7. Particle, volume, and insertion moves
Position in the course: Lesson 7 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Canonical sampling; Jacobians
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. State the ensemble before the acceptance rule
An acceptance rule is a consequence of a target density and proposal measure, not a formula transferable unchanged between ensembles. In canonical \(NVT\) sampling, a symmetric Cartesian displacement gives \(\min(1,e^{-\beta\Delta U})\). With a hard constraint, reject forbidden configurations while preserving symmetric reverse proposals. A truncated displacement resampled near a boundary is generally asymmetric because its allowed proposal volume depends on the starting point.
2. Derive the isothermal–isobaric volume factor
For fixed particle number and pressure \(P\), introduce scaled coordinates \(s_i=r_i/V^{1/3}\) in a cubic cell. The product measure transforms as \(dR=V^Nds\). Ignoring constants independent of \(V\), the target for \((s,V)\) is \(V^N\exp[-\beta(U(s,V)+PV)]\). A proposal symmetric in \(V\) while holding \(s\) fixed therefore has
The Jacobian is entropy from coordinate space, not an optional correction. If the proposal is symmetric in \(\eta=\ln V\), then \(dV=Vd\eta\) adds another factor \(V'/V\), producing exponent \(N+1\) under this convention. Constraints or removal of center-of-mass coordinates can change the exponent; derive the actual measure used by the algorithm.
3. Grand-canonical insertion and deletion
For classical indistinguishable particles at chemical potential \(\mu\), the grand partition contains \(z^NV^N/N!\) for an ideal gas, where \(z=e^{\beta\mu}/\Lambda^3\) and \(\Lambda\) is the thermal wavelength. Uniformly propose an inserted position in volume \(V\) and reverse by choosing one of \(N+1\) particles for deletion. With equal insertion/deletion attempt probabilities and no extra species factors, accepted insertion uses
Here deletion \(\Delta U\) is the new minus old energy. One must use a consistent labeled/unlabeled configuration convention so factorials and reverse particle selection are counted once. Dense liquids make uniform insertion inefficient because overlaps cause enormous energy penalties. Configurational bias changes the proposal, and its weights must be included explicitly.
4. Worked ideal-gas check
Set \(U=0\). The particle-number distribution is Poisson: \(P(N)=e^{-zV}(zV)^N/N!\). Its ratio \(P(N+1)/P(N)=zV/(N+1)\) exactly matches the insertion ratio. With \(zV=5\), insertion from \(N=9\) has acceptance \(1/2\), while deletion from \(N=10\) is accepted. This deterministic check catches wrong factorials or missing volume units.
Attempt probabilities also enter insertion and deletion. If insertion is chosen with probability \(p_{\mathrm{ins}}\) and reverse deletion with \(p_{\mathrm{del}}\), multiply the insertion ratio by \(p_{\mathrm{del}}/p_{\mathrm{ins}}\) under the actual forward and reverse states. Mixtures require species-specific chemical potentials and selection probabilities. Rules at particle-number boundaries can modify these factors; specify the full selection process.
5. Exercises and explained solutions
Exercise: For \(N=2\), \(V'/V=2\), and \(\Delta U+P\Delta V=0\), compare symmetric-\(V\) and symmetric-\(\ln V\) ratios.
Solution
They are \(2^2=4\) and \(2^3=8\). Both forward moves accept, but the reverse probabilities differ. They are different proposal measures sampling the same physical ensemble when derived consistently.
Exercise: Can one use grand-canonical insertion without specifying the units of chemical potential?
Solution
No. \(zV\) must be dimensionless and the thermal-wavelength and reference conventions matter. Chemical potential is not an arbitrary energy-only acceptance parameter.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
Related theory: Molecular dynamics · Quantum Monte Carlo
Software connection: CP2K · Quantum ESPRESSO
These software courses provide related background on energies, orbitals, or convergence management; they do not imply that the Monte Carlo or QMC examples on this page were executed there.
7. Related theory and practice
Molecular dynamics · Quantum Monte Carlo · RASPA3