Moves

PIMMS samples configurations with Metropolis Monte Carlo (MC). At each step it picks a move at random (according to the MOVE_* fractions, which must sum to 1.0), proposes the corresponding change, and accepts or rejects it so that the simulation converges on the correct Boltzmann distribution. This section has one page per move: each describes how the move works, why it preserves detailed balance, and the configuration relevant to it.

• Crankshaft
• Chain translate
• Chain rotate
• Chain pivot
• Head pivot
• Slither (reptation)
• Pull (cooperative reptation)
• Cluster translate
• Cluster rotate
• Temperature-switch Monte Carlo (TSMMC)
• Jump and relax
• Virtual-Move Monte Carlo (VMMC)

The move set at a glance

Code	Keyword	Status	Move
1	MOVE_CRANKSHAFT	core	Local single-bead perturbations (the workhorse)
2	MOVE_CHAIN_TRANSLATE	core	Rigid translation of a whole chain
3	MOVE_CHAIN_ROTATE	core	Rigid 90/180/270° rotation of a whole chain
4	MOVE_CHAIN_PIVOT	core	Pivot one half of a chain about an interior point
5	MOVE_HEAD_PIVOT	core	Pivot a single terminus (rarely useful)
6	MOVE_SLITHER	core	Reptation - the chain slides forwards/backwards
7	MOVE_CLUSTER_TRANSLATE	core	Rigid translation of a connected cluster of chains
8	MOVE_CLUSTER_ROTATE	core	Rigid rotation of a connected cluster of chains
9, 10, 12	MOVE_CTSMMC / MOVE_MULTICHAIN_TSMMC / MOVE_SYSTEM_TSMMC	stable	Temperature-switch (tempered-transitions) excursions
11	MOVE_PULL	stable	Cooperative reptation of a sub-segment (dense systems)
13	MOVE_JUMP_AND_RELAX	stable	Relax → jump → relax a single chain
14	MOVE_VMMC	experimental	Virtual-Move MC collective cluster move

(The three TSMMC variants share one page, since they are the same algorithm applied at different scopes.) Only the VMMC move is still experimental and requires EXPERIMENTAL_FEATURES : True; the temperature-switch, pull and jump-and-relax moves are stable and need no special flag.

A detailed-balance primer

Every move is constructed so that the simulation samples the Boltzmann distribution

\[\pi(x) \;=\; \frac{1}{Z}\, e^{-E(x)/T},\]

where \(x\) is a configuration, \(E(x)\) its energy, \(T\) the TEMPERATURE and \(Z\) a normalising constant. (PIMMS works in reduced units where Boltzmann’s constant is absorbed into \(T\).)

A sufficient condition for converging on \(\pi\) is detailed balance: for every pair of configurations \(x, y\), the move’s transition probability \(P\) must satisfy

(1)\[\pi(x)\, P(x \to y) \;=\; \pi(y)\, P(y \to x).\]

A move is built from a proposal \(g(x\to y)\) (the random change it suggests) and an acceptance probability \(A(x\to y)\), so \(P(x\to y) = g(x\to y)\,A(x\to y)\) for \(y\neq x\). The Metropolis-Hastings acceptance

(2)\[A(x\to y) \;=\; \min\!\left(1,\; \frac{g(y\to x)}{g(x\to y)}\, e^{-(E(y)-E(x))/T}\right)\]

satisfies (1) for any proposal. The ratio \(g(y\to x)/g(x\to y)\) corrects for an asymmetric proposal.

Two important special cases recur throughout this section:

• Symmetric proposal. If forward and reverse proposals are equally likely, \(g(x\to y) = g(y\to x)\), the ratio is 1 and (2) reduces to the plain Metropolis criterion \(A = \min(1, e^{-\Delta E/T})\) with \(\Delta E = E(y)-E(x)\). The local and rigid-body moves (crankshaft, translate, rotate, pivot, cluster moves) all use this.

• Composition of valid moves. A sequence of updates that each individually leave \(\pi\) invariant also leaves \(\pi\) invariant. Several PIMMS moves are “megamoves” - many sub-moves bundled into one step - or composites (e.g. Jump and relax) that rely on this fact.

Where a move needs more than these (a genuine Hastings ratio, a work-accumulation factor, or a link-probability product), the relevant page derives it explicitly.

Note

Hard-sphere overlaps are rejected outright: PIMMS never allows two beads on the same site, which is equivalent to assigning such configurations infinite energy (\(\pi = 0\)), consistent with (1).