Virtual-Move Monte Carlo (VMMC)

Keyword:: MOVE_VMMC
Move code:: 14
Status:: experimental (requires EXPERIMENTAL_FEATURES : True)

VMMC (Whitelam & Geissler, J. Chem. Phys. 127, 154101, 2007) is a collective move that lets correlated groups of chains move together, escaping the kinetic traps that single-chain moves hit in strongly-attractive / condensed phases.

How it works

A seed chain is chosen and given a trial rigid translation. PIMMS then builds a moving cluster around it by recruiting neighbours according to interaction-energy gradients: a neighbouring chain is pulled in when moving the seed alone would break their mutual attraction. Concretely, for a cluster member \(m\) and an interacting neighbour \(j\), a link is formed with probability

(1)\[p^\text{f}_{mj} = \max\!\Bigl(0,\; 1 - e^{-(E^{+}_{mj} - E^0_{mj})/T}\Bigr),\]

where \(E^0_{mj}\) is the current \(m\)-\(j\) interaction energy and \(E^{+}_{mj}\) is that energy after displacing \(m\) alone by the trial vector. If moving \(m\) alone would raise the energy (weaken the bond), \(j\) is recruited. Recruitment iterates outward until no further chains join, and the whole cluster is then translated rigidly. This way an attractively-bound group moves as a unit instead of being torn apart bond-by-bond.

Why detailed balance holds

VMMC is a Metropolis-Hastings move in which the recruitment is the proposal. The probability of proposing a particular cluster move is the product, over every neighbour pair tested, of forming the links that were formed and failing the links that were not:

\[g(x\to y) = \Bigl(\textstyle\prod_\text{formed} p^\text{f}\Bigr) \Bigl(\textstyle\prod_\text{failed} (1-p^\text{f})\Bigr).\]

The reverse move is generated by the same construction with the trial displacement reversed, giving link-formation/failure probabilities \(p^\text{r}\) and \(1-p^\text{r}\) evaluated from the \(-\) (reverse) virtual displacement. The Metropolis-Hastings acceptance (2) is then

\[A(x\to y) = \min\!\left(1,\; e^{-\Delta E/T}\, \prod_\text{formed}\frac{p^\text{r}}{p^\text{f}}\, \prod_\text{boundary}\frac{1-p^\text{r}}{1-p^\text{f}}\right),\]

where \(\Delta E\) is the total (interfacial) energy change of the rigid cluster move - intra-cluster interactions are unchanged - and the “boundary” product runs over the cluster-surface pairs whose link was tested but not formed. A formed link with \(p^\text{r}=0\), or a boundary link with \(p^\text{r}=1\), makes the reverse move impossible and is rejected outright. Because this is exactly the Hastings ratio for the recruitment proposal, the move satisfies detailed balance for any interaction model.

Two further ingredients keep it correct and efficient:

a cluster-size cutoff drawn from \(Q(n_c)\propto 1/n_c\) before growth, which is direction-independent (it cancels between forward and reverse) and enforces equal per-chain move frequency;
a hard-core check - if the rigid translation overlaps a non-cluster bead the move is rejected (an infinite-energy state).

The implementation computes \(\Delta E\) from a full, exact energy recompute and is validated by a detailed-balance test in a dense, strongly-attractive system (crank+VMMC reaches the same equilibrium as crankshaft alone while accepting many multi-chain cluster moves).

Configuration

MOVE_VMMCfloat: Probability of selecting a VMMC move (all MOVE_* must sum to 1.0).
VMMC_MAX_DISPLACEMENTint: Maximum magnitude (per dimension, lattice units) of the trial translation. Use small values (1-2): large rigid translations almost never clear hard-core clashes in dense phases. Default 3.
VMMC_MAX_CLUSTERint: Upper bound on the cluster-size cutoff draw (clamped to the number of chains at runtime). Default 1000.

VMMC is the move of choice for rearranging dense/condensed phases and studying self-assembly, where it dramatically outperforms single-chain moves at escaping kinetic traps.