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Equipartition theorem

At classical thermal equilibrium, each independent quadratic degree of freedom contributes the same average energy. This connects microscopic state dimensions to macroscopic heat capacity, but only after the model declares which modes are active and equilibrated.

Think like a programmer

Count modeled degrees of freedom from the data structure, not from a label. If a particle state gains rotation, vibration, or a constrained coordinate, update the count and its expected-energy test. Keep degrees of freedom separate from raw array length: a stored coordinate may be fixed, redundant, or represent a mode excluded by the current approximation.

Model checklist

Inputs
Active quadratic degrees of freedom f, temperature T, Boltzmann constant, particle/model type, constraints, and equilibrium criterion.
State
Expected mean energy, sampled kinetic/potential contributions, active-mode registry, and comparison residual.
Rule
Multiply one-half kBT by each active quadratic mode, then compare the expected value with statistics from an equilibrated sample.
Output
Classical reference energy, mode-by-mode expectation, and validity warning.
Check
Adding one active quadratic mode increases expected energy by one-half kBT; constraining a mode removes its contribution; zero temperature gives zero classical reference; invalid inputs are rejected.
\[\langle E\rangle=\frac{f}{2}k_BT\]

For example, a freely moving point particle in a three-dimensional classical box has three translational kinetic quadratic terms, so f equals 3 for that contribution. A rigid molecule may add active rotations; a harmonic vibration can contribute both kinetic and potential quadratic terms once that mode is thermally active. The count is a model decision backed by state representation and regime, not a number to memorize independently of the simulation.

Equipartition is useful as a regression reference. Divide a long equilibrium run into windows, estimate mean energy per particle, and compare it with f kBT divided by 2 under a tolerance that acknowledges finite-sample variation. A systematic mismatch can indicate a wrong degrees-of-freedom count, incomplete equilibration, non-conserving integration, or a regime where classical assumptions fail.

Try this experiment

Prediction: Adding an active quadratic degree of freedom raises expected mean energy by one-half kT.

Map each coordinate in a particle state type to its degree-of-freedom contribution.

Where this model breaks

Quantum effects freeze out modes at low temperature, and non-equilibrium distributions do not satisfy equipartition. Interactions, phase transitions, finite systems, constrained dynamics, and insufficient averaging can also make the reference inaccurate. Equipartition does not replace a material's measured heat capacity or prove a sampled system has equilibrated.

Summary

Use equipartition as a classical equilibrium reference and keep its active-mode registry synchronized with the model. Compare against sampled statistics with uncertainty, then treat disagreement as a diagnostic rather than forcing data to match the formula.

Glossary

Self-check

  1. What does f count?
  2. What does one extra mode add?
  3. When does classical equipartition fail?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Equipartition Theorem, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Kinetic Theory of Gases, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Equipartition Theorem as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Equipartition Theorem into a test

Connect explicitly counted model degrees of freedom to classical equilibrium energy expectations.

  1. Name the inputs and units that the kinetic theory of gases model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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