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Computational arrow of time

A computer program naturally has an execution order: read a state, calculate a later state, and overwrite or store the result. That order is a computational arrow of time, but it is not automatically the thermodynamic arrow of the modeled system. Forward-only code can represent reversible physics, and reversible-looking code can hide physical dissipation in an omitted environment.

The useful question is not merely whether a program can run backward. It is which information the physical law, numerical update, and data pipeline each preserve or discard.

An update index is not a physical arrow

A discrete calculation often has the form

\[x_{n+1}=F_{\Delta t}(x_n)\]

Here \(x_n\) is the complete stored state, \(\Delta t\) is the modeled time interval, and \(F_{\Delta t}\) is the numerical update.

The increasing index \(n\) records execution order. It does not prove entropy production. A program for ideal projectile motion also increments an index even though the underlying conservative mechanics is time-reversal symmetric.

To discuss reversibility, ask whether an inverse map exists and whether it corresponds to the physical reversal. If

\[F_{\Delta t}^{-1}(F_{\Delta t}(x))=x\]

then the stored update is mathematically invertible over its state space. Time symmetry usually asks for the stronger physically matched relation

\[F_{-\Delta t}=F_{\Delta t}^{-1}\]

An update can be invertible without satisfying this relation.

Forward Euler shows the difference

For a harmonic oscillator with angular frequency \(\omega\), forward Euler updates position and velocity as

\[x_{n+1}=x_n+v_n\Delta t, \qquad v_{n+1}=v_n-\omega^2x_n\Delta t\]

The update matrix has determinant

\[\det F_{\Delta t}=1+\omega^2\Delta t^2\]

The determinant is nonzero, so the exact algebraic matrix is invertible. However, replacing the time step by its negative does not give the inverse. Applying one positive step and then one negative step enlarges the state rather than restoring it. The method also produces systematic energy growth for this oscillator.

This example separates three ideas: invertibility, time symmetry, and physical accuracy. They are not synonyms. A symplectic, time-symmetric method such as velocity Verlet is usually a better reversible-mechanics reference, although finite-precision arithmetic can still prevent bit-for-bit recovery.

Many-to-one operations create information loss

A map is not invertible when different input states produce the same output. Rounding two nearby coordinates to the same stored number is many-to-one. A histogram discards particle identities and exact positions. Saving every hundredth step discards the intermediate trajectory. Rendering state as pixels discards far more than it displays.

These operations may be sensible, but their outputs answer narrower questions. A video can show qualitative motion; it cannot replace the full state needed to verify energy conservation or reverse a trajectory.

Loss occurs in layers:

import numpy as np

states = np.array([0.12341, 0.12344])
rounded = np.round(states, 3)
assert states[0] != states[1] and rounded[0] == rounded[1]
print("Distinct states map to the same displayed value:", rounded)

Every arrow should have a documented rule. If the raw layer is deleted, the reduced layer cannot later recreate distinctions it never retained.

Diffusion is easy forward and unstable backward

Macroscopic diffusion provides a strong example of a directional computation. A Fourier mode with wave number \(k\) under the diffusion equation decays as

\[A_k(t)=A_k(0)e^{-Dk^2t}\]

Here \(A_k\) is the mode amplitude and \(D\) is positive. Fine-scale modes have large wave numbers and decay fastest, so the field becomes smooth.

Formally reconstructing the initial mode requires

\[A_k(0)=A_k(t)e^{Dk^2t}\]

The reverse factor amplifies tiny measurement and roundoff errors most strongly at high wave number. The backward heat problem is therefore ill-conditioned: a mathematical expression exists, but finite data cannot recover the lost fine structure reliably.

This macroscopic equation already assumes coarse-grained local variables. Its practical irreversibility should not be confused with a claim that every underlying molecular collision violates time-reversal symmetry.

Chaos changes reproducibility requirements

In a chaotic model, tiny differences in initial state, floating-point order, or time step can grow until two microscopic trajectories disagree. Bitwise reproducibility is useful for auditing one implementation, but it is not the only scientific target.

Three levels should be distinguished:

A parallel sum may change floating-point addition order and alter the last few bits. Chaotic dynamics can amplify that difference. The later trajectory mismatch is not automatically a new physical effect. Compare conserved quantities, distributions, and declared observables as well as exact state hashes.

Randomness has hidden state

A pseudorandom generator is a deterministic state machine. A seed selects its initial state, but reproducibility also depends on the generator algorithm and the order in which random values are consumed. Adding one draw in an unrelated code path can shift every later result.

For checkpoint replay, save the complete generator state at the checkpoint, not only the original seed. External randomness from hardware, operating-system entropy, network events, or user input must be recorded if exact replay is required.

Replaying a random stream reproduces a sampled path. It does not make that path statistically independent evidence, and it does not invert a many-to-one stochastic transition.

Checkpoints localize divergence

Saving every state may be too expensive, while saving only the final image makes a failure difficult to diagnose. Checkpoints create a middle path. Each checkpoint can include a state hash, physical diagnostics, numerical settings, and random-generator state.

import hashlib


def checkpoints(seed):
    rng = np.random.default_rng(seed)
    x, saved = 0.0, []
    for step in range(100):
        x += rng.normal()
        if step % 10 == 0:
            digest = hashlib.sha256(np.float64(x).tobytes()).hexdigest()
            saved.append((step, x, digest))
    return saved


assert checkpoints(42) == checkpoints(42)
print("Replay matched all checkpoints in this environment.")

Hashes detect any byte-level difference but do not measure physical importance. Diagnostic tolerances must use units and scales. A tiny position difference and a large energy violation should not be collapsed into one unexplained boolean.

Storage policy creates an observational arrow

Overwriting the previous state is often sufficient to advance a calculation, but it makes the trajectory unrecoverable. Append-only event logs preserve history but can become large. Checkpointing plus deterministic replay trades storage for computation.

Compression can be lossless or lossy. A lossless codec restores the exact stored bytes; a lossy codec cannot. Even lossless storage of sampled checkpoints omits unsaved times. Descriptions should name both compression type and sampling cadence.

Derived data should be reproducible from a versioned recipe. Store plot axes, bin edges, filter settings, normalization rules, and source checkpoint identifiers. Inferring measurements back from pixels creates avoidable uncertainty.

Computational loss is not thermodynamic entropy

Deleting a file or rounding a coordinate reduces the information available to the analyst, but it is not by itself a calculation of thermodynamic entropy production. Thermodynamics requires a physical system boundary, reservoirs, temperatures, energy transfers, and an ensemble interpretation.

Likewise, deterministic replay does not validate the physical model. A perfectly reproducible program can implement the wrong force law. Reproducibility makes errors traceable; independent physical checks establish accuracy.

For a thermodynamic claim, use the total entropy balance

\[\Delta S_{\mathrm{total}} =\Delta S_{\mathrm{system}} +\Delta S_{\mathrm{surroundings}}\]

Then identify which physical interaction changes each term. Do not substitute storage bytes, state hashes, or visual complexity for entropy in joules per kelvin.

Evidence for an accountable computation

Begin with a small case having an analytic solution or conserved quantity. Confirm that the implementation approaches that result as the step is reduced. For reversible mechanics, test the reversal operation over one step before a long trajectory. Introduce rounding or downsampling only after the lossless reference works.

Record the first checkpoint where a replay differs. Determine whether the cause is an input change, generator-state mismatch, update-order change, dependency change, or platform-dependent arithmetic. Then ask whether the discrepancy changes the physical observable beyond its numerical and statistical tolerance.

The computational arrow is ultimately an audit of transformations. Forward state updates, lossy observation, finite precision, chaotic amplification, and thermodynamic irreversibility may all point in the same practical direction, but they are different mechanisms. Keeping their evidence separate makes both the code and the physics easier to trust.

A worked Python notebook

Compare a display spacing of 0.001 with the magnitude of a stored positive state. The question for this calculation is precise: how does display quantisation scale relative to value change when we change unrounded state value, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

This ratio describes representation; it does not supply a physical law for entropy production. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.01 to 1 dimensionless. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for unrounded state value, expressed in dimensionless. The output is display quantisation scale relative to value, expressed in dimensionless. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{0.001}{q}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.1 dimensionless, the expression gives 0.01 dimensionless. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("0.001/q", locals={"q": q})
reference_input = 0.1
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "dimensionless")
print(
    "Predicted display quantisation scale relative to value:",
    reference_output,
    "dimensionless",
)

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.1 dimensionless. The midpoint, at 0.505 dimensionless, gives 0.0019802 dimensionless. The upper endpoint gives 0.001 dimensionless. Read these together with the assumptions: they describe different choices of unrounded state value within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.01, 0.505, 1.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} dimensionless -> {output_value:.6g} dimensionless")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows unrounded state value; the vertical axis shows display quantisation scale relative to value. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.01, 1.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Unrounded state value (dimensionless)")
ax.set_ylabel("Display quantisation scale relative to value (dimensionless)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does display quantisation scale relative to value change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.1 dimensionless is -0.1 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \frac{0.001}{q^{2}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.10495
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "dimensionless")
print("Full expression:", full_prediction, "dimensionless")
print("Approximation difference:", full_prediction - linear_prediction, "dimensionless")

Here the changed input is 0.10495 dimensionless. The tangent estimate is 0.009505 dimensionless, while the full expression gives 0.00952835 dimensionless. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.99
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of unrounded state value inside the stated interval, predict whether display quantisation scale relative to value will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: This ratio describes representation; it does not supply a physical law for entropy production. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of dimensionless: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(1.0) - evaluate(0.01))
for points in [101, 201, 401]:
    grid = np.linspace(0.01, 1.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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