Finite-State Classical Mechanics
FFinite-State Classical Mechanics
Norman Margolus
Massachusetts Institute of Technology, Cambridge MA, USA. [email protected]
Abstract.
Reversible lattice dynamics embody basic features of physicsthat govern the time evolution of classical information. They have finiteresolution in space and time, don’t allow information to be erased, andeasily accommodate other structural properties of microscopic physics,such as finite distinct state and locality of interaction. In an ideal quan-tum realization of a reversible lattice dynamics, finite classical rates ofstate-change at lattice sites determine average energies and momenta.This is very different than traditional continuous models of classical dy-namics, where the number of distinct states is infinite, the rate of changebetween distinct states is infinite, and energies and momenta are nottied to rates of distinct state change. Here we discuss a family of classi-cal mechanical models that have the informational and energetic realismof reversible lattice dynamics, while retaining the continuity and math-ematical framework of classical mechanics. These models may help toclarify the informational foundations of mechanics.
The physics of continuous classical materials and fields is pathological. For ex-ample, in classical statistical mechanics, each degree of freedom of a system atthermal equilibrium has about the same finite average energy, proportional tothe temperature. This implies that a continuous material—which has an infinitenumber of degrees of freedom—will be able to absorb energy at a finite rateforever without heating up. Exactly this pathology, evident in the radiation fieldinside hot cavities (black bodies), led to the overturn of classical mechanics as afundamental theory and the advent of quantum mechanics [1].A similar pathology of the continuum exists in the classical mechanics ofparticles. Quantum mechanics provides a fundamental definition of energy interms of rate of change in time: frequency . What we call energy in the classicalrealm is (in fundamental units) just the average frequency of a large quantumsystem—and this is precisely the maximum rate at which the system can tran-sition between perfectly distinct states [2,3,4]. Since all physical systems havefinite energy, they all have a finite rate of distinct state change. But in contin-uous classical mechanics, each infinitesimal interval of time brings a perfectlydistinct new state, and so the rate of state change is infinite. Similarly, finitemomentum only allows a finite rate of distinct state change due to motion, notthe infinite rate required by infinite resolution in space.It would be nice to have a version of classical dynamics that avoids thepathologies of infinite state and infinite resolution in space and time, while still a r X i v : . [ n li n . C G ] J u l Norman Margolus being a subset of ordinary classical mechanics. Fredkin’s billiard ball model ofcomputation [5] illustrates that this is possible: a carefully designed classical me-chanical system with discrete constraints on initial conditions can be equivalent,at discrete times, to a reversible finite-state dynamics. In this paper, we discussFredkin’s model as well as others where the equivalence is even more direct. Inthese examples, local rates of state change in the finite-state dynamics play theroles of mechanical energies and momenta. This property is physically realistic,and arises from the fact that ordinary reversible computations, such as these, canbe interpreted as special cases of quantum computations [6], and hence inheritquantum definitions of energy and momentum based on rates of state change.
Some of the simplest models of physical systems are lattice models with classi-cal finite state. Dynamical models of this sort with local interactions are oftenreferred to as cellular automata , but here we will favor the more physical term classical lattice gas , which encompasses both deterministic and stochastic phys-ical models [7,8,9,10,11,12,13]. We discuss reversible classical lattice gases asfoundational models of both classical and quantum mechanics.By foundational models we mean here the simplest examples of systems thatexactly incorporate basic physical properties and principles ( cf. [14]). That aworld with finite entropy has classical lattice gas foundations is not surprising,and is well accepted in statistical mechanics [7, § The fundamental models of mechanics are continuous in space and time, andhence seem very non-discrete. In fact, though, all realistic physical systems havefinite resolution in space and time, usually described using continuous mathe-matics, and similar descriptions can be applied to discrete models.The finite resolution in space and time of quantum systems can be expressedin terms of uncertainty relations [15], but is better thought of as akin to theeffective discreteness of a finite-bandwidth classical signal [3,4]. Interestingly, thiskind of discreteness was recognized around the same time that the founders ofquantum mechanics discovered uncertainty [16]. The discoverer, Harry Nyquist,was thinking about how many dots and dashes could be put into a telegraphsignal, and he realized that bandwidth was the key quantity that set the bound.He gave a simple argument, first considering a signal periodic in time, andthen generalizing to the average rate for an infinitely long period. Consider acomplex valued periodic wave. This is composed of a discrete set of Fourier inite-State Classical Mechanics 3 components that fit the period: for period T the possible frequencies are 1 /T ,2 /T , 3 /T , etc. With a limited range of frequencies (a limited bandwidth ), theFourier sum describing the wave has only a finite number of terms with a finitenumber (say N ) coefficients. With N coefficients we can only freely choose thevalue of the sum at N times. Thus the minimum range of frequencies ν max − ν min needed to have N distinct values of the sum is given by the minimum separationbetween frequencies 1 /T , times the minimum number of separations N − ν max − ν min ≥ N − T . (1)For a long wave, the bandwidth ν max − ν min is the maximum average densityof distinct values, N/T . Turning the argument around, if we know the valuesof a periodic signal with finite bandwidth at enough discrete points, we candetermine all coefficients in the finite Fourier sum: the rest of the continuouswave is then determined and carries no additional information [17]. Thus waveswith finite bandwidth are effectively discrete.
The argument above goes through essentially unchanged for the wavelike evolu-tion of quantum systems. The wavefunction for an isolated system is expressedin the energy basis as a superposition of frequency components: ν n = E n /h ,where E n is the n -th energy eigenvalue, and h is Planck’s constant. To have N distinct (mutually orthogonal) states in a periodic time evolution, there mustbe a superposition of at least N distinct energy eigenfunctions, with distinctfrequencies. Again the minimum frequency separation is 1 /T if the period is T ,so the minimum range of energy eigenfrequencies is again given by (1).For systems that exactly achieve this bound on orthogonal evolution, it iseasy to show that the N equally spaced frequency components must be equallyweighted, and that the same minimum-bandwidth distribution minimizes allreasonable measures of frequency width [3,4]. For example, for the minimizingdistribution, average frequency ¯ ν minus the lowest ν min is half the bandwidth,so 2(¯ ν − ν min ) ≥ N − T . (2)In quantum mechanics, the average energy of an isolated system is E = h ¯ ν . Thelowest (ground state) energy E = hν min is like the lowest frequency used in aclassical signal: it is the start of the frequency range available for the dynamicsof the isolated system. If the ground state energy E is taken to be zero and N (cid:29)
1, then letting ν ⊥ = N/T be the average density of distinct states in time,and choosing units with h = 2 (so E = 2¯ ν ), (2) becomes E ≥ ν ⊥ . (3)Thus energy is the maximum average rate of distinct state change. This can alsobe regarded as an uncertainty relation between average orthogonalization time Norman Margolus τ = 1 /ν ⊥ and average energy width: ( E − E ) τ ≥
1. All uncertainty relationsbetween τ and any other width ∆ν of the energy-frequency distribution of thewavefunction are similar [3,4], with the choice of width and number of distinctstates changing the bound by only a factor of order one—a periodic oscillationbetween just two distinct states is the fastest [2,18]. The same kind of Fourieranalysis also applies to waves in space, rather than in time. All such boundsare attained simultaneously for minimum bandwidth, in which case quantumevolution becomes equivalent to a discrete evolution on a spacetime lattice: onlythe values of the wavefunction at lattice points in space and time are distinct[3,4,19,21]. The rest of the continuous state is redundant. A moving particle has both extra distinct states due to its distinct positions, andextra energy due to its motion. In the particle’s rest frame, it has neither. For alarge ( ≈ classical) system moving between two events, evolving at the maximumrate of distinct state change allowed by its energy, we can use (3) in the twoframes to count the extra distinct states due to the motion. If the system hasvelocity v and magnitude of momentum p , and the events are separated by time ∆t and distance ∆x in the laboratory frame, and ∆t r in the rest frame, with E and E r the corresponding energies, the invariant time-energy interval is E∆t − p∆x = E r ∆t r . (4)But from (3), E∆t is simply the number of distinct states seen in the laboratoryframe, and E r ∆t r the number seen in the rest frame. The difference is the numberof distinct states due to the motion, which from (4) is simply p∆x . Thus if µ ⊥ is the average density in space of states distinct due to the motion, p ≥ µ ⊥ . (5)If we multiply (5) by v = ∆x/∆t , we get that vp ≥ ∆x µ ⊥ /∆t ≡ ν motion , thenumber of distinct states per unit time due to motion. So we also have p ≥ ν motion /v . (6)Thus motion and speed define a minimum p . In a quantum realization of areversible lattice gas, a hop of an isolated particle from one lattice site to anotheris a distinct state change, and speed is just distance over time. In our discussionof energy and momentum conserving lattice gases, we will use the minimumpossible momentum for any quantum system, from (6), as our estimate of themomentum of a freely moving isolated particle in an ideal realization [19,20]. Choosing units with the speed of light c = 1, it is always true relativistically fora freely moving particle that E = p/v , so we can compute the energy of a free inite-State Classical Mechanics 5 particle once we know its momentum and velocity. From (6), E ≥ ν motion /v , soenergy is smallest when v is as large as possible—we can treat ν motion as constanthere, since it doesn’t depend on the distance between lattice sites, and we canmake our particles travel faster by increasing only the distance between latticesites without changing the time. Now, given a lattice gas dynamics with a set ofparticle velocities related by the lattice geometry, there is a family of equivalentevolutions that only differ in the choice of the fastest particle speed. Of these, theevolution with the least possible energy has its fastest-moving particles travelingat the speed of light. This makes sense physically, since a system with a non-zero rest-frame energy has a non-trivial internal dynamics—time passes in therest frame. If we want to just model a logical evolution and nothing extra, thefastest-moving particles should have no internal dynamics. We discuss three reversible lattice gases, each with a finite-state dynamics thatreproduces discrete samples of a classical particle evolution, and one model thatsamples a classical field. In its discrete form, the field example also turns intoa reversible lattice gas. Although these models could represent macroscopic sys-tems with any given energy, we’re interested here in looking at intrinsic minimumenergy and momentum defined by state change on the lattice. We analyze thisfor just the last two models—the first two are introductory. The field example isparticularly interesting because energy and momentum are bound together onthe lattice, moving as a relativistic particle with a discrete set of possible speeds.This behavior is intimately related to a biased random walk. t t + .5 t +1 .5 t +2 .5 t +1 t +2 t +3 Fig. 1. Lattice gas molecular dynamics.
Left:
Particle and momentum conservingcollisions in a single-speed four-velocity lattice gas. Particles are at lattice points atinteger times. At half integer times, they are midway between.
Right:
A single-speedsix-velocity lattice gas run on two million sites of a triangular lattice, with obstaclesand a visualizing “smoke” gas added to the model. It exhibits realistic fluid behavior.
Lattice gas fluids.
The first lattice gases that reproduced samples of a classicalmechanical evolution were models of fluids [22]. Lattice gas fluids are stylized
Norman Margolus molecular dynamics models, with particles started at points of a lattice, movingat one of a discrete set of velocities, and colliding at points of the lattice ina manner that conserves momentum and guarantees particles will again be onthe lattice at the next integer time. We illustrate this in Figure 1 (Left). Weshow two particles of a four-velocity 2D lattice gas. In the top row we showthe particles at integer times, in the bottom row halfway between integer times.The dynamics shown is invertible and is momentum and particle conserving, butis too stylized to be a realistic fluid. Four directions aren’t enough to recoverfully symmetric Navier Stokes fluid flow in the large scale limit—but six are[23,11,12,13]! Figure 1 (Right) is a snapshot of a simulation of a six-velocitylattice gas, with obstacles and a second tracer fluid (smoke) added, showing flowpast an obstacle [24].The reason we can talk here about momentum conserving collisions is be-cause the discrete lattice gas is, conceptually, embedded in a continuous dy-namics where we know what momentum is. In classical mechanics, a dynamicswith continuous symmetry under translations in space defines a conserved linearmomentum, and a continuous rotational symmetry defines a conserved angularmomentum [25]. The embedding allows us to define discrete conserved quantitiesthat derive from continuous symmetries which cannot exist on a discrete lattice.The full continuous symmetries associated with the conservations [25] can onlyemerge in a lattice model in the macroscopic limit. This makes conservationmore fundamental than continuous symmetry in lattice models [26].In order to emulate the reversibility of microscopic physics, a local latticedynamics must have a structure where data on the lattice are partitioned intoseparate groups for updating [27,28,29]; then if the transformation of each groupis invertible, this property is inherited by the overall dynamics. At least twodifferent partitions, used at different times, are required—with only one, eachgroup would be forever isolated. In a continuous dynamics that has been initial-ized to act discretely, the alternation of partitions does not involve any explicittime dependence. For example, at the integer-times of Figure 1 (Top Left), wesee that all collisions happen at lattice locations, and the data at each lattice siteare transformed independently of all other sites—this constitutes one partition.Not only invertibility, but particle and momentum conservation are guaranteedby the collision rule. In between collisions, particles travel straight to adjacentlattice locations without interacting. This constitutes the second partition.We get a different view of partitioning for the same continuous dynamicsif we define a lattice gas from the half-integer-time states of Figure 1 (BottomLeft). In this case, we catch all particles when they are going straight, in betweenlattice sites. Particles are spread out in space, rather than piled up at lattice sites,and we can tell which way they are going from where they are, when. Groupsof four locations that can contain particles converging on a lattice site define apartition—for example, the middle 2 × inite-State Classical Mechanics 7 ABABABABAB
ABABABABAB
A A
A AB B
Fig. 2. Fredkin’s billiard ball model of computation.
Perfect billiard balls movingon a lattice can perform reversible computation. Ball presence is a 1, absence a 0. Leftto right, we show (a) a right-angle collision where different logical combinations comeout at different places, (b) a logical collision that happens in between integer times, (c)infinitely massive mirrors used to route signals, and (d) mirrors used to allow signalpaths to cross, regardless of the signal values. of view of the continuous dynamics, the 2 × The billiard ball model.
In Figure 2 we illustrate the lattice gas that EdFredkin invented [5] to try to silence skeptics who claimed reversible computingwas physically impossible. This is a perfectly good reversible, particle and energyconserving classical mechanical model that uses the collisions of hard spheres toperform computation: ball or no-ball is a one or a zero, and collisions separatedifferent logical cases into separate paths. The model uses infinitely massivemirrors, which are allowed in non-relativistic classical mechanics, to route signals.The equivalence to a discrete lattice gas is incomplete, though, since certaincollisions (such as head on ones) will take the balls off the lattice—it is notenough to merely start all balls at lattice locations. The model can be completedby simply mandating that all problematic cases cause balls to pass through eachother without interacting. This is a general feature of sampled classical dynamics:it is typically necessary to add a form of classical tunneling to the continuousdynamics, in order to maintain its digital character. From the point of view ofthe mathematical machinery of classical mechanics, there is really nothing wrongwith doing this: it doesn’t impact invertibility, conservations, relativity, etc.There is still a problem, though, with turning the billiard ball model into alattice gas that acts as a faithful sampling of a classical mechanical dynamics.Because of the hard collisions, the number of locations that need to be updatedas a group in order to ensure invertibility and particle conservation is ratherlarge: in Figure 2 (b), when the ball coming in at B is in the second column andabout to interact, the next value of the location marked AB at the top dependson the presence or absence of the ball coming from B . To implement this as alattice gas requires lattice sites that hold many particles, or the use of ratherlarge partitions. This structure also implies extra constraints, not present in thecontinuous classical model, on the positions where collisions can occur, so thatparticles only converge on places where all particles can be updated as a group. Norman Margolus
ABABABABAB
AB ABABABAB
AB BA
Fig. 3. Soft sphere model of computation.
Compressible balls collide. Left toright we show (a) collisions displace the colliding balls inwards, putting the AB caseson different paths, (b) we recast this as a lattice gas, with particles located at the tailsof the arrows, (c) there is interaction in only two cases (and their rotations); otherwiseall particles move straight, (d) adding a rest particle to the model allows particle pathsto cross; this still follows the rule “otherwise all particles move straight.”
Soft sphere model.
We can avoid all of these issues with a simple modificationof the classical billiard ball dynamics, illustrated in 3 (a). If we make the collisionsvery elastic, rather than hard, colliding particles spend a finite amount of timecolliding, and are deflected onto inward paths, rather than outward as in thebilliard ball model. This soft sphere model [30] is equivalent to a lattice gaswhere interactions can happen at a point, as in Figure 1 (left-top). The discreteand continuous models can exactly coincide everywhere at integer times.Figure 3 (b) shows a direct translation of (a) into a lattice gas. As in (a), wehave two streams of particles coming in at A and B and depict the state at aninteger time, so we see particles at each stage of the collision at different pointsin space. In the lattice gas diagram, the particles are located at the tails of thearrows, and the arrows indicate direction information carried by the particles.The rule (c) is very simple: diagonal particles colliding at right angles turninto horizontal particles and vice versa—plus 90 ◦ rotations of these cases. In allother cases, the particles pass through each other unchanged. In (d) we add anunmoving rest particle to the model, so we can place it at any signal crossingto prevent interaction: the rule is unchanged, since moving straight is alreadythe behavior in “all other cases.” This allows the model to perform computationwithout the addition of separate infinite-mass mirrors [30]. Similar computinglattice gases can be defined on other lattices, in 2D or 3D.We can analyze the minimum energy and momentum for a unitary quantumimplementation of this reversible classical dynamics. Looking at Figure 3 (b), wecount state change and direction of motion in the middle of each arrow—duringthe time when the particle is moving freely between lattice sites. In this waywe always see a single isolated particle moving with a definite velocity, and canapply (6) directly to get the minimum momentum. Taking the time betweenlattice sites as our unit of time, each particle motion constitutes one change perunit time. The particles moving diagonally are the fastest moving particles, sowe take their speed v = 1 to get a minimum energy model—they each have ideal(minimum) momentum p = 1. From the geometry of the model, we see that thehorizontal particle must then be moving at speed 1 / √
2, and there is again one inite-State Classical Mechanics 9 change in a unit of time as it moves, so its ideal momentum from (6) is √
2. Thisagrees with conservation of momentum, since each of the two incoming particleshas a horizontal component of momentum of 1 / √
2. The horizontally movingparticle is moving slower than light, and so has a mass. By energy conservation,since the sum of the incoming energies is 2, that must be the energy of thehorizontal particle. Then m = (cid:112) E − p = √ cf. [20]).
10 20 30 40 - - - =+ tt +1 =+=+ else, no changeUpdate blocks alternately at even and odd coordinates. Fig. 4. Continuous wave dynamics equivalent to a finite-state dynamics.
Left:
Any one-dimensional wave obeying the continuous wave equation is a superposition ofa rightgoing and a leftgoing wave. We constrain the two components to always give adiscrete sum at integer times, so we can sample then.
Middle:
Each component wavealternates flat intervals with intervals that have a slope of ±
1. There are only fourcases possible at integer times.
Right:
Block rule for directly evolving the sum wave.
We discuss here a simple classical field dynamics in which constraints on thecontinuous initial state make it equivalent to a reversible lattice gas at integertimes and positions ( cf. [20,31,32,33]). This example illustrates the mechanismbehind a phenomenon that was discovered experimentally in reversible cellularautomata models: the spontaneous appearance of realistic waves [9,33,34].In Figure 4 (left) we show, at the top, a continuous wave that is the su-perposition of continuously shifting rightgoing and leftgoing waves—we assumeperiodic boundaries so what shifts off one edge reappears at the other. It is ageneral property of the one dimensional wave equation that any solution is asuperposition of an unchanging rightgoing waveform and an unchanging leftgo-ing waveform. In this case, each of the two component waves contain segmentsthat have slope 0, alternating with segments that have slope +1 or −
1. As thecomponent waves shift continuously in space, at certain moments the slope 0’s ofeach wave align with the ± ± converging towards the center of each partition. This is illustrated in Figure 4(Middle). At these times, the next integer-time configuration for each pair ofcolumns is completely determined by the current configuration, and doesn’t de-pend on any information outside the pair. We see, from Figure 4 (Middle), thatthere are only four distinct cases, and only two of them change the sum: slopes \/ turn into /\ and vice versa. We flip hills and troughs; nothing else changes.Figure 4 (Right) is thus the evolution rule for a discrete string dynamics thatexactly follows the continuous wave equation at integer times and positions. Therightgoing and leftgoing waves can be reconstructed from the sum—only the sumis evolved as a lattice gas. This flip rule must be applied alternately to the twopossible partitions into pairs of columns. The rule works just as well, though, ifwe also partition pairs of rows, so the rule is separately applied to 2 × Transverse motion.
It is interesting to analyze the energy and momentum fora discrete string, evolving under the flip rule, that has a net motion up or down.If we decompose such a string into a superposition of a rightgoing and a leftgoingwave, we find that each has a net slope across the space. As long as these twonet slopes add to zero, the string itself will have no net slope and so meets itselfcorrectly at the edges. The rightgoing and leftgoing waves should be thought ofas infinite repeating patterns of slopes, rather than as meeting themselves at theedges of one period of the string.Let 2 N be the width of one period of the string, in units of the width of a slopesegment. The repeating pattern of the string can be decomposed into the sumof a repeating pattern of N non-zero rightgoing slopes and N non-zero leftgoingslopes. Let R + be the number of the N rightgoing slopes that are positive, R − the number that are negative, and similarly with L + and L − for the leftgoingwave. The net slope of the string will be zero as long as R + + L + = R − + L − .Our unit of distance is the width (= height) of a slope segment; our unit oftime the time needed for a slope segment to move one width. Thus the rightgoingwave will be displaced upward a distance R − − R + in time 2 N , and the leftgoingwave up by a distance L + − L − in the same time, so the net upward displacementof the string will be D = R − − R + + L + − L − in time 2 N .We can calculate this another way. Consider a partition of the rightgoingand leftgoing waves into pairs of columns at an integer time. A pair containingslopes \\ or // contributes a net of zero to D , since R + and L + enter withdifferent signs, as do R − and L − . Thus all of the net motion can be attributedto the \/ and /\ cases, which respectively contribute +2 and −
2. If there are N \ / partitions ready to flip up, and N / \ partitions ready to flip down, the netupward velocity of the string is v = D N = N \ / − N / \ N . (7)
String momentum and energy.
Using (6) we can assign minimum momenta.We take the maximum of v to be the speed of light (assuming this string speed inite-State Classical Mechanics 11 is physically possible), and get one unit of momentum upward for each isolatedmotion \/ → /\ , one downward for each /\ → \/ , so net momentum upward is p = N \ / − N / \ . (8)Treating the overall motion of the string as that of a relativistic particle, thetotal energy E = p/v , so from (7) and (8), the total relativistic energy E = N . (9)We can compare this to the energy of the string, treated as a continuousclassical mechanical system. Using the notation Ψ t to denote partial derivativewith respect to t , and with c = 1, the continuous 1D wave equation is Ψ tt = Ψ xx ,and the classical energy density of the string is proportional to Ψ t + Ψ x . Since allof the slopes are ± N ,twice the length of the string. The integral is also the same at all intermediatetimes. This follows from Ψ tt = Ψ xx but we can also see this directly. Let l ( x, t ) bethe amplitude of the leftgoing wave, and r ( x, t ) the rightgoing. Then Ψ = l + r , Ψ x = l x + r x and Ψ t = l x − r x , so Ψ t + Ψ x = 2( l x + r x ), and the integral is 4 N .Similarly, the classical Lagrangian density L ∝ Ψ t − Ψ x can be compared.In this case, again using Ψ = l + r etc., L ∝ l x r x . At integer times the integralover the width of the space is zero, since every non-zero slope is aligned witha zero slope. Halfway between integer times, half the columns contain pairs ofnon-zero slopes that are passing each other, the other half contain pairs of zeroslopes, which don’t contribute to the integral. Each non-zero pair of equal slopescontributes +1 to the integral; each pair of unequal slopes contributes −
1. Thisis a count of potential minus kinetic energy: halfway through a transition \/ → /\ or /\ → \/ of a continuous string, all of the energy would be kinetic, and for anunchanging length of stretched string // or \\ all energy would be potential. Rest frame energy.
For a string moving according to this discrete wave dy-namics, there is a maximum amount of internal evolution of string configurationswhen the net vertical velocity is zero, and no internal evolution when the stringis moving vertically as fast as possible, since then there is a unique configuration · · · \/\/\/\/\/\/\/\/\/\/\/\/\/ · · · (10)that moves upward at the speed of light. The rest frame energy of the stringcharacterizes how much dynamics can happen internally. This is given directlyby (cid:112) E − p as E r = (cid:113) N − ( N / \ − N \ / ) . (11)The more the string has a net motion, the less rest energy it has. This makesthe string a strange relativistic particle, since its total energy is independent ofits speed! In the relation E r = E/γ , we have E r → /γ → down. A realistic model of particle motion would have to include interactionswith other systems, that can add or remove particle energy while conserving en-ergy and momentum overall. We might still use the simple string model, though,as a proxy for a more realistic finite-state model of inertia, by simply specifyinga statistical interaction that can change the length of the string to change E . Statistical inertia.
To contemplate a statistical coupling to another system,it is helpful to recast the analysis of the string model in a population-statisticsformat. Recall that, for a given population of rightgoing and leftgoing slopes,the number going each way is the same and, for the string to meet itself atthe edges, the sum of all the slopes is zero. Together, these two constraintsimply that L − = R + and L + = R − : the population statistics for rightgoingand leftgoing waves are mirror images. Therefore we can analyze the motionlooking at the statistics for just one of the component waves. From (7), using N \ / − N / \ = D/ L + − L − and letting p + = L + /N and p − = L − /N , v = p + − p − , (12)with p + + p − = 1. Here p + is the average frequency of upward steps per unittime, and p − the frequency of downward. To see this for p + = L + /N , notice thateach leftgoing / contributes one unit of upward motion in the course of 2 N stepsof evolution, and so does the mirror image \ moving rightward, so together theytake us one position up in N steps; p − is similar. Thus the transverse motionof the string is like a one-dimensional random walk—which is known to exhibitsimilarities to relativistic particle motion [35,36,32].These frequencies could become true probabilities if the populations were sta-tistically coupled to some environment. For example, imagine the string acting asa mass coupled to a spring, to form an oscillating system. As the spring stretchesit slows down the mass, removing energy and changing the bias p + − p − = v .Eventually it turns the mass around and speeds it up, etc. If the populationsare stochastic the velocity and energy determine the entropy of the string, whichwould change cyclically with time in an oscillator. This introduces a rather ther-modynamic flavor into a discussion of inertia in classical mechanics. Reversible lattice gas dynamics derived by sampling classical mechanical evolu-tion are foundational models for all of mechanics, in the same way that classicallattice gases have long been foundational for statistical mechanics. They are par-doxically both continuous and discrete, both classical and quantum. They havean intrinsic energy and momentum based on counting classical state change givenby the minimum allowed by the general properties of energy and momentum inquantum mechanics. They are non-trivial models and can in fact be computationuniversal.These models are foundational rather than fundamental. We are not at allsuggesting that nature is, at base, a classical cellular automaton [26,37,10], but inite-State Classical Mechanics 13 rather that these reversible classical systems that are also special cases of unitaryquantum systems provide a simplified context in which to study the foundationsof mechanics. This is exactly the role that classical lattice gases play in sta-tistical mechanics: classical special cases of quantum systems [7, § cannot be mod-eled in this manner may tell us something about the essential role that quantummechanics plays, that couldn’t be played by a classical informational substra-tum. These finite-state classical mechanical models also turn some foundationalquestions on their head, since they can be regarded as special cases of unitaryquantum evolution, rather than macroscopic decoherent limits. Acknowledgments.
I thank Ed Fredkin and Tom Toffoli for pioneering andinspiring these ideas, and Gerald Sussman for many wonderful discussions.
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