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Pressure from collisions

Pressure emerges from momentum transferred to a wall per unit time and area. In a particle simulation, it is an event reduction over wall impacts, not an arbitrary property attached to a particle cloud. The averaging window and wall orientation are part of the result.

Pressure from collisions in Python

Read the relationship as a short program. Begin with measured or chosen values, keep only the state needed for the next calculation, apply the stated rule, and inspect a result that can be checked independently.

import numpy as np

impulses_ns = np.array([0.02, 0.03, 0.01])
wall_area_m2, duration_s = 0.1, 0.5
pressure = impulses_ns.sum() / (wall_area_m2 * duration_s)
print(f"Mean compressive pressure: {pressure:.3f} Pa")
\[p=\frac{\Delta p_\perp}{A\Delta t}\]

Impulse has units kg·m/s, so impulse divided by time is force and then divided by area is pascals. Tangential momentum transfer relates to shear stress, not the normal pressure estimate. A collision resolver should return the impulse it applied to each body; the wall receives the opposite impulse by Newton's third law. Logging this event avoids inferring a noisy time derivative from only particle positions.

At equilibrium, many short-window estimates fluctuate because collision times are discrete. Increase duration, area, or ensemble size to reduce statistical noise, but report the choice: averaging can hide genuine transients. Compare pressure estimates across windows only when wall area, orientation, interaction law, particle number, and units are compatible.

A useful check starts with a prediction: More frequent or harder normal wall collisions raise average pressure, while tangential impulse does not enter the normal-pressure reducer. Specify an event record with time, wall id, normal, and impulse. Write a reducer for normal impulse over one interval, then double accumulated impulse while holding area/duration fixed. State what a short window can change in the estimate even if the long-time gas state is unchanged.

The explanation above has deliberate limits. A tiny averaging window is noisy. Wall models, particle size, finite contact time, intermolecular forces, non-equilibrium flows, and spatial gradients can make simple ideal-gas interpretations inaccurate. A particle simulation also needs a separate temperature/energy model before pressure statistics can justify thermodynamic claims.

One particle in a rectangular box

Take a particle of mass \(m\) moving between two walls separated by \(L_x\). An elastic hit on a wall reverses its normal velocity, so the particle momentum change has magnitude

\[|\Delta p_x|=2m|v_x|\]

The time between hits on the same wall is \(2L_x/|v_x|\). The average force magnitude contributed to that wall is therefore

\[\overline F_x=\frac{2m|v_x|}{2L_x/|v_x|}=\frac{mv_x^2}{L_x}\]

If the wall area is \(A\) and \(V=AL_x\), the pressure contribution is \(mv_x^2/V\).

Many particles and the ideal gas law

Adding independent particle contributions gives

\[pV=\sum_i m_i v_{x,i}^2\]

At isotropic equilibrium, the three Cartesian mean-square components are equal, so

\[pV=\frac13\sum_i m_i v_i^2=\frac23K_{\mathrm{trans}}\]

Combining this with

\(K_{\mathrm{trans}}=3Nk_BT/2\)

produces \(pV=Nk_BT\). Pressure depends on particle number density and temperature, not on one representative collision.

The factor one third assumes an isotropic three-dimensional distribution. A driven gas can have unequal normal stresses on different walls, so averaging them prematurely hides nonequilibrium behavior.

Signed collision-event reduction

Let \(\widehat{\mathbf n}\) point out of the gas domain. If a collision resolver returns impulse on the wall \(\mathbf J_{\mathrm{wall}}\), its outward-normal projection is compressive with a convention-dependent sign. Preserve that sign internally and convert to a positive pressure magnitude only at the reporting boundary.

events = [
    {"wall": "right", "time": 0.1, "impulse": np.array([0.02, 0.0])},
    {"wall": "left", "time": 0.2, "impulse": np.array([-0.03, 0.0])},
    {"wall": "right", "time": 0.3, "impulse": np.array([0.01, 0.0])},
]
normal = np.array([1.0, 0.0])
selected = [e for e in events if e["wall"] == "right" and 0 <= e["time"] < 0.5]
impulse = sum(np.dot(e["impulse"], normal) for e in selected)
print("Right-wall pressure (Pa):", impulse / (wall_area_m2 * duration_s))

Use a half-open time interval so an event on a shared boundary is not counted in two adjacent windows. Require positive finite area and duration. Tangential impulse belongs to shear stress, not normal pressure.

Units and absolute pressure

Impulse divided by time is force, and force divided by area is pressure:

\[\frac{\mathrm{kg\,m/s}}{\mathrm{s\,m^2}}=\frac{\mathrm N}{\mathrm{m^2}}=\mathrm{Pa}\]

The collision calculation gives pressure exerted by the modeled particles on the wall. Gauge pressure subtracts a reference environmental pressure; absolute pressure does not. The ideal-gas law uses absolute pressure. A simulation with vacuum outside its walls should not subtract atmospheric pressure unless that exterior is explicitly part of the model.

Statistical windows

Wall impacts are discrete, so short-window estimates fluctuate and can even be zero between collisions. Increasing duration or wall area collects more events and usually reduces relative noise, but it also smooths genuine transients. Report event count and window limits with every pressure value.

Successive windows can remain correlated. Use independent runs or blocks longer than the collision-correlation time when estimating uncertainty. Pressure from six walls of a finite box provides a useful isotropy comparison, not six automatically independent measurements.

Interacting particles and the virial contribution

For particles with pair forces in a periodic equilibrium volume, pressure includes interparticle forces. A common three-dimensional virial form is

\[pV=Nk_BT+\frac13\left\langle\sum_{i<j}\mathbf r_{ij}\cdot\mathbf F_{ij}\right\rangle\]

The exact implementation depends on boundary, cutoff, long-range treatment, and force convention. The ideal-gas expression is recovered when interaction forces vanish. A hard collision model can estimate pressure from impulses; a smooth-potential periodic model usually uses a virial estimator because there may be no physical wall.

Verification

For one particle in a one-dimensional box, compare accumulated impulse over many round trips with the analytic average force. Double its normal speed: impulse per collision and collision rate both double, so pressure contribution increases by four. Double the number of otherwise identical independent particles at fixed volume and velocity distribution: pressure doubles.

Reverse the stored wall normal and confirm the signed projection reverses before any positive-pressure presentation. Use a collision containing only tangential impulse and require zero normal contribution. Sum all wall impulses with particle momentum change to test Newton's third law.

At equilibrium, compare the long-time collision estimate with \(Nk_BT/V\) and check convergence as event count grows. For an interacting model, compare wall and virial estimators when both are valid.

Finite particle size, attractive or repulsive interactions, inelastic walls, moving boundaries, gradients, and quantum statistics change the simple derivation. The central definition—normal momentum transfer per time and area—remains, but the appropriate estimator must match the actual model.

A worked Python notebook

Reduce a wall's compressive impulse over area 0.1 square metres and duration half a second. The question for this calculation is precise: how does mean pressure change when we change accumulated normal impulse, 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 time window and chosen wall normal must be stated before impulses are summed. 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 0.1 N 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 accumulated normal impulse, expressed in N s. The output is mean pressure, expressed in Pa. 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.

\[f(q) = 20.0 q\]

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 0.06 N s, the expression gives 1.2 Pa. 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("q/(0.1*0.5)", locals={"q": q})
reference_input = 0.06
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "N s")
print("Predicted mean pressure:", reference_output, "Pa")

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 Pa. The midpoint, at 0.05 N s, gives 1 Pa. The upper endpoint gives 2 Pa. Read these together with the assumptions: they describe different choices of accumulated normal impulse 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, 0.05, 0.1])
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} N s -> {output_value:.6g} Pa")
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 accumulated normal impulse; the vertical axis shows mean pressure. 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, 0.1, 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("Accumulated normal impulse (N s)")
ax.set_ylabel("Mean pressure (Pa)")
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 mean pressure change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.06 N s is 20 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 20.0\]

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 = 0.0605
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, "Pa")
print("Full expression:", full_prediction, "Pa")
print("Approximation difference:", full_prediction - linear_prediction, "Pa")

Here the changed input is 0.0605 N s. The tangent estimate is 1.21 Pa, while the full expression gives 1.21 Pa. 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 = 0.1
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 accumulated normal impulse inside the stated interval, predict whether mean pressure 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 time window and chosen wall normal must be stated before impulses are summed. 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 Pa: 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(0.1) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.1, 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.

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