LESSON NOTES · 03

Ensembles, pressure, and trustworthy averages

Prerequisites: Hamiltonian dynamics; elementary statistical mechanics

1. A target distribution, not just a temperature

NVE fixes particle number, volume and energy. Canonical NVT targets \(e^{-\beta H}\), \(\beta=1/(k_BT)\); a thermostat must preserve this distribution, not merely its mean temperature. NPT allows volume fluctuations. Ergodicity is an assumption: a trapped trajectory need not represent equilibrium.

2. Count independent thermal modes

Subtract independent constraints and removed center-of-mass motion consistently. Streaming velocity is not thermal motion. Equipartition is classical; instantaneous kinetic temperature fluctuates and quantum high-frequency vibrations need not obey it.

\[ K=\sum_i\tfrac12m_iv_i^2,\quad T_{kin}=\frac{2K}{fk_B},\quad\langle K\rangle=\tfrac f2k_BT. \]

3. Derive canonical pressure by scaling coordinates

For an unconstrained 3D canonical system with differentiable pair potential, set \(r_i=V^{1/3}s_i\) in its partition integral. The Jacobian contributes \(V^N\). Differentiating \(\ln Z\) gives an ideal term and a potential derivative. Define \(r_{ij}=r_i-r_j\) and \(F_{ij}\) as force on \(i\) from \(j\); repulsion has positive virial.

\[ \begin{aligned}P&=k_BT\frac{\partial\ln Z}{\partial V}=\frac{Nk_BT}{V}-\left\langle\frac{\partial U}{\partial V}\right\rangle,\\\frac{\partial U}{\partial V}&=-\frac1{3V}\sum_{i<j}r_{ij}\cdot F_{ij},\\P&=\frac{Nk_BT}{V}+\frac1{3V}\left\langle\sum_{i<j}r_{ij}\cdot F_{ij}\right\rangle.\end{aligned} \]

4. Use the estimator consistently

The instantaneous mechanical ideal term is \(2K/(3V)\). In periodic boxes use image-consistent pair separations; blindly summing wrapped positions times total forces is unsafe. Constraints, many-body forces, long-range and explicit-volume terms need their own contributions. The partition derivation above specifically assumed an unconstrained system.

Ensembles, pressure, and trustworthy averages — schematic under the stated model assumptions

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked degrees-of-freedom example

Three 3D particles without internal constraints, after removing center-of-mass momentum, have \(f=6\) and \(\langle K\rangle=3k_BT\). Using \(f=9\) underestimates temperature by one third at the same kinetic energy. Do not blindly substitute this constrained finite-system bookkeeping into the unconstrained partition derivation.

Convergence and check yourself

Separately halve timestep, increase box size, tighten force/constraint tolerances, compare independent initial states, discard equilibration and estimate correlated errors. In NVE inspect drift as well as fluctuations. Stable energy does not prove structural equilibration or force-field accuracy. Can a thermostat hide integration heating?

Answer and reasoning

Yes, by removing extra energy while masking its cause.

References and further reading

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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