A Byte of PhysicsLogo

Microscopic vs macroscopic views

A microscopic description follows individual particles and their interactions. A macroscopic description uses a small set of collective quantities such as pressure, volume, temperature, density, and internal energy. The macroscopic state is a deliberate compression: an enormous number of distinct particle arrangements produce the same measured thermodynamic values.

What a microstate contains

For \(N\) classical point particles, a microstate at one instant contains positions and momenta

\[\Gamma=\{\mathbf r_1,\ldots,\mathbf r_N;\mathbf p_1,\ldots,\mathbf p_N\}\]

plus particle identities, masses, internal degrees of freedom, and any field or boundary state required by the model. The Hamiltonian or force rules determine how this state evolves.

Tracking a microstate exactly is not feasible for a macroscopic gas, which may contain on the order of Avogadro's number of particles per mole. Even a simulation with millions of particles represents a tiny sample by laboratory standards. Statistical mechanics asks which collective predictions are stable despite this missing microscopic detail.

Particle labels often have no macroscopic significance. Permuting equal-particle storage does not change density, pressure, or temperature. An implementation whose result depends on array order may contain an asymmetric update or reduction bug.

What a macrostate contains

An equilibrium macrostate can be specified by a small set of state variables. For a fixed amount \(n\) of ideal gas,

\[pV=nRT\]

connects pressure \(p\), volume \(V\), and absolute temperature \(T\). Only two are independent after \(n\) is fixed.

This equation does not identify one microstate. Many arrangements of particle positions and velocities share the same values. That many-to-one map is why a macrostate cannot be used to reconstruct the microscopic trajectory.

Macroscopic variables are reproducible because relative fluctuations become small in a large equilibrium system. They are not perfectly motionless: pressure and energy fluctuate microscopically even when their long-time averages are steady.

Density is a spatial reduction

Whole-system mass density is

\[\rho=\frac{\sum_i m_i}{V}\]

A local density needs a selected cell or smoothing kernel:

\[\rho_{\mathrm{cell}}=\frac{1}{V_{\mathrm{cell}}}\sum_{i\in\mathrm{cell}}m_i\]

Small cells resolve sharp spatial detail but contain fewer particles and fluctuate more. Large cells have smoother statistics but erase gradients. The cell size is therefore a physical resolution choice, not merely a display setting.

import numpy as np

positions = np.array([0.1, 0.3, 0.8, 1.2])
masses = np.ones(4)
velocities = np.array([2.0, -1.0, 1.0, 3.0])
inside = (positions >= 0) & (positions < 1)
cell_mass = masses[inside].sum()
cell_volume = 1.0
flow_velocity = np.sum(masses[inside] * velocities[inside]) / cell_mass
print("Density and mean flow:", cell_mass / cell_volume, flow_velocity)

An empty cell has undefined flow velocity, not zero flow. Code must report insufficient samples rather than divide by zero or invent a physical state.

Temperature removes organized flow

For a monatomic classical ideal gas in three-dimensional equilibrium,

\[\left\langle\frac12m|\mathbf v-\mathbf u|^2\right\rangle=\frac32k_BT\]

where \(\mathbf u\) is removed before averaging. A container translating uniformly can have large laboratory kinetic energy without being hotter.

For unequal masses or constraints, accumulate energy with the actual masses and use the correct number of active quadratic degrees of freedom:

\[T=\frac{2K_{\mathrm{random}}}{f k_B}\]

Here \(f\) is not automatically \(3N\). Removing center-of-mass momentum, fixing particles, or imposing constraints changes it.

Temperature is a statement about a distribution in equilibrium or local equilibrium. Two beams moving in opposite directions can have zero mean flow and a large mean-square speed while remaining far from a thermal Maxwell distribution.

Pressure is momentum flux

Microscopically, ideal-gas pressure arises from particle momentum transferred to walls. If a particle's wall-normal momentum changes by \(\Delta p_{\perp}\), the impulse delivered to the wall has the opposite sign. Over wall area \(A\) and interval \(\Delta t\),

\[p=\frac{1}{A\Delta t}\sum_{\mathrm{collisions}}\Delta p_{\mathrm{wall},\perp}\]

A single snapshot contains positions and velocities but no completed interval of wall impulses, so it cannot provide a stable collision-based pressure estimate. Record collision events and average long enough to sample many impacts.

For an equilibrium monatomic ideal gas, kinetic theory gives

\[pV=Nk_BT=\frac13Nm\langle v^2\rangle\]

for identical particles with bulk motion removed. Interacting gases require force or virial contributions beyond the ideal collision picture.

Internal energy and the loss of detail

For a monatomic ideal gas,

\[U=\frac32Nk_BT\]

This one scalar does not reveal which particles are fast, where they are, or whether spatial correlations exist. Two microstates can have equal total random kinetic energy but different velocity histograms. Equilibrium theory predicts that overwhelmingly many accessible microstates display the familiar macroscopic behavior, but special nonequilibrium arrangements are possible.

Coarse-graining is useful because most measurements do not resolve every particle. Information discarded by a reducer cannot be recovered from its result. Keep raw or adequately checkpointed microscopic state when later questions may depend on correlations or rare events.

Ensemble averages and time averages

An ensemble average considers many possible microstates prepared under the same macroscopic conditions. A time average follows one system for a long interval. Under ergodic-like assumptions, the two may agree for suitable observables, but this is not guaranteed for every system or every time scale.

Successive simulation steps are correlated. Recording one million closely spaced samples does not necessarily provide one million independent observations. Estimate correlation times or use block averages before assigning uncertainty to temperature or pressure.

rng = np.random.default_rng(42)
# Independent illustrative temperature observations, in kelvin.
samples = rng.normal(300, 5, 1000)
block_means = samples.reshape(20, 50).mean(axis=1)
standard_error = block_means.std(ddof=1) / np.sqrt(len(block_means))
print(f"Mean temperature: {block_means.mean():.3f} +/- {standard_error:.3f} K")

Local equilibrium and gradients

A gas with temperature, density, or velocity gradients is not in global equilibrium. It may still admit local macroscopic fields if each cell contains enough particles and relaxes internally faster than the fields change. This separation of scales supports fluid mechanics.

If a mean free path is comparable to cell size or apparatus size, local equilibrium can fail. Then pressure may be anisotropic, velocity distributions non-Maxwellian, and a single temperature field inadequate. More particles in the code do not fix an invalid continuum assumption.

Verification

Reorder particle storage and require every symmetric macroscopic reducer to remain unchanged. Add one constant velocity to all particles: density and temperature after bulk removal should stay fixed while laboratory momentum changes. Duplicate a noninteracting sample and double the volume: density and intensive equilibrium estimates should stay fixed while particle number and internal energy double.

Compare wall-impulse pressure with \(Nk_BT/V\) over progressively longer intervals. Subdivide the box and verify that cell masses sum to total mass. Repeat with larger particle counts; relative equilibrium fluctuations should generally shrink.

Real particles have finite size, interactions, rotational and vibrational modes, quantum statistics, reactions, and phase changes. A microscopic simulation is itself a model. Agreement among its reducers checks internal consistency, while comparison with independent theory or experiment checks whether its assumptions describe the intended matter.

A worked Python notebook

Coarse-grain equal molecular masses in a cell with a volume of 10⁻²¹ cubic metres. The question for this calculation is precise: how does mass density change when we change particle count in a cell, 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 count is discrete; the continuous curve is only a guide to the proportional relation. 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 1000 to 100000 count. 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 particle count in a cell, expressed in count. The output is mass density, expressed in kg/m^3. 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.

\[f(q) = 4.65 \cdot 10^{-5} q\]

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 10000 count, the expression gives 0.465 kg/m^3. 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("4.65e-26*q/1e-21", locals={"q": q})
reference_input = 10000.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "count")
print("Predicted mass density:", reference_output, "kg/m^3")

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 0.0465 kg/m^3. The midpoint, at 50500 count, gives 2.34825 kg/m^3. The upper endpoint gives 4.65 kg/m^3. Read these together with the assumptions: they describe different choices of particle count in a cell 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([1000.0, 50500.0, 100000.0])
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} count -> {output_value:.6g} kg/m^3")
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 particle count in a cell; the vertical axis shows mass density. 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(1000.0, 100000.0, 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("Particle count in a cell (count)")
ax.set_ylabel("Mass density (kg/m^3)")
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 mass density change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10000 count is 4.65e-05 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.

\[f'(q) = 4.65 \cdot 10^{-5}\]

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 = 10495.0
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, "kg/m^3")
print("Full expression:", full_prediction, "kg/m^3")
print("Approximation difference:", full_prediction - linear_prediction, "kg/m^3")

Here the changed input is 10495 count. The tangent estimate is 0.488018 kg/m^3, while the full expression gives 0.488018 kg/m^3. 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 = 99000.0
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 particle count in a cell inside the stated interval, predict whether mass density 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 count is discrete; the continuous curve is only a guide to the proportional relation. 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 kg/m^3: 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(100000.0) - evaluate(1000.0))
for points in [101, 201, 401]:
    grid = np.linspace(1000.0, 100000.0, 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

Share to: