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3.1 Introduction

Despite appearances to the contrary, we live in a finite-resolution world. A finite-sized physical system with finite energy has only a finite amount of distinct detail, and this detail changes at only a finite rate [1–3]. Conversely, given a physical system’s finite rates of distinct change in time and space, general principles of quantum mechanics define its minimum possible average energy and momentum. We apply these definitions to classical finite-state lattice dynamics.

3.1.1 Ideal Energy

It was finiteness of distinct state, first observed in thermodynamic systems, that necessitated the introduction of Planck’s constant h into physics [4]. Quantum mechanics manages to express this finiteness using the same continuous coordinates that are natural to the macroscopic world. Describing reality as superpositions of waves in space and time, finite momentum and energy correspond to effectively finite bandwidth; hence finite distinctness. For example [3], the average rate \(\nu \) at which an isolated physical system can traverse a long sequence of distinct states is bounded by the average (classical) energy E:

$$\begin{aligned} \nu \le 2\,E/h\,, \end{aligned}$$
(3.1)

taking the minimum possible energy to be zero. Here E / h is the average frequency of the state, which defines a half-width for the energy frequency distribution. If we compare (3.1) in two frames, we can bound the average rate \(\mu \) of changes not visible in the rest frame, and hence attributable to overall motion:

$$\begin{aligned} \mu \le 2\,pv/h\,. \end{aligned}$$
(3.2)

Here p is the magnitude of a system’s average (classical) momentum, which is also a half-width for a (spatial) frequency distribution; v is the system’s speed.

These kinds of constraints are sometimes referred to as uncertainty bounds, but they in no way preclude precise finite-state evolution. Given rates of change, these bounds define ideal (minimum achievable) average energy and momentum for finite state systems, emulated as efficiently as possible (with no wasted motion or state) by perfectly-tailored quantum hamiltonians [3, 5].

Clearly there can never be more overall spatial change \(\mu \) than total change \(\nu \) in a physical evolution: this is reflected in \(pv/E = (v/c)^2\). From this and (3.2),

$$\begin{aligned} E\ge (h\mu /2)/(v/c)^2\,. \end{aligned}$$
(3.3)

Thus for a given rate \(\mu \) of overall motional change, E can only attain its minimum possible value if the motion is at the speed of light; then no energy is invested in rest-frame dynamics (rest energy). In a finite-state dynamics with several geometrically related signal speeds, to minimize all energies (3.3) the fastest signals must move at the speed of light. If we then want to realize the dynamics running faster, we must put the pieces of the system closer together: we can increase p in (3.2), but not v. Of course in finite-state models of particular physical systems, realistic constraints on speeds and separations may require higher energies.

These bounds can be used to define ideal local energies and momenta for some invertible lattice dynamics, determined by rates of distinct change.

3.1.2 Local Change

We restrict our attention to finite-state lattice dynamics that emulate the locality, uniformity and microscopic invertibility of physical law: invertible cellular automata (CA). We assume the dynamics is defined as a regular arrangement of invertible interactions (logic gates), repeated in space and time, each of which independently transforms a localized set of state variables.

This kind of CA format, where the state variables are always updated in independent groups, has sometimes been called partitioning CA, and encompasses a variety of lattice formats that have been used to model physical dynamics [6–13]. It is interesting that all globally invertible CA can be recast in this physically realistic format, as a composition of independent invertible interactions, even if the CA was originally defined as a composition of non-invertible operations on overlapping neighborhoods [14–16]. Historically, CA originated as physics-like dynamics without invertibility [17–19].

Now, in the energy bounds above, only rates of change matter, not the amount of state updated in a single operation. This is unrealistic. We can define a large-scale synchronous dynamics, where the global rate of state change is independent of the size of the system. Physically, total energy must be bounded by the total rate of local changes, since each independent local update also obeys an energy bound. We resolve this conflict by allowing synchronous definition, but counting the global average rate of distinct change as if local updates were non-synchronous—which would in fact be true in most relativistic frames.

There is also an issue of what not to count. For a dynamics defined by a set of gate operations, it might seem natural to include, in the minimum, energy required to construct the gates and to turn them on and off. This is the energy needed to construct a perfectly-tailored hamiltonian. Here we ignore this construction energy, and discuss the ideal case where the hamiltonian is given for free (as part of nature), and we only need to account for energy required by state change within the dynamics.

3.1.3 Two Examples

In the remainder of this paper, we introduce and discuss two 2\(\times \)2 block partitioning CA (cf. [20]). These dynamics are isomorphic to classical mechanical systems, and are simple enough that it is easy to compare energetic quantities, defined by local rates of state change, with classical ones.

The first example is a scalable CA version of the Soft Sphere Model [21], which is similar to Fredkin’s classical mechanical Billiard Ball Model [22]. This digital system emulates the integer time behavior of an idealized classical mechanical system of elastically colliding balls, and is computation universal. The CA is scalable in that square blocks of ones (balls) of any size can be collided to simulate a billiard ball computation. This model has not been published before.

The second example is a CA model of an elastic string that exhibits simple-harmonic motion and exactly emulates the continuum wave equation at integer times, averaged over pairs of adjacent sites. This model has been discussed before [7, 23–25], but the analysis of overall translational motion, ideal energy, and their relativistic interpretation, have not been previously published.

3.2 Scalable Soft Sphere CA

Many CA dynamics can be interpreted as the integer-time behavior of a continuous classical mechanical system, started from an exactly specified initial state. This is true, for example, for lattice gas models of fluids. Such stroboscopic classical mechanical CA inherit, from their continuous counterparts, conserved quantities such as energy and momentum that we can compare to ideal quantities determined by local rates of state change. Of course the continuum models we have in mind would be numerically unstable if actually run as continuous dynamics, but this issue is not inherited by the finite-state CA [26 ] .

A famous stroboscopic dynamics of this sort is Fredkin’s billiard ball model of computation, in which hard spheres moving in a plane, each with four possible initial velocities, are restricted to a square lattice of initial positions. At each integer time, the system is again in such a configuration. To guarantee this property without additional restrictions on initial states, we let billiard balls pass through each other in some kinds of collisions, without interacting .

Figure 3.1 shows a variant of this model in which the balls are much more compressible, so collisions deflect paths inward rather than outward. This variant has the advantage that it is more directly related to a simple partitioning CA (cf. [6]). In the collision illustrated in Fig. 3.1a balls enter from the left with a horizontal component of velocity of one column per time unit, so consecutive moments of the history of a collision occur in consecutive columns.

Fig. 3.1
figure 1

Scalable Soft Sphere dynamics. a Stroboscopic view of continuous classical collision, one time step per column. b Finite state collision, with particles drawn lighter at odd time-steps. c 2D partitioning rule. Only these cases (and their rotations) interact. Otherwise, all particles move diagonally, unchanged. d 1D version of the rule

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The collision shown is energy and momentum conserving, and compression and rebound take exactly the time needed to displace the colliding balls from their original paths onto the paths labeled AB. If a ball had come in only at A with no ball at B, it would have left along the path labeled \(A\bar{B}\): the collision acts as a universal logic gate.

Figure 3.1b shows a realization of the collision as a simple partitioning dynamics. Each time step in (a) corresponds to two in (b), and again particles are shown at each integer time—drawn dark at even times and light at odd. The rule (c) is inferred from (b), interpreting that diagram as showing the positions of two streams of colliding particles at one moment (dark), and their positions at the next moment (light). All particles move diagonally across a block, unchanged, in the cases not defined explicitly in (c). If this rule is applied to just the dark particles in each of the dark-bordered \(2\times 2\) blocks in (b), ignoring the light particles, it moves them to the light positions; applied to just the light particles in each of the light-bordered blocks, it moves them to the dark positions. The dynamics alternately applies the rule to these two partitions. To also allow collisions like (b) for streams of balls arriving from the right, top, or bottom, we define the rule (c) to have discrete rotational symmetry: in each of the cases shown in (c), each of the four \(90^\circ \) rotations of the pattern on the left turns into the corresponding rotation of the pattern on the right.

Note that (b) can also be interpreted as showing a time history of a collision of two particles in a one-dimensional partitioning dynamics (the center of mass dynamics). Then we get the rule (d), with the cases not explicitly shown interchanging the two cell values. Three dimensional versions of the dynamics can be constructed as in [21].

It is not surprising that a time-independent continuous dynamics turns into a time-dependent discrete partitioning. In the continuous model, balls approach a locus of possible collision, interact independently of the rest of the system, and then move away toward a new set of loci. The partitions in the continuous case are just imaginary boxes we can draw around places within which what happens next doesn’t depend on anything outside, for some period of time. Thus it is also not surprising that we can assign a conserved energy to partitioning dynamics.

3.2.1 Ideal Energy and Momentum

For a physical realization of the SSS dynamics, let \(\tau \) be the time needed for gate operations to update all blocks of one partition, and let \(v_0\) be the average speed at which the physical representation of a fastest-moving particle travels within the physical realization of the CA lattice (assuming discrete isotropy).

Equations (3.2) and (3.3) define an ideal (minimum) momentum and energy for a block in which there is a distinct overall spatial change and direction of motion. Clearly these ideal quantities are conserved overall in collisions, since freely moving particles move diagonally at \(v_0\) before and after. Are they also conserved in detail during collisions?

When two freely moving particles enter a single block in the collision of Fig. 3.1b, the number of block changes is reduced: one instead of two. The ideal magnitude of momentum for each freely moving particle before the collision is \(p_1=(h/2\tau )/v_0\). For two colliding particles moving horizontally within a block the ideal is \(p_2=(h/2\tau )/(v_0/\sqrt{2})=2 p_1/\sqrt{2}\), which is the same as the net horizontal momentum before the collision. Ideal energy is similarly conserved.

Note, however, that the separate horizontal motions of the \(+\) and − particles during the next step of the collision of Fig. 3.1b imply an increase in the minimum energy and momentum for that step. This effect becomes negligible as we enlarge the scale of the objects colliding.

3.2.2 Rescaling the Collision

If two columns of k particles are collided in the SSS dynamics, then the resulting collision just shifts the output paths by k positions along the axis of the collision. This is illustrated in Fig. 3.2a for \(k=3\). Thus \(k\times k\) blocks of particles collide exactly as in the classical collision of Fig. 3.1a: the SSS CA can perform logic with diagonally-moving square “balls” of any size. When two balls of equal size meet “squarely,” moving together along a horizontal axis, each pair of columns evolves independently of the rest; colliding along a vertical axis, pairs of rows evolve independently. Square balls can participate in both kinds of collisions.

Fig. 3.2
figure 2

Multiple collisions in the SSS dynamics. a Colliding columns of particles are displaced horizontally by the height of the column. b Each column is slowed down by the collision. c The fraction of energy that is mass during a collision decreases with increasing initial particle speed \(v_0\) (from top, 20 % of c, 40 %, 60 %, 80 %, 100 %)

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During such a collision, from the blocks that change we can infer a net momentum and hence a velocity for the motion of each colliding ball: Fig. 3.2b illustrates this for \(k=100\), with the time unit being the time for a freely moving \(k\times k\) ball to travel its length (and width). Looking at just the changes in the top half of (a), we determine the magnitude and direction of minimum average momentum for each block that changes using (3.2), and hence determine a total momentum. Half of the conserved total energy is associated with each ball, so gives the magnitude of velocity of a ball as a function of time, as number and type of changes evolve. This is plotted in (b).

The fraction \(1/\gamma \) of the total energy E that is mass energy depends on \((v/c)^2 = (v/v_0)^2(v_0/c)^2\). Thus given (b), it depends on an assumption about the value of \(v_0/c\). The fraction \(1/\gamma \), as a function of time in the \(k=100\) collision, is plotted in Fig. 3.2c under different assumptions. The bottom case, \(v_0=c\), has the greatest range but the smallest value at all times. The top case is \(v_0=.2c\). As expected, the faster the speed of the fastest signals, the less the energy tied up in mass, hence the smaller the total energy. Ideally, \(v_0=c\).

3.3 Elastic String CA

In this second example we discuss a classical finite state model of wave motion in an elastic string. This stroboscopic classical mechanical model exactly reproduces the behavior of the time-independent one-dimensional wave equation sampled at integer-times and locations. As in the SSS example, a continuous model is turned into a finite state one by restricting the initial state (in this case the initial wave shape) to a perfect discrete set of possible initial configurations, and this constraint reappears at each integer time. In the continuum limit the discrete constraint on the wave shape disappears; the exactness of the wave dynamics itself (at discrete times) is independent of this limit.

The elastic string CA uses partitioning, but in a different way than the SSS CA: here the partitioning actually constrains the continuous classical dynamics used to define the CA, but in a way that never affects the classical energy. In the SSS case, the time dependence associated with the partitioning completely disappears in the continuous classical-mechanical version of the dynamics.

3.3.1 Discrete Wave Model

Consider an ideal continuous string for which transverse displacements exactly obey the wave equation. Figure 3.3a shows an initial configuration with the string stretched between equally spaced vertical bars. The set of initial configurations we’re allowing are periodic, so the two endpoints must be at the same height.Footnote 1 Any configuration is allowed as long as each segment running between vertical bars is straight and lies at an angle of \(\pm 45^\circ \) to the horizontal.

Initially the string is attached at a fixed position wherever it crosses a vertical bar. We start the dynamics by releasing the attachment constraint at all of the gray bars. The attachment to the black bars remains fixed. In Fig. 3.3b the segments that are about to move are shown with dotted lines: the straight segments have no tendency to move. Under continuum wave dynamics, the dotted segments all invert after some time interval \(\tau \). This will be our unit of time for the discrete dynamics. The new configuration at the end of this interval is shown in Fig. 3.3c, with segments that have just moved dotted. At this instant in time all points of the string are again at rest and we are again in an allowed initial configuration. Now we interchange the roles of the black and gray bars and allow the segments between adjacent gray bars to move for a time interval \(\tau \). The dynamics proceeds like this, interchanging the roles of the black and gray bars after each interval of length \(\tau \). Since attachments are always changed at instants when all energy is potential and the string is not moving, the explicit time dependence of the system doesn’t affect classical energy conservation.

Fig. 3.3
figure 3

Discrete wave dynamics. Elastic string is held fixed where it crosses black bars

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We express this dynamics as a purely digital rule in Fig. 3.4. In Fig. 3.4a we show a wave with the black bars marking the attachments for the next step. To simplify the figure we have suppressed the gray bars—they are always situated midway between the black bars and so don’t need to be shown. We have also added a grid of \(45^\circ \) dotted lines that shows all of the segments that the string could possibly follow. In Fig. 3.4b we add in horizontal black bars, in order to partition the space into a set of \(2\times 2\) blocks that can be updated independently. Note that in all cases the segments that are allowed to change during this update step, as well as the cells that they will occupy after the update, are enclosed in a single block. The long box below Fig. 3.4b contains just the slope information from the string. This array of gradients is clearly sufficient to recreate the wave pattern if the height at one position is known. This is not part of the 2D dynamics: it will be discussed as a related 1D dynamics.

Fig. 3.4
figure 4

Discrete wave dynamics. a A wave configuration. Possible wave paths are indicated by dotted lines. b Horizontal and vertical lines indicate one of two partitions used for discrete update rule. A 1D array summarizing wave gradients is shown below (not part of the 2D dynamics). c Top, dynamical rule for 2D wave. Presence of wave-path segments is indicated by 1’s. Bottom, equivalent 1D dynamical rule for gradients

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Figure 3.4c shows the dynamical rule for a block. Since the dotted lines indicate the direction in which segments must run if they appear in any cell, the state information for each segment is only whether it is there or not: this is indicated with a 1 or a 0. The only segments that change are peaks or valleys , and these are represented by two 1’s at the top of a block or at the bottom of a block respectively. The rule is that peaks and valleys turn into each other, and nothing else changes. We apply the rule alternately to the blocks shown, and to a complementary partition shifted half a block horizontally and vertically.

3.3.2 Exact Wave Behavior

At the bottom of Fig. 3.4c we’ve presented a dynamics for the gradients of the wave. The full 2D dynamics just turns peaks into valleys and vice versa, leaving straight segments unchanged: we can do that equally well on the array of gradients. As the 2D dynamics interchanges which blocking to use, the dynamics on the gradients also alternates which pairs of gradients to update together. In all cases, the dynamics on the gradients duplicates what happens on the string: if the two dynamics are both performed in parallel, the gradient listed below a column will always match the slope of the string in that column.

The dynamics on the gradients has an interesting property. Turning a peak into a valley and vice versa is exactly the same as swapping the left and right elements of a block. Leaving a or \({ \backslash \backslash }\) unchanged is also exactly the same as swapping the left and right elements of a block. In all cases, the dynamics on the gradients is equivalent to a swap.

This means that the left element of a block will get swapped into the right position, and at the next update it will be the left element of a new block and will again get swapped into the right position, and so on. Thus all of the gradients that start off in the left side of a block will travel uniformly to the right, and all that start in the right side of a block will travel uniformly to the left.

This shows that the system obeys a discrete version of the wave equation. Half of the gradients constitute a right-going wave, and half constitute a left-going wave. At any step of the dynamics, the 2D wave in the original dynamics is just the sum of the two waves: it is reproduced by laying gradients end to end.

If we refine the lattice, using more and more cells to represent a wave of given width, smoother and smoother waves can be represented. Of course even without going to a large-scale limit, the CA dynamics is already exactly equivalent to a continuous wave equation with constrained initial wave shapes, sampled at integer times: simply stretch the rightgoing and leftgoing waves constructed out of gradients to the full width of the lattice. This just amounts to drawing the wave shape corresponding to each block of the current partition a little differently.

3.3.3 Overall Transverse Motion

Assume the string carrying a discrete wave wraps around the space. We’ve discussed the horizontal motion of waves along such a string, but the string itself can move vertically. For example, a pattern such as all the way around the space reproduces itself after two partition update steps, but shifted vertically by two lattice units. This is clearly the maximum rate of travel for a string: one position vertically per update step. Call this \(v_0\).

We can express the net velocity of the string in terms of the populations of rightgoing and leftgoing gradient segments. Let be the number of rightgoing segments with slope \(+1\) (rightgoing /’s), and similarly for , \(L\!\!_{\texttt { +}}\) and . If the width of the space is B blocks, then there are segments forming the rightgoing wave, and forming the leftgoing one.

For the rightgoing or leftgoing wave, periodically repeating its sequence of gradients corresponds to an unbounded wave with the same average slope. When both waves have shifted horizontally the width of one period (after 2B partition update steps), the net vertical shift is the sum of the slopes of the leftgoing gradients, minus the sum for the rightgoing ones: . We can compute this by summing the differences for each pair of slopes grouped together in the columns of one partition. Only columns containing or contribute a non-zero difference, and so we only need to count the numbers of blocks and that are about to change, to compute the constant velocity

(3.4)

3.3.4 Ideal Energy and Momentum

Only blocks that change have overall motion, and with \(\tau \) the time taken to update one partition, the frequencies of positive and negative motion are and . Thus from (3.2), attributing a momentum to each changing block, the total ideal momentum up is , and down is , so the net ideal momentum . From (3.4), the corresponding relativistic energy is \(E=c^2 p/v = (h B/2\tau )/(v_0/c)^2\). Letting \(v_0 \rightarrow c\) to minimize energy, and choosing units with \(h=2\) and \(c=1\) and \(\tau =1\), this becomes

(3.5)

Energy is the constant width (in blocks) of the string, and momentum is the constant net number of blocks moving up.

There is an interesting subtlety involved in letting \(v_0\rightarrow c\) in the 2D dynamics. We interpret all gradient segments as always moving, swapping in pairs in each update in order to recover the wave equation—even though some paired segments are in different blocks when they “swap” identical values. If all block motion forward or backward is at the speed c, each segment must be interpreted as traveling at the speed \(c\sqrt{2}\) as it swaps diagonally. If instead we interpret segments as moving up and down (or not moving), none travel faster than light, but the interaction is non-local at the scale of an individual block.

3.3.5 Rest Frame Energy

For the transverse motion of the string to approach the maximum speed, almost all of the block updates must contribute to overall motion, and almost none to just internally changing the string. This slowdown of internal dynamics is a kind of time dilation, which is reflected in the rest frame energy \(\sqrt{E^2 - p^2}\). From (3.5),

(3.6)

The energy \(E_r\) available for rest-frame state-change decreases as more blocks move in the same direction. In this model total energy E is independent of v, hence rest energy \(E_r= E/\gamma \) must approach 0 as \(1/\gamma \rightarrow 0\). This contrasts with a normal relativistic system that can never attain the speed of light, which has a constant rest energy \(E_r\) and a total energy E that changes with v.

The analysis up to here applies equally well to both the 1D and 2D versions of the dynamics of Fig. 3.4. In 2D, however, there is an additional constraint: there must be an equal number B of positive and negative slopes, so that the string meets itself at the periodic boundary. Since there are also an equal number B of right and left going gradients, and . Thus from (3.4) and (3.6),

(3.7)

If were the probability for a walker to take a step to the right, and the probability to the left, then (3.7) would be the standard deviation for a 2B-step random walk. Related models of diffusive behavior that make contact with relativity are discussed in [24, 27, 28]. None of these define relativistic objects that have an internal dynamics, however.

3.4 Discussion

Given the definition of a finite-state dynamics, we could try to assign intrinsic properties to it based on the best possible implementation. For example, programming it on an ordinary computer, a basic property is the minimum time needed, on average, to simulate a step of the dynamics. It would be hard, though, to be sure we’ve found the most efficient mapping onto the computer’s architecture, and the minimum time would change if we used a different computer, or built custom hardware using various technologies. The true minimum time would correspond to the fastest possible implementation allowed by nature! Such a definition seems vacuous, though, since we don’t know the ultimate laws of nature, and even if we did, how would we find the best possible way to use them?

Surprisingly, a fundamental-physics based definition of intrinsic properties is not in fact vacuous, if we base it on general principles. Assuming the universe is fundamentally quantum mechanical, we couldn’t do better than to simply define a hamiltonian that exactly implements the classical finite-state dynamics desired at discrete times, with no extra distinct states or distinct state change. This ideal hamiltonian identifies the fastest implementation that is mathematically possible, with given average energy.

This procedure assigns to every invertible finite-state dynamics an ideal energy that depends only on the average rate of distinct state change. This is generally not much like a physical energy, though, since we haven’t yet included any realistic constraints on the dynamics. For example, each state change might correspond to a complete update of an entire spatial lattice, as in the synchronous definition of a CA. Then the energy would be independent of the size of the system. We can fix this by constraining the finite-state dynamics to be local and not require synchrony: defining it in terms of gates that are applied independently.

We expect the ideal energy, and distinct portions of it, to become more realistic with additional realistic constraints. For this reason, we studied invertible lattice dynamics derived from the integer-time behavior of idealized classical mechanical systems. In the examples we looked at, ideal energies and momenta defined by local rates of state change evolve like classical relativistic quantities.

It seems interesting and novel to introduce intrinsic definitions of energy and other physical quantities into classical finite-state systems, and to use these definitions in constructing and analyzing finite-state models of physical dynamics. Since all finite-energy systems in the classical world actually have finite state, and since classical mechanics doesn’t, this may be a productive line of inquiry for better modeling and understanding that world. Moreover, inasmuch as all physical dynamics can be regarded as finite-dimensional quantum computation, finite-state models of classical mechanics may play the role of ordinary computation in understanding the more general quantum case.