of the system. Isotropy , entropy , and energy scaling 20 The left side of Equation (18) is the chan ge of en tropy per ch ange in volume at absolute tem perature T . Since an ideal g as volume chan ges in propor tion to T , the left side measures how entropy changes relativ e to a s caling factor, ∂ V , propo rtional to T . u T on th e rig ht side measures th e nu mber of scalings in u ( energy density r eceived) based on T . Hence , implicitly Equ ation ( 18) says that for uniformly r adiating energy the number of scalings on the left s ide is 4 3 those on the right side, a ratio that connects to metabolic scaling. If 4 3 scaling applies to radiating energy then 4 3 scaling should apply at all scales. Isotropic radiation explains the sphericity of the space ﬁllers used in WBE 1997: Deg ( s phere ) = 3. I sotropy also is a large scale f eature of the universe (Fixsen, 1 996). Connect isotropy at all scales to energy d istribution in or g anisms. Treat S as uniformly scaled and ne sted. For the circulatory system , the ao rta is the ﬁrst generation and the capillaries are the η th . Identify the av erag e radius for a part of a c one of radiation with the radius of a tube to obtain: The 4 3 Degrees of Freedom Theorem : I n R ’ s reference fr ame, Deg ( S ) = 4 3 Deg ( R ), where S is an isotro pic supply of energy and R r eceiv es S ’ s energy . Proof: The unif ormly n ested scaled mod el m ust be extended to acco unt fo r an initial energy source. An initial en ergy source is externa l to a system . A system’ s degrees of freedom are within it. Hence, a 0 th generation en ergy supp ly is r equired. Designate a point source 0 as the 0 th generation of an energy supply S . Uniformly scaled nesting o r d egrees of freed om cor respond s to S isotrop ically ra- diating energy at a ll scales at ρ ( s ) = s energy units p er time unit into a Receip t R . Consider 0 and a ll points in a cone o f radiated energy originating fr om it as comprising a Supply . Let ℓ represent a radial distance traveled b y radiation at the rate s ene rgy units per time unit or scaling. L et V k , ∀ k ≥ 1, be th e portio n o f the con e ( V k contains sub-Supp lies) with average radial leng th ℓ k . ( The radial length is averaged since the ends of V k are curved surfaces.) Let s scale V k , su ch that V k + 1 = sV k . Since energy den sity D k + 1 = 1 s D k , ρ ( E k + 1 ) = ρ ( V k + 1 ) ρ ( D k + 1 ) = s ρ ( V k ) 1 s ρ ( D k ) = ρ ( V k ) ρ ( D k ) = ρ ( E k ); the rate of radiation is constant. T o be ab le to comp are the number of degrees of freedom in S relati ve to R , let γ ‘scale’ the average radial length ℓ k of V k : γ ≡ ℓ k + 1 /ℓ k = 1 . From 0 to the far end of V k + 1 , the ra diation front has η ( ℓ k + 1 ) = log γ ( γ k ) = k scalin gs; the radiation front is ( k + 1) × ℓ from 0. In S , De g s ( s k V 1 ) = Deg γ ( γ k ℓ 1 ) = Deg s ( s k ℓ 1 ) and if Deg ( V ) = 1, then Deg ( ℓ ) = 1 . I n S , γ = s . Let r k be the av era ge radius of V k = π ( r k ) 2 ℓ k . Cones have a unifo rm slope. Let β ≡ r k + 1 / r k represent the scaling of the average radii for V k . Scaling factors s , β and γ are in strumental variables fo r comparin g the degrees of freedom in Supply S relative to the degrees of freedom in its Receipt R . In S , sinc e ℓ i = ℓ, ∀ i > 0, V k + 2 V k + 1 = s = s k + 1 V 1 s k V 1 = π ( β k + 1 r 1 ) 2 ( ℓ k + 2 ) π ( β k r 1 ) 2 ( ℓ k + 1 ) = β 2 , (19) so β = s 1 2 . If Deg s ( V k ) = 1, then Deg β ( r k ) = 1 2 . Isotropy , entropy , and energy scaling 21 Since radiation is isotropic, for e very V k transmitting energy at the rate ρ ( E k ) let θ k be a corresp onding spherical Receip t re ceiving energy at the same r ate a nd scaling by σ with radius ξ k = 1 2 ℓ k scaling by α ≡ ξ k + 1 /ξ k . σ an d α ar e in strumental variables for determinin g, in R , the d egrees of freedo m of th e sphe re θ k in R relativ e to θ k ’ s radiu s ξ k . If Deg σ ( θ k ) = 3, then Deg α ( ξ k ) = 1 and θ k + 2 θ k + 1 = σ = σ k + 1 θ 1 σ k θ 1 = 4 3 π ( α k + 1 ξ 1 ) 3 4 3 π ( α k ξ 1 ) 3 = α 3 , (20) so α = σ 1 3 . If Deg σ ( θ k ) = 1, th en for ξ k = 1 2 ℓ k in R Deg α ( ξ k ) = Deg γ ( ℓ k ) = 1 3 . But in S Deg γ ( ℓ k ) would be 1. Compare the relativ e num ber of de g rees of freedom of V k and θ k . Use the relation- ship between th e radius ξ k of θ k and the average radial le ngth ℓ k of V k : ξ k = 1 2 ℓ k . In S , Deg s ( ℓ ) = 1. Since ξ k in R has α = σ 1 3 , in R Deg s ( ℓ ) = 1 3 . In th e third line of Equation (21), γ canno t have both 1 and 1 3 degrees of freedom in ter ms o f s . Calculating in th e third line the relativ e number of degrees of freedom of V k scaling by s 1 in S co mpared to θ k scaling by σ 1 in R requ ires specify ing in which reference fr ame, S or R , the calculation is taking place. In S ’ s reference frame in the ﬁrst two lin es, and in R ’ s reference frame with γ = s 1 3 in the third and fourth lines: V k + 1 = s k V 1 ( in S ) = π ( β ) 2 k ( r 1 ) 2 ( γ k ) ℓ 1 ( in S ) = π ( s 1 2 ) 2 k ( s 1 3 ) k ( r 1 ) 2 ℓ 1 ( in R ) = π s 4 3 k ( r 1 ) 2 ℓ 1 . ( in R ) (21) When s scales V k in S , σ 4 3 scales θ in R , so in R Deg s ( S ) = 4 3 Deg σ ( R ). The extra 1 3 degree o f freedo m o f S in R ’ s r eference frame is du e to the e ﬀ e ct o f radial mo tion in S . QED. 3.0.2 The 4 3 DFT : Observations, implications, speculations Comments below about energy , quantum mechanics and gravity are speculations. On metabolic sca ling. Assume that for o rganisms k an d k + 1, their masses M k < M k + 1 , and that e very organism R isotropically recei ves ene rgy from a circulatory system with energy supply capacity S . Assume that R ’ s circulatory s ystem volume V is prop ortional to R ’ s v olume θ . By The 4 3 DFT , in R , S k + 1 ∝ V k + 1 = s 4 3 V k ∝ s 4 3 θ k , since V ∝ θ . Assume that R ’ s average number of cells N ∝ M , its mass, an d that M ∝ θ ∝ ρ ( θ ) = Y , its metabolism. Then, for a Receipt θ k + 1 = s θ k ∝ s M k . The expon ent of the s factor of the Supply s 4 3 V k must be 1 to ma tch the degrees of Isotropy , entropy , and energy scaling 22 freedom of M k . A 3 4 power of the Supp ly’ s volume V k + 1 , ( V k + 1 ) 3 4 = ( s 4 3 V k ) 3 4 = s ( V k ) 3 4 ∝ s ( θ k ) 3 4 ∝ s ( M k ) 3 4 = ( s 1 3 ( sM k )) 3 4 = ( s 1 3 ) 3 4 ( M k + 1 ) 3 4 (22) supplies energy at the (Receipt) rate Y k + 1 to the Receipt M k + 1 . Hen ce, when V ∝ θ , the energy supplied by V is proportional to M 3 4 so Y ∝ M 3 4 . Di ﬀ erent mathem atical reaso ning giv es the same r esult and implies that 3 4 metabolic scaling occu rs a t the cellular le vel. Le t ρ ( r ) b e an organism’ s average cellular metabolic rate. Modify The N RT by add ing factors x , y and Deg m ( ρ ( r )), m a scaling factor: ρ ( R ) = x Deg σ ( R ) × y Deg m ( ρ ( r )) × ρ ( r ) . (23) The 4 3 DFT im plies x = 4 3 in Equ ation (23). Assume th at N d oes no t vary (which can be shown to imply V ∝ M ). That is the id ealized case for a mature organ ism. Then in Equatio n ( 23) R ∝ θ , ρ ( R ), an d Deg σ ( R ) ar e constan ts; ρ ( r ) as an average is constant. Since ρ ( R ) = Deg σ ( R ) × ρ ( r ) by The NRT , y in Equation (23) must be 3 4 . Scaling up of an organism’ s energy supply cap acity is o ﬀ set by scaling down of its av erag e cellular rate of en ergy use. Thu s the metabo lic capac ity , which is th e prod uct 4 3 Deg R ( R ) × 3 4 Deg ρ ( r ) ( ρ ( r )), is inv ariant: 4 3 × 3 4 = 1. This observation uses the fact that for ρ ( r ) = k σ , log σ ( σ η ) = log k σ (( k σ ) η ). The metab olism of an organism’ s N cells is Y = ρ ( r ) × N , N tim es average cellular metabolism. Apply Equation (21) and the value of y in Equation (23). Then ρ ( R k + 1 ) = Y k + 1 = σ 4 3 N k × m 3 4 ρ ( r k ) . (24) If in Equatio n (23) y instead equals 1, then θ k + 1 = σ 4 3 θ k . Space e xp ands. 4 3 DFT a nd eco nomics. For th e same rea son as in metab olic scaling , in eco nomic enterprise there are economies of scale. On the other hand, increasing the e ﬃ ciency of individuals frees up energy t h at can increase η , boosting econom ic growth. T urnstile Analogy . A stadiu m has fo ur seating levels, each with r ows of n seats. It has three exit le vels each with n turnstiles. Each s tad ium level empties one row per tim e unit. Each turnstile level o nly allows a maxim um o f one ro w to e xit p er time un it. If the stadiu m is full, th e e mptying rate of th e stadiu m levels is 4 3 the exiting c apacity of the exit turn stiles. T wo possible rem edies are: (1) incre ase the nu mber of tur nstiles (the Receip t) by 1 3 ; (2) decrease the rate of exiting per sons by 3 4 . Th e seco nd solutio n applies to m etabolic scaling. The ﬁr st solution is co nsistent with the expan sion of the universe. Tha t is, if ρ ( r ) in Equation (23) does not scale down, then θ must scale up. A th eory of emer gence. T ogether , The 4 3 De grees of F r eedom Theor em a nd The N etwork Rate Theo r em explain emergence. In R ’ s refere nce frame, S with 1 3 more d egrees of fre edom than R initiates Deg ( R ) and causes Deg ( R ) to in crease. T hat in creases the multip licativ e e ﬀ ect o f η in The Network Rate Th eor em . Structur es (stars, and organisms) and processes ( ecosystems, langu ages, markets, and mathema tics) emerge at all scales. Michael Stum pf a nd Mason Porter recently (2012) sugge sted that allometric scaling for metabolism ha s, of all putative scale-free po wer laws, the most e videntiar y supp ort. Isotropy , entropy , and energy scaling 23 (The ratio, π , o f the circ umferen ce to the diam eter of a circ le is an example of a scale- free relationship in Euclidean geo metry .) T he metab olism power law is a manifestation of The 4 3 De grees of F reedom Theor em , which may be the uni verse’ s mo st fundamen tal scale-free power la w . If the universe is ﬁnite, then there are smallest and largest scales for The 4 3 De grees of F reedom Theor em . S scaling creates sp ace in R . R having been created , S increases its degrees o f freedom to ﬁll R . Perhap s a push-pull mechanism makes time one directional. The fractal dimension o f iso tropic energy distribution (th e Sup ply) is ( 4 3 ) rd that of the Receipt. Fra ctality of a Supply S ind uces fractality in its Receipt R at all scales. Another natural lo g arithm theor em. Isotropy suggests the following. The Natural Logarithm Theorem 2 F or a ﬁnite isotr opic network R , the base of the logarithmic fu nction describing R ’s intrinsic de grees of fr eedom is the natural loga- rithm. Proof: The c ontribution o f networking to th e multiplicatio n o f ca pacity per transmit- ting node of R ’ s n = σ η nodes as a pr opo rtion of η is d η d n = d  log σ σ η  d ( σ η ) = 1 ln ( σ ) σ η . (25) The per node r eception of the incr ease in capacity η due to networking as a pr oportion of η is 1 n = 1 σ η . F or an iso tr opic n etwork, th e co ntribution to the in cr ease in capacity per transmitting no de, as d escribed in Equation ( 25), e quals the incr ease in capacity per r eceiving nod e, so 1 σ η = 1 ln ( σ ) σ η ⇒ ln( σ ) = 1 ⇒ σ = e . (26) An informa tion network wh ere e very node has an equ al ca pacity to transmit an d re- ceiv e is i sotr opic. Dunb ar’ s optim al au dience o f three is slightly more than e ≈ 2 . 7 1828 hearers. On Clausius’s 3 4 Mean P ath Length Theo r em. Clausius in his paper introdu cing the concept of mean path length (Clausius, 1858, p. 1 40 of translation in Brush) notes: The mean lengths of path for the tw o cases (1) wher e the remaining molecules move with the same velocity as the one watched, a nd (2) where they ar e at rest, bear th e propo rtion of 3 4 to 1. It would not be di ﬃ cult to prove the correctne ss of this relation; it is, howev er, unnecessary for us to dev ote our time to it. One can prove Clausius’ s theorem using The NRT an d The 4 3 DFT . Theorem : The mean path length of an isotropic Supply S is 3 4 of the Receipt’ s ( R ’ s). Proof: Per time unit, every k th generation Su pply S k isotropically supp lies E k energy to a corr espondin g Receipt R k . If no energy is lost in tr ansmission, then ρ ( E ) = ρ ( S ) = Isotropy , entropy , and energy scaling 24 ρ ( R ). Let S ’ s mean path length s scale S and R ’ s mea n path length σ scale R . Th en, ρ ( E ) = ρ ( S ) = Deg s ( S ) × s ( N etwork Rat e T heor em ) = 4 3 ! Deg σ ( R ) × s 4 3 Degr ee s o f F reed o m T heo rem ! = Deg σ ( R ) × 4 3 ! s = Deg σ ( R ) × σ ( ρ ( S ) = ρ ( R )) = ρ ( R ) ( N etwork Rat e T heo r em ) . (27) It follows that 4 3 s = σ and so s = 3 4 σ . QED. The mean path length of a social network—a Receipt—r eceiving isotropically trans- mitted information should be 4 3 of e ( = 2 . 71828 ), about 3 . 624. This is close to the 3.65 found by W a tts and Str ogatz fo r acto rs (T able (1)). Using cur rent values fo r th e mean path length fo r th e 1 657 p opulatio n and lexicon in T able (1) is justiﬁed since tho se values are close to what theory suggests they should be. A speculatio n abou t ener gy . Use dime nsional analysis. A cluster in a generatio n is analogo us to a m ass (fr om a particle poin t of view). Let one clu ster width also be one unit of distance (fro m a wave point of vie w). The number of clusters in a generation is a distance per unit of η . Since η is pro portion al to time, the mass of clusters in a generation per time is ( ma s s )( length ) / ( t ime ) . S radiates on e generation p er unit of η or time, ( leng th ) / ( time ). Th e dim ension o f a radiating m ass o f c lusters p er g eneration is ( ma s s )( lengt h ) / ( time ) × ( lengt h ) / ( t ime ) = ( ma s s )( len gth ) 2 / ( time ) 2 . (28) Energy ha s the same dimensions as the right side of Equation ( 28). That imp lies th at energy is due to the excess 1 3 Deg s ( S ) in R ’ s reference frame. A 0 th generation (the sun) supplies energy to planets circling it, a black hole supplies energy to stars circling it, and a singularity supplies energy for the universe that emerges from it. On Hu ygens principle. T he physicist Christiaan Huygens in 169 0 had the id ea th at “ev ery point on a propag ating wa vefro nt serves as the source of spherical secondary wa vefron ts” and “the secondary wa velets hav e the same fre quency and speed” (Hecht, 2002, p. 1 04), consistent with uniform ly scaled nested degrees of freedom. On dimensions. In S , Deg ( S ) = 3 comp ared to a radial length’ s Deg = 1. In R , Deg ( R ) = 3 compa red to a r adial le ngth’ s Deg = 1. In R ’ s re ference fram e, Deg ( S ) = 4 3 Deg ( R ). In R ’ s re ference f rame, assign S a 4 th dimension. Time as a 4 th dimension appears in the special theory of relati vity . Time in R may be due to radiative motion in S . On the er godic hypo thesis. T o deri ve the H Theorem , Boltzmann assumed th at every combinato rial state (ph ase) was equally likely and would o ccur ev entu ally—the ergodic hypoth esis. Th e physicist Shang-Keng Ma comments (p. 44 2): Isotropy , entropy , and energy scaling 25 This argumen t is wr ong, because this in ﬁnite lo ng tim e must be much longer than O ( e N ), while the usual observation time is O (1). ( O (1) means the or der of magn itude of a counting numbe r . In princip le there is not enoug h time for the ergodic hypothesis to be true.) T heory now postulates equal prob- ability for all points in phase space. Neither the h ypothesis nor a postulate is necessary . All th at is necessary is the capacity to bin odally conn ect or collide. Then the n etwork is scaled b y its mean path leng th σ in tim e proportion al to log e ( O ( e N )) = O ( 1). The network’ s de gr ees of freedom are equally a vailable in an isotropic system. The equip artition th eor em. In Statistical Me chanics the eq uipartition theorem (which also relies on the equal probability of all points in phase space) provides: In the me an each degree of freedo m of the system at a tem peratur e T h as the therm al energy 1 2 kT (Gr einer, 199 5, p. 19 8), where k is Boltzmann’ s constant and T is absolute temp erature. The ratio of 3 degrees of freedo m in S to 2 d egrees o f freedom in motion throu gh a plane in R is 3 2 , o r 1 2 kT per S ’ s degrees of freedom , a conjectur al explanation. On a similarity to Schr ¨ odinger’s Equatio n . A succeeding gener ation in a u niform ly scaled S has moved one s -scaling that is orthogonal to the p receding g eneration . Let ψ represent a function that counts intrinsic degrees of freedo m. Bearing in min d that η appears to be propo rtional to time t (in R ’ s reference frame), ψ ( k + 1 ) = i × s × d ψ ( k ) d η . (29) Apply Equation (29) to k scalings of s along all radii fr om a 0. The n the k + 1 st genera- tion makes a rig ht angle to th e k th generation , because of the factor i in Equ ation (29), and form s a ring k + 1 radial scalin gs from 0 . Eq uation (29) r esembles Schr ¨ odinger’ s wa ve equation. On special r elativity and qu antum scaling. In Hermann B o ndi’ s ‘ k calculus’ (1962), time for one of two inertial tra velers is scaled by k re lativ e to the other . I n Bondi’ s one dimensiona l par adigm (Bondi, p. 102) , Brian and Alfred are on u naccelerated paths in space and p ass by ea ch other at point 0. Alfred send s a rad ar p ulse at time t which reaches Brian at tim e kt . Brian’ s response to Alfred ’ s signal reaches Alf red at time k ( k t ) = k 2 t . Bondi ﬁn ds a fo rmula for k (p. 1 03) and derives the L orentz transfo rmation s that characterize the principle of relativity for inertial referen ce frames. Generalize for a ﬁn ite universe: for all pair s of inertial path s ﬁnd the 0 co mmon to all. Eac h pair of paths has a k factor . The smallest is a quantum scaling factor . Instead o f co nnecting poin ts on a pair of in tersecting inertial paths with straight lines as Bondi does, connect them with part of the arc of a circle. Conside r a set of radial lines intersecting at 0 with conc entric circles superimposed . t i + 1 = k t i = k i t 1 . This resemble s the model u sed in The 4 3 De grees of F reedom Theorem . Suppose that ev ery local 0 acts as a gravity po int relative to the set of inertial li n es eman ating f rom it. A quantum scaling factor helps d escribe the geometry of S , and is connecte d to gra vity . An ev ent perceived by two inertial observers with a common 0 as di ﬀ erent in R occurs in the same gener ation in S . The per ceiv ed relativity of time and spac e may Isotropy , entropy , and energy scaling 26 occur due to the refer ence fr ame adopted . W e perceive space as a single refer ence frame. The 4 3 De grees of F r eedom The or em su ggests space has two. They are d i ﬃ cult to distinguish because they each ha ve 3 dimensions within their own domain. On the Arr ow . Aristotle describes Z eno’ s Arr ow p aradox in his Physics : “ . . . if every- thing when it occup ies an equal space is at rest, and if that which is in locomotion is always occu pying such a space at any m oment, the ﬂying arrow is therefor e motion - less”. An inertial o bject is n ot m oving relative to its cluster in S . I nertial mo tion m ay be the perception in R of S ’ s backg roun d moving relative to the inertial object 3 . The two slit e xp eriment. Light w aves tra veling throug h two parallel slits in a thin plate appear to in terfere; if light consisted of particles, the impression left b y the p articles should add. Compare this to stereoscopic vision. T wo eyes provide stereoscopic depth perception . Similarly , two slits m ay enable stereosco pic depth perceptio n in R of S ’ s unifor m nested s caling ; the observer sees the moving backgrou nd as wa ves with ampli- tudes of nested heights. Per haps the two slit e xp eriment is a reference frame problem. On Entangleme nt. Consider a scaled generation of clusters: 0 s z } | { − → s 1 − → s 2 − → s 3 . . . − → s η . (30) In (30) a node in s 1 connects to a node in s η in s steps and in η × s g eneration s or steps, counting the ﬁrst g eneration as a step. In (30) the number of scalings out of η scalings is like a distance and is also a proportion of s . In S ’ s reference frame, from the perspective of the average scaling f acto r s : s ⊃ k a 1 , b 1 k 1 ⊃ k a 2 , b 2 k 2 ⊃ . . . ⊃    a η , b η    η , (31) where A ⊃ B mean s A contains B , and k a i , b i k i = k a i , . . . , b i k i is a rep resentative cluster in the i th generation . In R ’ s r eference frame, distance is propo rtional to the number of scalings by s : k a 1 , b 1 k 1 + k a 2 , b 2 k 2 + . . . +    a η , b η    η = η × s . (32) In R ’ s referen ce frame, th e distance η × s spans the system, but in S ’ s ref erence frame, s as the a verage least distance between pairs of nodes s p ans the system. Is s > 0 a distance or a scaling factor? Supp ose the answer is, both. The Natural Logarithm Theorem 3 If s spanning the network in one generation (d η ) in S is eq uivalent to η se gments ea ch of s un its spanning the n etwork in R , th en s = e. Proof: The equivalen ce r educes to a di ﬀ erential equation per unit of η : d s d η = η × s . (33) Consider s in Equa tion (33) as if it is a function. The solution for Equa tion (33) is s = e η . (34) If η = 1 , then s = e. Equivalence of (31) and (32) leads to the natural logarithm. 3 Scotty in the 2009 Star Trek movie: Imagine that! It neve r occurred to me to think of SP A CE as the thing that was mo ving! Isotropy , entropy , and energy scaling 27 In o ther words, th e natu ral logarithm is evidence of space’ s du al referen ce frames. W av e p article duality may also be. Aspect, Dalibar d an d Ro ger ( 1982) perform ed an experiment th at p recluded co m- munication b etween sepa rated p articles, a nd f ound en tanglemen t: the r esult is co nsis- tent both with quan tum mechanics and n on-lo cality (Einstein, 1935 ; Bell, 19 64). Th is may result from the equiv alence of (31) and (32). 4 Discussion The section on The Network Rate Theory found an inn ate ra te of le xic al change that enables dating the beginning of languag e. I f the same phy sical principles apply to div er se phenomen a, then generalizing the same method should lead to a general theory of emergence. As conjectur ed abov e. The univ er se consists o f m any kind s of structures and p rocesses, com plex at all scales. One mechan ism fo r cr eating complexity is many r ules a pplied to simple co m- ponen ts. An other mech anism is to have a simple initiatin g p rocess, such as is otr opic radiation or scaling, with an eno rmous number of degrees of freedom. If the two theo- rems describ ed above a re valid, they are consistent with the seco ndly describ ed mech- anism. That st atistical mechanics ﬁrst d ealt with gas m olecules is perhaps an ac cident o f history . Statistical mechan ics m ight also ha ve developed by asking how mu ch intelli- gence a collective in telligence contributes to its compo nent intelligences. Refer ences [1] Achar d, S., Salvador , R., Whitcher, B., Suck ling, J. & Bullmor e, E . (200 6). 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