Irreversibility in simulation
Every numerical model advances from an earlier state to a later state, but forward execution alone is not a physical arrow of time. A model can fail to return to its initial state because the physical process is thermodynamically irreversible, because the microscopic trajectory is chaotic, because the numerical update has no exact inverse, or because information was discarded. Those causes must be separated.
Three meanings of reversible
Microscopic time-reversal symmetry concerns equations of motion. For an isolated classical system with position-dependent conservative forces, reversing all momenta maps a valid trajectory into one that retraces the motion in reverse time.
Numerical reversibility concerns an update algorithm. If a discrete update maps state \(x_n\) to \(x_{n+1}\), it is reversible only if the previous state can be recovered under the declared arithmetic and inverse rule. A good approximation to reversible differential equations need not be exactly invertible after floating-point rounding.
Thermodynamic reversibility is stricter. A reversible thermodynamic process can restore both the system and its surroundings to their initial states through an infinitesimal change in external conditions. It must be quasistatic and free from dissipative effects. A quasistatic process with friction is still irreversible.
Confusing these definitions creates false conclusions. A time-symmetric particle model does not make a rapidly expanding gas thermodynamically reversible, and an irreversible computer operation does not prove that a physical system produced thermodynamic entropy.
Entropy production needs a boundary
For a process involving a system and its surroundings, the second law can be written
\[\Delta S_{\mathrm{universe}} =\Delta S_{\mathrm{system}} +\Delta S_{\mathrm{surroundings}}\ge0\]The equality applies to an ideal reversible process. A macroscopic irreversible process has a positive total change. The system entropy alone can decrease if a larger entropy increase occurs in the surroundings, as in a refrigerator. Therefore a calculation that tracks only visible particles but omits a thermostat or frictional reservoir cannot establish the total entropy balance.
Entropy is a state function. For a reversible path between equilibrium states,
\[\Delta S=\int_i^f\frac{\delta Q_{\mathrm{rev}}}{T}\]The subscript on heat is essential. For an actual irreversible path, dividing the actual heat transfer by temperature does not generally produce the system's entropy change. Instead, choose a hypothetical reversible path connecting the same equilibrium endpoints or use an appropriate state equation.
Free expansion exposes the distinction
Consider an ideal gas initially confined to part of a rigid insulated container. Removing a partition lets it expand into a vacuum. The gas does no boundary work, receives no heat, and has no change in internal energy:
\[Q=0,\qquad W=0,\qquad\Delta U=0\]For an ideal gas, unchanged internal energy means unchanged temperature. Yet the entropy increases:
\[\Delta S=nR\ln\left(\frac{V_f}{V_i}\right)>0\]Here \(n\) is the amount of gas, \(R\) is the gas constant, and \(V_f>V_i\) gives the positive sign. The entropy change is evaluated using a reversible isothermal path between the same endpoints, not by substituting the actual zero heat into the reversible formula.
A particle calculation may preserve every microscopic collision and still show the gas spreading. If every momentum were reversed with impossible precision, the microscopic trajectory could reconcentrate. The reversed state is extraordinarily special among the states compatible with the expanded macrostate, so ordinary uncontrolled preparations do not produce it.
Fine-grained and coarse-grained descriptions
Hamiltonian dynamics preserves phase-space volume. Under ideal exact evolution, the fine-grained Gibbs entropy of a complete probability density remains constant. The density can stretch and fold into structures too fine for a practical observer to resolve.
Coarse-graining groups many nearby microstates into cells or histogram bins. As the fine-scale structure becomes unresolved, the coarse-grained distribution can spread over more cells and its entropy can rise. This describes lost resolution in the macroscopic description, not destruction of the exact microscopic information by the Hamiltonian equations.
A plotted histogram therefore needs its bin boundaries, sampling interval, and state variables. Changing the binning during a run can create an artificial entropy trend. Compare like with like.
Physical dissipation is energy transfer
Friction, viscosity, inelastic deformation, and electrical resistance transform organized macroscopic energy into less organized internal energy. A physical model should conserve total energy when all relevant reservoirs are included, even though the visible mechanical energy decreases.
Deleting a fraction of velocity at every step creates damping, but it does not explain where the removed energy goes. A better accounting records the lost mechanical energy as energy gained by an internal reservoir or an external bath. Momentum exchange also needs a destination when the boundary exerts an impulse.
For a damped mechanical subsystem,
\[\Delta E_{\mathrm{mechanical}} +\Delta E_{\mathrm{internal}} +E_{\mathrm{boundary}}=0\]The signs depend on the chosen boundary convention, so it must be stated. Entropy production additionally depends on temperatures and reservoir changes; energy loss from one displayed variable is not enough.
Numerical sources of irreversibility
Finite-precision arithmetic maps many exact real numbers to one stored value. Rounding is many-to-one and cannot be inverted without extra information. Clipping, filtering, histogramming, image rendering, state compression, and deleting old checkpoints are also lossy.
Common time integrators introduce truncation error. Some, such as properly implemented velocity Verlet for conservative forces, are time-symmetric, while ordinary forward Euler is not. Adaptive step selection can make reversal difficult because the backward run may choose a different step sequence. Floating-point operation order can also change rounding.
Chaotic sensitivity amplifies these small differences. A failed long-time trajectory reversal may therefore indicate tiny numerical discrepancies amplified by the dynamics, not macroscopic entropy production. Conserved quantities and convergence of statistical observables are stronger diagnostics than visual retracing alone.
Random processes require more than a seed
A stochastic update consumes random values. Reusing the same initial seed reproduces a forward stream only if the generator algorithm, state, draw order, and implementation are unchanged. It does not automatically supply the random choices needed for the inverse transition.
If the transition is many-to-one, no seed can reconstruct information that the update discarded. Exact replay needs checkpoints or a log of choices. Statistical reversibility, such as detailed balance in an equilibrium Markov chain, is a statement about transition probabilities and is different from reversing one sampled path.
A layered reverse test
Start with the smallest deterministic case. Test one forward update followed by its declared inverse before attempting a long chaotic trajectory.
import numpy as np
def verlet(state, dt):
x, v = state
new_x = x + dt * v - 0.5 * dt**2 * x
new_v = v - 0.5 * dt * (x + new_x)
return np.array([new_x, new_v])
initial = np.array([1.0, 0.2])
state = initial.copy()
for _ in range(1000):
state = verlet(state, 0.01)
state[1] *= -1
for _ in range(1000):
state = verlet(state, 0.01)
state[1] *= -1
print("Reversal error:", np.linalg.norm(state - initial))
assert np.allclose(state, initial)
Then repeat while adding one potentially irreversible element at a time: rounding, damping, a thermostat, a stochastic collision, or coarse-grained storage. This isolates the first loss rather than attributing every discrepancy to "entropy."
Use a dimensionless recovery measure only after defining physical scales. One option is
\[\epsilon_{\mathrm{rec}} =\frac{\lVert x_{\mathrm{recovered}}-x_{\mathrm{initial}}\rVert} {x_{\mathrm{scale}}}\]State which components enter the norm and how positions, momenta, and other quantities are nondimensionalized. Adding meters directly to momentum units would make the norm meaningless.
Evidence for a physical conclusion
A strong analysis reports the system boundary, initial macrostate, microscopic dynamics, integrator, numerical step, precision, reversal rule, and every modeled reservoir. It checks energy and momentum where they should be conserved and includes the surroundings in the entropy balance.
It also distinguishes three outcomes. Exact state recovery tests the discrete algorithm. Convergence under smaller steps tests numerical approximation. Stable distributions and macroscopic averages across independent preparations test the statistical prediction. Passing one test does not imply the others.
The central lesson is that irreversibility is not a single symptom. Thermodynamic entropy production, coarse-grained information loss, physical dissipation, chaotic amplification, and non-invertible computation can coexist, but they are different claims requiring different evidence.
A worked Python notebook
Use a reversible harmonic trajectory as a reference for a forward-and-reverse computation. The question for this calculation is precise: how does ideal oscillator position change when we change time, 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.
The equation omits damping; failure to recover the initial state can then reveal numerical information loss. 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 to 6 s. 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 time, expressed in s. The output is
ideal oscillator position, expressed in m. 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.
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 2 s, the expression gives -0.234287 m. 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("cos(q)+0.2*sin(q)", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted ideal oscillator position:", reference_output, "m")
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 1 m. The midpoint, at 3 s, gives -0.961768 m. The upper endpoint gives 0.904287 m. Read these together with the assumptions: they describe different choices of time 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([0.0, 3.0, 6.0])
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} s -> {output_value:.6g} m")
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 time; the vertical axis shows ideal oscillator position. 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(0.0, 6.0, 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("Time (s)")
ax.set_ylabel("Ideal oscillator position (m)")
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 ideal oscillator position change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is -0.992527 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.
\[f'(q) = - \sin{\left(q \right)} + 0.2 \cos{\left(q \right)}\]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 = 2.03
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, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")
Here the changed input is 2.03 s. The tangent estimate is -0.264063 m, while the full expression gives -0.263953 m. 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 = 6.0
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 time inside the stated interval, predict whether ideal oscillator position 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: The equation omits damping; failure to recover the initial state can then reveal numerical information loss. 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 m: 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(6.0) - evaluate(0.0))
for points in [101, 201, 401]:
grid = np.linspace(0.0, 6.0, 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.