Path-analysis of metric-first & entropy-first approaches P. Fraundorf Physics & Astronomy/Center for NanoScience, U. Missouri-StL (63121) and ∗ Physics, Washington University (63110), St. Louis, MO, USA (Dated: February 23, 2012) Established idea-sets may not update seamlessly. The tension between new and old views of na- tureis documented in Galileo’s dialogs and now present in many fields. Oneevolutionary response may be to consider the simplicity of paths from various starting points to one goal. We illustrate with a look at two simplifications: The move from Lorentz-transform to metric-equation de- 2 scriptions of space-time, and themovefrom classical to statistical thermodynamics with help 1 from Boltzmann’s choice-multiplicity &Shannon’suncertainty. Connectionsofthelattertocorre- 0 lationmeasuresbehindavailablework,modelselection,andlayeredcomplexityarealsoexplored. 2 New strategies are exemplified with Appendices on: anyspeed vector-velocity addition, the energy- b momentumhalf-planelosttofinitelightspeed,themoderndistinctionbetweenproperandgeometric e accelerations, single map-frame views of anyspeed acceleration, quantifying risk with a handful of F coins, available work in bits,quantitativemodel-selection, and theevolution ofanalog/digital com- plexity. 2 2 PACSnumbers: 05.70.Ce,02.50.Wp,75.10.Hk,01.55.+b ] h p Contents I. INTRODUCTION - n e I. Introduction 1 Evolution of well-worn approaches naturally encoun- g ters resistance from experts in the old1. For instance s. II. Principles 1 Martin Gardner in his book on parity inversion2 cites c Hermann Kolbe’s negative reaction to the prediction si III. Metric-based motion 2 of carbon’s tetrahedral nature by Jacobus van’t Hoff y (Chemistry’s first Nobel Laureate). The hullabaloo3 h IV. Multiplicity-based thermodynamics 3 about the Nowak et al. paper4 on models for evolving p insect social behavioris a more recentresearchexample, [ V. Collateral connections 3 while participants in the content-modernization branch 3 A. surprisals 3 of physics education research(PER) have engaging tales v B. average surprisals 4 on the education side5. For text publishers, however, 8 C. net surprisals 5 even funerals may not mark progress since choosers of 9 a course text might understandably like to teach that 6 4 VI. Discussion 6 course the way they learned it, whether they own the . strategy or not. 6 VII. Conclusions 6 One way to objectively assess new approaches is 0 1 perhaps to examine the algorithmically-shortest path Acknowledgments 6 1 to quantitative insight from each given starting point. : For experts in the old, traditional approaches may be v A. Proper-velocity 6 i algorithmically-shortesteven if they are not shortest for X 1. vector addition 6 newcomers to a given subject. Differing perceptions, in 2. energy vs. momentum 7 r this context, might thus be put onto a rational footing. a In this context here, we examine textbook trends to- B. Proper-acceleration 7 ward“metric-first”approachestorelativisticmotion,and 1. proper or geometric? 7 “entropy-first” approaches to statistical inference about 2. accelerated roundtrips 8 physical systems, in hopes of helping individual teachers chart their own path through the evolving terrain. C. Choice multiplicities 8 1. quantifying risk 9 D. Matchup multiplicities 9 II. PRINCIPLES 1. work’s availability 9 2. model selection math 10 From a given starting point, the strategy for putting 3. layered correlations 11 together a concept map may be to minimize the number of: (i)assumptionsand(ii)newconceptsneededtomake References 12 agivensetofquantitativepredictionspossible. Drawings 2 nized clocks, as simply an equation for time-elapsed on the clocks of a traveler. The shift may seem uninteresting if one has mastered space-time throughLorentz transforms,andsees relativ- ityasanextensionofNewtonianphysicstoextremesitu- ations. Ontheotherhandifonehasbeenexposedmainly tosinglemap-framecalculationsand/orseesrelativityas aneverydaycauseofmagnetism&gravity,thenthetwo- frame approach serve up un-needed complication. The traditional approach for instance: (i) emphasizes symmetrybetweenframesevenwhenthehome-framee.g. ofatravelingclockoryardstickisquitespecial,(ii)raises thedissonantspectre10–13 ofrelativisticmass,(iii)avoids use of proper-acceleration14,15 as an integrative com- plement to geometric-accelerations(affine-connection ef- fects) at low and high speeds, and (iv) misses out on in- sightsthatproper-velocity16,17 offerse.g. intorelativistic velocity-addition(AppendixA1)andthelightspeedlimit (Appendix A2). The single map-frame approach avoids these problems, with the result that elements of it are finding their way into texts on all levels. In context of this evolving strategy we recommend drawing concept-interconnection maps (like those in the sections below) showing the steps from what you, and whatseparatelyyourstudents,understandtowhatyou’d FIG. 1: Proper-kinematics from themetric-equation first. like them to master in a given course. Intro-physics and relativity texts over the past couple of decades will give you an idea about how the Lorentz-transform ap- ofsuchpathsfromvariousstartingpoints,inthiscontext, proach is already evolving to consider privileged frames might inspire one to evolve one’s own starting point in e.g. likethatofthetravelingclockintime-dilationanaly- teaching a given class over time. sisandthetravelingyardstickinlength-contractionanal- Note that we are weighing self-consistent approaches ysis. Becauseitdoesn’tyetappearintexts,however,let’s for their compactness, portability, and appropriateness showbrieflyhowametric-firstapproachcanbuildsignif- in much the same way that different variable-changes in icant insight from what intro-physics students already calculus, and coordinate-system choices in analytic ge- know. ometry, buy more advantage for some tasks and less ad- BeginwithMinkowski’sversionofPythagorastheorem vantage for others. In that sense, we seek to apply the i.e. the flat-space metric equation shown in the top of science of Bayesian model-selection to the evolution of Figure 1. This introduces time-elapsed on the traveler’s what we teach. clock i.e. proper-time τ, and hence two new rate-of- Belowweillustratethis withafew examples,basedon travel parameters: namely the 3-vector proper-velocity contentchangesalreadyunderwayintheevolvingphysics w~ ≡ d~x/dt and Lorentz-factor γ ≡ dt/dτ in terms of curriculum. Similar charts for your own approach to a already-familiar coordinate-velocity~v ≡d~x/dt. given class, as it relates to the textbook in hand as well For anyspeed work, proper-velocity instead of as the larger picture, may be worth putting together for coordinate-velocity: (i)equalsmomentumperunitmass, sharingwithyourstudentsandperhapsinelectroniccol- (ii) adds vectorially (with an out-of-frame rescale) as laboration spaces with the larger physics teaching com- shown in Appendix A1, and (iii) with no upper-limit munity as well. most elegantly (at 1 [ly/ty]) parameterizes the transi- tionfromsubtohyperrelativistic. Moreoverthismetric- first analysis can be extended to treat constant proper- III. METRIC-BASED MOTION acceleration with equations that in the unidirectional case are particularly simple, and connected to the low- The traditional path via Lorentz transforms, by using speed equations for constant coordinate-acceleration. twoseparatecoordinate-frameswiththeirownyardsticks The big caveat here is that simultaneity is defined en- and synchronized clocks, likely provides the most direct tirely by the single reference map-frame. Robust quan- routetolengthcontraction. Ontheotherhandshorter titative insight into the meaning of that, as well as paths to time dilation6, accelerated motion7, and of length-contraction, likely will have to await Lorentz gravitation8,9 follow by invoking the flat-space metric transforms. equation to a single map-frame of yardsticks & synchro- Asteachersweshouldprobablychooseapaththrough 3 space-time for students which draws strength from our second law, whose physics actually comes not from sta- prior training and acquired insight into both paths, as tistical inference but from the assumption that mutual well as the path’s connection to the past and future of information available on the state of an isolated system students in each given course. For instance, with intro- will not increase over time. ductory physics students it’s quite easy to tell students Finallytheverydefinitionofreciprocaltemperatureas (even if the book doesn’t) that time passes differently anuncertainty slope will convince many that the change on different clocks so that, unless otherwise stated, time in state-uncertainty about any finite system, per unit will be measured on a set of synchronized “map-clocks” change in energy, is likely to be finite. Hence reciprocal- affixed to the yardsticks used to measure position. Even temperature’sinfinity(theabsolute-zerooftemperature) better ifwecanalsogivethemanupdatedviewofforces is likely inaccessible. This natural definition of tempera- in non-rain frames (Appendix B1), and of constant ac- ture has the added advantage that it prohibits one from celeration at any speed (Appendix B2). approaching absolute-zero from negative or positive di- rections, and shows that the negative absolute-zero ap- proachablee.g. by spinsystems with a populationinver- IV. MULTIPLICITY-BASED sionisasfarawayfrompositiveabsolute-zeroasyoucan THERMODYNAMICS get. Examples of the power in this recasting of famil- Thequestionhereis: DoIstartbyintroducingtemper- iar rules include many senior undergraduate thermal atureinhistoricalunits andthezeroethlawwhilesaving physics texts, like those by Kittel & Kroemer18, Dan entropy to the end, or do I startwith choice-multiplicity Schroeder19, and Claude Garrod20 (who refers to recip- andentropysothattheassumptionsbehindtheidealgas rocal temperature21 as coldness), Tom Moore & Dan law,equipartition,andmassactionareexplicitfromday Schroeder’s AJP paper22, Tom Moore’s introductory one? Senior undergraduate texts almost all now do the physics Unit T6, etc. latter, while only a small number of introductory texts have made the switch so far. Thesimplestaxiomaticpathto: (a)theidealgasequa- V. COLLATERAL CONNECTIONS tion, (b) equipartition, and (c) the law of mass action is likely Boltzmann’s choice-multiplicity We’ve now covered two paradigm-shifts that have a well-defined place in the physics curriculum. The ap- N 1 pi proachtakenwithrespecttotheminagivenclassshould W = , (1) p inform itself to both teacher & student backgrounds, as iY=1(cid:18) i(cid:19) well as to course objectives. The second paradigm-shift makes contact with other developments of interest to from which entropy S =kln[W] and its derivatives may physics students as well. be defined. This choice-multiplicity,ofcourse,isjustthe dimensionless count which underlies the familiar use of To explore this we step back from uncertainties to both informationunits andJoules per degreeKelvin[12]. probability measures, and then forward from uncertain- Historical approaches introduce these consequences as ties to correlation measures, to show how the second empirical and/or informally-useful relationships, with- simplification also allowsphysics to make contactwith a out clear definition of their underlying mechanism numberofotherlivelydisciplines. Becauseofthephysics and assumptions and typically with discussions of en- in between, out-of-discipline students may never hear tropy/multiplicity (the horse) following these conse- abouttheseconnectionsiftheyaren’tatleastmentioned quences (the cart). Such approaches do not provide in- in one of their physics classes. sight into: (i) the quantitative limitations of these con- cepts, or (ii) strategies for moving beyond those limi- tations e.g to systems in which subsystem correlations A. surprisals cannot be ignored. Another reason to introduce multiplicity first is that Recallthatinformationunitscanbeintroducedbythe the laws of thermodynamics (short of two physical pos- statement that # choices equals 2#bits. Also very small tulates) follow therefrom as well. The zeroth law follows probabilities p can be put into everyday terms as the fromthefactthatthelargestnumberofstatesisavailable surprisal23 s = n bits of tossing n coins all heads up when the uncertainty slopes of two subsystems (recipro- since p = 1/2#bits, with the added advantage that sur- cal temperatures for the energy observable) equilibrate prisals add whenever their probabilities multiply (Ap- as a conserved quantity is shared. pendix C1). Evidence in bits24 for a true-false propo- The first law, oft described as a statement of energy sition can similarly be written as e[p] = s[1−p]−s[p], conservation, in fact arises from maximum entropy in- where surprisal is s[p]=ln [1/p]. 2 ference as a relation between ordered and disordered All of these applications rely on the fact that proba- changes in any observable, whether they are conserved bilities between 0 and 1 can be written as multiplicities on transfer between subsystems or not. Likewise for the w = 1/p between 1 and +∞ or as surprisals between 0 p 4 FIG. 2: Choice multiplicity ⇒ gas law, equipartition & mass action. and +∞using informationunits determined by the con- set p, can be written: stant k in the expression s = kln[1/p]. This surprisal p ⇔multiplicity⇔probabilityinter-conversionissumma- N 1 rized by: 0≤Sp/p ≤Sq/p ≡kln Wq/p ≡k piln q ≤∞. i=1 (cid:20) i(cid:21) (cid:2) (cid:3) X (3) 1 0≤sp ≡kln[wp]≡kln ≤∞ (2) ThusSq/p e.g. foranobservationis inbits the average- p surprisal if the expected model q differs from the (cid:20) (cid:21) operating-model p. where of course the units are bits if k=1/ln[2]. Althoughwrittenforadiscreteprobability-set,theex- pression is naturally adapted to continuous as well as quantum mechanical probability-sets25,26. In this con- text natural as distinct from historical units for temper- ature become energy per unit information, and for heat B. average surprisals capacity become bits27. Note that the upper limit on S is ln [N]. Also the p/p 2 The treatments of the ideal gas law, equipartition, fact that S ≤ S , i.e. that measurements us- q/p p/p massaction,andthe lawsofthermodynamicsinthe pre- ing the wrong model q are always likely to be vious sectionconnectto this traditionby defining uncer- more surprised by observational data than those taintyorentropySasanaverage surprisale.g. inJ/K using the operating-model p, underlies maximum- between0and+∞,Boltzmann’smultiplicityWbetween likelyhood curve-fitting and Bayesian model-selection as 1 and +∞ as eS/k where k is Boltzmann’s constant, and well as the positivity of the correlation and thermody- 1/W as a reciprocal-multiplicity between 0 and 1. Their namic availability measures discussed below. relevance to the thermal side of physics education has Thus in this two-distribution case, 1 ≤ W ≤ +∞ q/p been discussed above. is an effective choice-multiplicity for expected prob- Moregenerallytheinterconversionfortheaveragesur- abilitysetqinthe face ofoperating-probabilitysetp. In prisal, uncertainty, or entropy associated with predicted generalW ≤W . For the uniform N-probabilityset p/p q/p probability-set q, as measured by operating probability- u =1/N forirunning from1to N,wecanalsosaythat i 5 FIG. 3: Cross-disciplinary applications for log-probability measures in statistical inference. W ≤W =N =W ≤W . connecting together today, based on their relevance to p/p u/p u/u p/u cross-disciplinaryinterests of students in physics classes. The specific application areas are: (i) thermodynamic C. net surprisals availabilityas inAppendix D1, (ii) algorithmicmodel selectionasinAppendixD2,and(iii)theevolutionof complexityasinAppendixD3. Thesurprisal⇔multi- The tracking of subsystem correlations has taken a plicity⇔probabilityinterconversionforthesecorrelation back seat in traditional thermodynamic use of log- analyses may be written: probability measures. This is illustrated e.g. by the traditionaltreatmentofsubsystementropiesasadditive, in effect promising that correlations between e.g. be- N p i 0≤I ≡kln M ≡k p ln ≤∞ (4) tween gas atoms in two volumes separated by a barrier q/p q/p i q can be safely ignored. More generally, however, subsys- (cid:2) (cid:3) Xi=1 (cid:20) i(cid:21) tem correlations (e.g. between a sent and a received Log-probability measures are useful for tracking message, or between traits of a parent and of a child) subsystem-correlations in digital as well in analog are of central importance. In fact the maximum en- complex systems. In particular tools based on tropy discussed above is nothing more than minimum Kullback-Leibler divergence I ≥ 0 (the negative of q/p KL-divergence with a uniform prior28, so that physicists Shannon-Jaynes entropy) and the matchup-multiplicity expert in its application to analog systems can play a or choice-reduction-factor M associated with refer- q/p pivotal role informing students who take physics courses ence probability-set q have proven useful: (i) to engi- about these connections across disciplines. neers for measuring available-work or exergy in thermo- In particular the foregoing are backdrop to the dynamic systems30, (ii) to communication scientists and paradigm-shift which broke out of physics into the wide geneticists for studies of: regulatory-proteinbinding-site world of statistical inference in the mid-20th century29. structure31, relatedness32, network structure, & replica- We’ll touch on only three of the many areas that it’s tion fidelity33,34, and (iii) to behavioral ecologists want- 6 ingtoselectfromasetofsimple-modelstheonewhichis leastsurprisedbyexperimentaldata35,36fromacomplex- reality. In context of this idea-set, the logical schematic in Figure 3 illustrates connections that often go unmen- tioned between what are now-classical application areas in their specialized fields. It thus suggests that physi- cists, particularly thanks to their long experience with log-probability measures in analog systems, can play a keyroleinthe cross-disciplinaryapplicationofinformat- ics to complex systems. These multi-moment correlation-measures have 2nd law teeth making them relevant to quantum computing37, and they enable one to distinguish pair from higher-order correlations making them relevant to the exploration of order-emergence in a wide range of biologicalsystems38,39. They may be especially useful in addressing challenges associated with the sustainability of multi-layer complex systems40. FIG. 4: Addingw/c=1 proper-velocity vectors. Bayesianinformaticsdevelopmentcoursewhichhelpedto VI. DISCUSSION nail down some of the inter-connections discussed here. Similar analyses might also help each of us decide when it is (and is not) appropriate to spend time Appendix A: Proper-velocity in the educational arena e.g. on: (i) geometric- algebra approaches41–43 to complex numbers & cross- Minkowski’s flatspace metric-equation naturally de- products, (ii) energy6 & least-action44 based introduc- fines the relationbetween traveler-timeelapsed (δτ) and tions to mechanics, (iii) vector potential introductions the distance/time between events defined with respect to magnetism45, (iv) explore-all-paths introductions to to the yardsticks (δ~x) and synchronized-clocks (δt) of a quantum mechanics46. The approach may even come in singlemap-frame,thusdefiningLorentzfactor&proper- handy for mediating differences in research strategy as velocity as alternate ways to describe rate of travel. well, e.g. in deciding how much time to spend (in con- Proper-velocity, referred to by Shurcliff as the text of a particular problem) on: (a) CPT approaches “minimally-variant” parameter for describing position’s to the application of non-Hermitian Hamiltonians47, (b) rate of change, can simplify our understanding of many molecule-code as distinct from kin-selection models of relativistic processes. We choose two ways here that are evolving eusocial or altruistic behavior4, etc. relevant to an introductory physics course. VII. CONCLUSIONS 1. vector addition In short both quantitative and schematic considera- A useful mnemonic for relative motion in the Newto- tions of the algorithmic path to key deliverables from nian world is: your & your audience’s conceptual starting point may help point you toward approaches that help your ~vAC =~vAB +~vBC (A1) students become maximally-informed in minimum time. These may not lessen “the detailed work of content where e.g. ~vAB is the vector velocity of object A with modernization”5, but they may help provide the process respect to object B. Note that in general ~vAB = −~vBA, with useful direction tayloredto our individual points of and in sums “a common middle letter cancels out”. reference. What would your concept maps look like in For a relation that works for uni-directional velocity- this context? addition even at coordinate-speeds v near lightspeed c, one might use the similar relationship: w ≡γ v =γ γ (v +v ) (A2) Acknowledgments AC AC AC AB BC AB BC where the proper-velocity w~ ≡ d~x/dτ = p~/m is map- I would like to thank graduate students Bob Collins, distance ~x traveled per unit time τ on traveler-clocks, Zak Jost, and Pat Sheehan for their participation in a coordinate-velocity ~v ≡ d~x/dt with v ≤ c is the usual 7 bydefiningLorentz-factorintermsofcoordinate-velocity in effect lowers a curtain on kinetic-energy/momentum space by making only the lowerrighthalfof it accessible to moving objects. Thelog-logplotinFigure5,whichalsohaslinesofcon- stant mass and constant coordinate-velocity, thus pro- vides students with an integrativeview of kinetic-energy andmomentumspaceforawiderangeofobjectsin(and beyond) everyday experience. Thus for example if one points these relationships out as early as possible in an intro-physicscourse (perhaps as early as the kinematics- section on relative velocities if one takes the time to distinguish traveler-time τ from map-time t in defining Lorentzfactorγ ≡dt/dτ asintheprevioussection),then onemayfindopportunitiesagain-and-againtoreferback to it as new phenomena come up in the course. Appendix B: Proper-acceleration Bystickingwithasinglemap-frametodefineextended FIG. 5: Kinetic energy vs. momentum plot. simultaneity, one finds that equations for accelerated motion also extapolate nicely from low to high speed. map-distancetraveledperunittimetonmap-clocks,and The organizing parameter for this extension is the three Lorentz-factor γ ≡ dt/dτ = 1/ 1−(v/c)2 = E/mc2 ≥ non-zero(spatial)componentsofanobject’sacceleration 1 is the “speed of map-time per unit traveler-time” pro- four-vector as seen from the vantage point of the object vided that the map-frame definpes simultaneity. itself. Thus uni-directional coordinate-velocities add but This 3-vector is referred to as the object’s proper- Lorentz-factors multiply when forming the proper- acceleration α~. Again we discuss two uses for this quan- velocity sum. This allows colliders (sometimes with titythataremostrelevanttostudentsinanintro-physics Lorentz-factors well over 105) to explore much higher- course. speed collisions than would be possible with a fixed- target accelerator! In the any-speed and any-direction case this becomes: 1. proper or geometric? w~ ≡γ ~v =(w~ ) +w~ (A3) AC AC AC AB C BC Forintro-physicsstudentsevenatlowspeedsonemight point out that there are experimentally two kinds of where C’s view of the out-of-frame proper-velocity acceleration: proper-accelerations associated with the (w~ ) is in the same direction as w~ ≡ γ ~v AB C AB AB AB push/pullof externalforces,and geometric-accelerations but rescaled in magnitude by a factor of (γ +w~ · BC AB w~ /(c2(1+γ ))) ≥ 0, as illustrated for “unit proper causedbychoiceofareference-framethatisnotgeodesic BC AB i.e. a local reference coordinate-system that is not“in velocities” in Fig. 4. Hence the original low-speed equa- free-float”. tion generalizes nicely when proper-velocity w~ ≡ d~x/dτ Typically proper-accelerations are felt through their is used instead of coordinate-velocity~v ≡d~x/dt. points of action e.g. through forces on the bottom of your feet, or through interaction with electromagnetic 2. energy vs. momentum fields. On the other hand, geometric-accelerations as- sociatedwithone’scoordinatechoiceareassociatedwith affine-connectionforces(anextendedversionoftheNew- Vector proper-velocity w~ like vector momentum p~ = tonian concept of inertial force) that act on every ounce mw~ = mγ~v has no size upper-limit. Likewise for scalar of an object’s being. Lorentz-factor γ, which like kinetic energy K = (γ − 1)mc2 has no intrinsic upper-limit. Affine-connection effects either vanish when seen from the vantage point of a local free-float or geodesic frame HenceNewtonlikelyimaginedthatbychoosingtheap- (anextendedversionoftheNewtonianconceptofinertial propriatemassm,objectsmaybefoundwithanydesired frame),orgiverisetonon-localforceeffectsonyourmass mix of kinetic energy K and translational momentum p. distribution which cannot be made to disappear. Some AsshowninFigure5,however,Minkowski’smetricequa- of these are summarized in Table I. tion: Although the following need not be shared, the as- (cδτ)2 =(cδt)2−δ~r·δ~r (A4) sertion above contains the essence of general relativity’s 8 TABLE I:Acceleration typesand various forces. force name type proper geometric: non-free non-local normal ⊕ string ⊕ spring ⊕ friction ⊕ drag ⊕ centripetal ⊕ electromagnetic ⊕ gravity ⊕ reaction “gees” ⊕ centrifugal ⊕ Coriolis effect ⊕ tidal ⊕ equivalence principlewhichguaranteesthatNewton’s Laws can be helpful locally in accelerated frames and curved space time, provided that we invoke inertial forces to explain the geometric-accelerations which op- FIG. 6: 1-gee proper-acceleration roundtrips. erate in those frames. The mathematics of geometric accelerations comes from the fact that in general relativity an object’s co- ordinate acceleration (as distinct from only its proper- 1dw dv acceleration 4-vector A) is equal to: α≡ =γ3a, where a≡ (B3) γ dτ dt dUλ =Aλ−Γλ UµUν (B1) This yields three integrals of constant proper- µν dτ acceleratedmotion that reduce to the familiar equations of constant coordinate-accelerationat low speeds: where geometric-accelerations are represented by the affine-connection term Γ on the right hand side. These manadypboesitthioensudmepoenfdaesnmt atenrymass. sCixotoeerdninseaptearaactceevleerlaotciiotny α= ∆w =c∆η =c2∆γ v=≪c ∆v v=≪c 1∆(v2) (B4) ∆t ∆τ ∆x ∆t 2 ∆x goestozerowheneverproper-accelerationisexactlycan- celed by that connection term, and thus when physical These in turn allow one for example to write out ana- and inertial forces add to zero. lyticalsolutions(cf. Fig. 6)forround-tripsinvolvingcon- stant 1 gee ≃ 1.03[ly/y2] accelerated/decelerated travel between stars. 2. accelerated roundtrips Appendix C: Choice multiplicities For unidirectional (1+1)D motion, the rapidity or hyperbolic velocity angle η simply connects the inter- Senior physics courses have for already been re- changable velocity parameters Lorentz-factorγ ≡dt/dτ, arranged considering the fundamental role that multi- proper-velocity w ≡ dx/dτ and coordinate-velocity v ≡ plicity (and its logarithm, namely entropy) play in un- dx/dt via: derstanding and predicting behaviors. Although intro- physics courses are weaker in this context, books like w v Tom Moore’s “Six Ideas”6 have put choice-multiplicity η ≡sinh−1 =tanh−1 =±cosh−1[γ] (B2) where it belongs at the start of the thermo-chapters. c c h i h i Hencethe only sectioninthis Appendix isoneforstu- These parameters may then be used to express the dents with virtually no math background. The hope is proper-acceleration α experienced by an object travel- that teachers will individually explore ways to introduce ing with respect to a map-frame of co-movingyardsticks the connection between bits and J/K, while at the same andsynchronizedclocksinflatspacetime,intermsofits time nurturing an appetite for textbook revisions that coordinate-accelerationa which cannot be held constant bettercommunicatetherelationbetweenthermalphysics at high speed, as: and information theory downstream. 9 FIG. 7: Logarithmic measures of probability and odds. 1. quantifying risk Surprisal in bits (defined by probability = 1/2#bits) 0F◦IGan.d8:1A00v◦aiwlaabtleerwinorvka/rgiroaums svtsa.teasm.bienttemperatureTo for might be useful to citizens in assessing risk and/or stan- dards of evidence (cf. Fig. 7), because of its simple, intuitive, and testable ability to connect even very small costsofmedicalmalpracticeinthelongrunbyempower- probabilities with one’s experience at tossing coins. For ingpatientswithtoolstomakeinformedandresponsible example, the surprisalof dying froma smallpoxvaccina- choices, making the need for legal redress less frequent. tion (one in a million) is about 19.9 bits (like 20 heads Thus the media for risk-assessing public could play a in 20 tosses), while the surprisal of dying from smallpox keyroleinreducingthecostsofdefensivemedicine. Some onceyouhaveit(oneinthree)isonlyabout1.6bits(i.e. mightevenenjoysurprisaldataonthesmallprobabilities more likely than 2 heads in 2 tosses). associated with some gambling opportunities. After all, Thussurprisal: (i)hasmeaningwhichiseasytoremind there really is more to the lottery than simply knowing yourself of with a few coins in your pocket, (ii) reduces “the size of the pot”. hugenumberstomuchmoreintuitivesize,and(iii)allows onetocombinerisks”fromindependentevents”withad- dition/subtraction rather than multiplication/division. Appendix D: Matchup multiplicities Forinstance(fromthe numberssuggestedabove)your chance of dying is decreased by getting the vaccination, Of the integrative concepts discussed in this paper, as long as the surprisal of getting smallpox without the the least familiar to physicists (judging from textbooks, vaccinationislessthan20−2≃18bits. Thatmeansthat at least) may be those associated with the logarithmic vaccination is your best bet (absent other information) correlation-measure often referred to as KL-divergence if your chances of being exposed to smallpox are greater thanthoseofgetting18headsin18tosses(1outof218 ≃ and its multiplicity: in effect a kind of normalized choice-reduction factor which is never less than 1. 333,333). Hence in this Appendix we discuss matchup-multiplicity Given the large difference between something with 2 connections to: (i) available work, (ii) model-selection bits ofsurprisalandsomethingwith 18,communications math important at least for physicists involved in cross- bandwidth might be better spent by newsmedia provid- disciplinarywork,and(iii)theevolutionofcomplexana- ing us with numbers on observed surprisal, rather than log as well as digital systems. by reporting only that “there’s a chance” of something bad (or good) happening. Saying the latter treats your audience as consumers of spin rather than information. 1. work’s availability Likewise,useofsurprisalsincommunicatingandmon- itoring risks to medical patients could make patient de- cisions about actions with a small chance of dire out- Best-guess states (e.g. for atoms in a gas) are inferred comes as informed as possible. This could reduce the by maximizing the average-surprisal S (entropy) for a 10 given set of control parameters (like pressure P or vol- ume V). This constrained entropy maximization, both classically and quantum mechanically, minimizes Gibbs availability in entropy units A = −kln[Z] where Z is a constrained multiplicity or partition function. When absolute temperature T is fixed, free-energy (T times A) is also minimized. Thus if T, V and number of molecules N are constant, the Helmholtz free energy F =U−TS(whereU isenergy)isminimizedasasystem “equilibrates”. If T and P are held constant (say during processes in your body), the Gibbs free energy G=U + PV−TSisminimizedinstead. Thechangeinfreeenergy undertheseconditionsisameasureofavailableworkthat mightbedoneintheprocess. Thusavailableworkforan ideal gas at constant temperature T and pressure P is o o W =∆G =NkT Θ[V/V ] where V = NkT /P and by o o o o o Gibbs inequality Θ[x]≡x−1−ln[x]≥0. Moregenerally,theworkavailablerelativetosomeam- bientisobtainedbymultiplyingambienttemperatureT o byKL-divergenceornet-surprisal∆I ≥0,definedasthe average value of kln[p/p ] where p is the probability of o o agivenstateunderambientconditions. Forinstance,the FIG. 9: Probabilities of constant vs. linear fitsusing AIC. work available in equilibrating a monatomic ideal gas to ambient values of V and T is thus W = T ∆I, where o o o KL-divergence ∆I =Nk(Θ[V/Vo]+(3/2)Θ[T/To]). The ering only the most likely set {Aˆk} of k = 1 to K fit- resulting contours of constant KL-divergence put lim- parametervalues{A }associatedwithagivenmodelM. k its on the conversion of hot to cold as in flame-powered One might then use this to predict the surprisal of data air-conditioning, or in the unpowered device to convert generated by reality’s unknown operating probability q, boiling-water to ice-water (Fig. 8). Thus KL-divergence namely S ≡hln 1 i given that model: (also known to engineers as available-work or exergy in p/q p q units of kT ) measures thermodynamic availability in h i o bits. 1 1 1 S ≃ln =ln +ln p/q (cid:20)p[D|A,M](cid:21) "p[D|Aˆ,M]# (cid:20)ΩM(cid:21) (D1) 2. model selection math Although we can’t calculate KL-divergence I directly p/q because we don’t know S , the argument is that the q/q Inspite of: (i) the centralroleofmodel-selectioninall best model M available (in the absence of inside infor- observational science, and (ii) the math background re- mation about reality q) is the one that is least surprised quiredtodophysics,themathematicsofmodel-selection by available data D i.e. for which S is least. p/q is sometimes hardly an afterthought in physics tests and Note that the first term in the minimized quantity on assessments. Introductory physics texts sometimes even the right-hand-side of Equation D1 is just the negative definemodelsassimplifiedversionsofphysicalsystems48, log-likelyhood of the fit i.e. the quantity traditionally instead of as idea-based representations that (like the minimized when choosing the optimum set of param- molecule-basedcode-stringsofone-celledorganisms)help eters for a given model. For example, that first term us correlate our behaviors with the world around. when least-squares fitting N data points with normally- Hence the background of physicists, e.g. on disser- distributed errors having a mean µ and standard devi- tation committees of biophysics students whose project- ation σ is a non-varying constant plus a second con- literature requires a mathematical approach to model- stant times the average square of data deviations from selection, may lead them to think that parameter- the meani.e. the model’svarianceormean-square-error. estimation and model-selection are one and the same. When choosing one model over another, however, the In this section we follow astronomer Phil Gregory’s dis- second “Occam-factor” term must also be considered in cussionofOccamfactorstoshowhowthetworelate,and the analysis. Gregory provides a lovely general expres- pointtoaninterestingstrategyforgettingquickanswers sion for Occam factors in the linear least-squares case, that so far seems to be better known in ecology than in in terms of prior probabilities for the fit parameters and physics. the parameter-function covariance matrix. On the other We begin with Gregory’s Occam factor28, defined as handstatisticians haveset-up simpler rules-of-thumbfor Ω ≡ p[D|A,M]/p[D|Aˆ,M] i.e. as the factor by which this Occam factor, also grounded in Bayesian inference M the likelihood of a set of data D is increased by consid- and the connection described here to KL-divergence.