Temperature as an emergent quantity

Temperature is a macroscopic property that summarizes the thermal state of a system. It helps predict the direction of spontaneous energy transfer when systems can exchange energy. A single molecule does not carry the equilibrium temperature of an entire gas; temperature emerges from the statistical behavior of many microscopic degrees of freedom.

Thermal equilibrium and the zeroth law

Two systems are in thermal equilibrium when placing them in thermal contact produces no net macroscopic thermal energy transfer after transients settle. The zeroth law states: if system \(A\) is in thermal equilibrium with system \(C\), and system \(B\) is also in equilibrium with \(C\), then \(A\) and \(B\) are in thermal equilibrium with each other.

This transitive property makes thermometers possible. A thermometer has some reproducible thermometric property—volume, electrical resistance, voltage, pressure, or emitted radiation—that changes with thermal state. Calibration maps that property to a temperature scale. The thermometer must interact weakly enough and be given enough time that its own presence does not substantially change the system being measured.

Temperature is therefore defined operationally before a molecular model is introduced. The molecular explanation must agree with this equilibrium behavior; it does not replace the need for a measurement procedure.

Temperature is not heat

Temperature is a state property. Heat is energy transferred across a system boundary because of a temperature difference. A body does not “contain heat” in the same sense that it has internal energy. Once transferred energy is inside the chosen system, it contributes to internal energy according to the system's microscopic degrees of freedom.

A small cup of boiling water has a higher temperature than a large bathtub of warm water, but the bathtub can have much larger internal energy. Temperature is intensive: joining identical equilibrium systems does not double it. Internal energy and volume are extensive: they scale with system size under ordinary conditions.

Kelvin and Celsius

The kelvin is the SI unit of thermodynamic temperature. Celsius and kelvin intervals have the same size, but their zeros differ:

\[T_{\mathrm K}=T_{^\circ\mathrm C}+273.15\]

A temperature difference of \(1\,\mathrm K\) equals a difference of \(1\,{}^\circ\mathrm C\). Ratios such as “twice the absolute temperature” require kelvin, not Celsius, because Celsius has an offset zero. Absolute zero is \(0\,\mathrm K\), equal to \(-273.15\,{}^\circ\mathrm C\).

Absolute zero is not the statement that every microscopic motion classically vanishes in every physical system. Quantum systems retain zero-point behavior, and the third law places important limits on reaching absolute zero.

The ideal-gas kinetic interpretation

For a dilute monatomic ideal gas in classical thermal equilibrium, the mean translational kinetic energy per particle is

\[\langle K_{\mathrm{trans}}\rangle=\frac32 k_B T\]

where \(k_B\) is Boltzmann's constant and \(T\) is in kelvin. The factor \(3/2\) comes from three independent translational velocity components, each contributing \(k_BT/2\) under classical equipartition.

For particle mass \(m\),

\[\frac12m\langle v^2\rangle=\frac32k_BT\]

\[v_{\mathrm{rms}}=\sqrt{\langle v^2\rangle}=\sqrt{\frac{3k_BT}{m}}\]

Temperature sets a distribution scale, not one shared molecular speed. At equilibrium, particles continually exchange kinetic energy in collisions and have a broad range of speeds.

Remove bulk motion before estimating temperature

Suppose a gas as a whole moves with center-of-mass velocity \(\mathbf V_{\mathrm{cm}}\). Its particles may have large laboratory kinetic energy even when their random motion relative to the flow is modest. For particles of masses \(m_i\) and velocities \(\mathbf v_i\),

\[\mathbf V_{\mathrm{cm}}=\frac{\sum_i m_i\mathbf v_i}{\sum_i m_i},\qquad K_{\mathrm{thermal}}=\sum_i\frac12m_i|\mathbf v_i-\mathbf V_{\mathrm{cm}}|^2\]

Using the raw laboratory speeds would count organized translation as heat. In a fluid with spatially varying flow, subtract a suitable local mean velocity rather than one global value.

import numpy as np

rng = np.random.default_rng(42)
N, mass, k_b, temperature = 10_000, 4.65e-26, 1.380649e-23, 300.0
velocities = rng.normal(0, np.sqrt(k_b * temperature / mass), (N, 3))
velocities += np.array([100.0, 0.0, 0.0])
thermal = velocities - velocities.mean(axis=0)
kinetic = 0.5 * mass * np.sum(thermal**2)
active_degrees = 3 * N - 3
estimate = 2 * kinetic / (active_degrees * k_b)
print(f"Temperature after removing centre-of-mass motion: {estimate:.2f} K")

The active degree count should remove constrained collective degrees consistently. For a finite isolated three-dimensional particle system whose center-of-mass momentum has been removed, blindly using \(3N\) can bias the estimator; the removed constraints reduce the independent quadratic degrees.

Equipartition has a domain of validity

The classical equipartition result

\[\langle E_{\mathrm{quadratic}} angle=\frac12k_BT\]

applies to quadratic contributions that are thermally active in classical equilibrium. Diatomic and polyatomic molecules can store energy in rotation and vibration as well as translation. Quantum level spacing can freeze out some modes at lower temperatures, so the classical heat capacity and degree count fail.

Interacting systems can also store substantial potential energy. A kinetic estimator may still define temperature in equilibrium molecular dynamics, but internal energy is not then only kinetic energy. Near phase transitions or far from equilibrium, one scalar may hide essential structure.

A deeper statistical definition

In equilibrium thermodynamics, temperature relates entropy \(S\) to internal energy \(U\) at fixed volume \(V\) and particle number \(N\):

\[\frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)_{V,N}\]

This says temperature measures how rapidly the logarithmic abundance of accessible microstates changes when energy is added. When two large systems exchange a fixed total energy, the most probable equilibrium division maximizes total entropy and makes their temperatures equal.

This definition also shows why “temperature is average kinetic energy” is not universal. The kinetic formula is a result for a particular classical model; the entropy derivative is broader.

Temperature fields and nonequilibrium

A system with a gradient is not in global thermal equilibrium. It may still admit a local temperature if each small region relaxes internally much faster than macroscopic conditions change. The region must contain enough particles for stable statistics but be small enough to resolve the gradient.

An anisotropic velocity distribution can produce different directional kinetic temperatures. A bimodal distribution may have the same mean kinetic energy as an equilibrium Maxwell distribution while behaving very differently. Always inspect at least a velocity histogram, directional moments, sample count, and time history before trusting one temperature number.

Verification and uncertainty

Duplicate an identical set of particles: the mean temperature should remain unchanged, while total random kinetic energy doubles. Multiply every thermal velocity by two: kinetic energy and the kinetic temperature estimate should increase by four. Add the same constant velocity to every particle: after center-of-mass removal, temperature should not change.

Estimate uncertainty from independent samples, not raw time steps that remain strongly correlated. A longer trajectory does not provide proportionally more information if neighboring states are nearly identical. Compare block averages and report fluctuations, especially for small systems.

Finally check equilibrium evidence: stationary distributions, consistent directional temperatures, stable averages, and agreement among defensible thermometers. A numerical value with kelvin units is not automatically a thermodynamic temperature.

A worked Python notebook

Connect an equilibrium temperature with the mean random kinetic energy of a monatomic molecule. The question for this calculation is precise: how does mean translational energy per molecule change when we change temperature, 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.

Bulk flow is excluded and the classical translational degrees of freedom are active. 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 100 to 600 K. 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 temperature, expressed in K. The output is mean translational energy per molecule, expressed in J. 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.

\[f(q) = 2.0709735 \cdot 10^{-23} q\]

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 300 K, the expression gives 6.21292e-21 J. 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("3*1.380649e-23*q/2", locals={"q": q})
reference_input = 300.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "K")
print("Predicted mean translational energy per molecule:", reference_output, "J")

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 2.07097e-21 J. The midpoint, at 350 K, gives 7.24841e-21 J. The upper endpoint gives 1.24258e-20 J. Read these together with the assumptions: they describe different choices of temperature 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([100.0, 350.0, 600.0])
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} K -> {output_value:.6g} J")
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 temperature; the vertical axis shows mean translational energy per molecule. 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(100.0, 600.0, 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("Temperature (K)")
ax.set_ylabel("Mean translational energy per molecule (J)")
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 translational energy per molecule change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 300 K is 2.07097e-23 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.

\[f'(q) = 2.0709735 \cdot 10^{-23}\]

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 = 302.5
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, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 302.5 K. The tangent estimate is 6.26469e-21 J, while the full expression gives 6.26469e-21 J. 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 = 500.0
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 temperature inside the stated interval, predict whether mean translational energy per molecule 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: Bulk flow is excluded and the classical translational degrees of freedom are active. 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 J: 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(600.0) - evaluate(100.0))
for points in [101, 201, 401]:
    grid = np.linspace(100.0, 600.0, 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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