Temperature-switch Monte Carlo (TSMMC)

Ordinary local moves relax a system efficiently but struggle to cross large energy barriers. Once a chain - or the whole system - has fallen into a deep basin, almost every proposed move is uphill and gets rejected, and the simulation can sit there essentially stuck, sampling one basin while never discovering a lower one on the other side of a barrier. TSMMC moves are built to get over exactly these barriers.

The idea is a temperature excursion. Instead of proposing a single perturbation, a TSMMC move takes part of the system (or all of it) on a round trip in temperature: heat it along a schedule from the simulation temperature \(T\) up to a high “jump” temperature, let it rearrange while it is hot, then cool it back to \(T\). While hot, moves that are effectively impossible at \(T\) become easy, so the system can climb out of its basin and wander; on the way back down it settles into whatever basin it has found. The entire excursion - which is internally thousands of ordinary sub-moves - is then accepted or rejected as a single Monte Carlo move.

This page explains the acceptance correction that makes such a move valid, then describes the three variants that differ in what is taken on the excursion.

Why a naive excursion would be wrong

The tempting scheme - “heat, move around, cool, then accept on the final energy with the usual Metropolis rule” - is not correct. The moves made at the elevated temperatures are drawn from hot distributions, not from the target distribution at \(T\), so they bias the proposal in a way the ordinary Metropolis criterion does not account for. Using it would distort the sampled ensemble: the simulation would no longer converge on the Boltzmann distribution at \(T\).

TSMMC fixes this with a tempered-transitions acceptance (Neal, 1996; closely related to nonequilibrium candidate Monte Carlo, Nilmeier et al., 2011). The trick is to account for the thermodynamic work done as the temperature is switched along the schedule, and to fold that work into the accept/reject decision. Done correctly, the elevated-temperature exploration is corrected for exactly: TSMMC changes how the system explores, but not the distribution it converges on.

The temperature schedule

PIMMS works in reduced units where the inverse temperature is simply \(\beta = 1/T\). A TSMMC excursion follows a fixed schedule of inverse temperatures

\[\beta_0 \;=\; \tfrac{1}{T},\quad \beta_1,\ \beta_2,\ \dots,\ \beta_M,\]

built once at the start of the move:

an up-ramp of TSMMC_NUMBER_OF_POINTS linearly-spaced steps from just above \(T\) to the jump temperature TSMMC_JUMP_TEMP,
a short hold at the jump temperature (a fixed number of points at the top of the ramp), and
a down-ramp that mirrors the up-ramp back toward \(T\).

So the schedule is a closed loop in temperature - it leaves \(T\), reaches the jump temperature, and returns - of total length \(M \approx 2\times\texttt{TSMMC\_NUMBER\_OF\_POINTS}\) plus the hold. At each rung \(\beta_k\) the system is propagated by a burst of ordinary Monte Carlo moves that are themselves reversible at \(\beta_k\) (they individually satisfy detailed balance for the Boltzmann distribution at that temperature). The number of sub-moves per rung is set by TSMMC_STEP_MULTIPLIER (scaled by system size for the chain-based variants - see below).

The acceptance correction

Let \(x^{(k)}\) be the configuration reached after the propagation at rung \(\beta_k\), with \(x^{(0)}\) the starting configuration, and let \(U(x)\) be its energy. The excursion accumulates a single scalar, the tempered-transitions work

(1)\[W \;=\; \sum_{k=1}^{M}\bigl(\beta_{k-1}-\beta_k\bigr)\,U\!\bigl(x^{(k-1)}\bigr) \;+\;\bigl(\beta_{M}-\beta_{0}\bigr)\,U\!\bigl(x^{(M)}\bigr).\]

Each term is the change in inverse temperature at one switch, multiplied by the energy of the configuration at the instant of that switch (i.e. before the system is propagated at the new temperature). The sum runs over every temperature change in the excursion, including the initial departure from \(\beta_0\) and the final return to it. The whole excursion is then accepted with

(2)\[A \;=\; \min\!\bigl(1,\; e^{W}\bigr).\]

Two features are worth drawing out:

Only the temperature switches contribute. The propagation within a rung uses a kernel that already preserves the Boltzmann distribution at \(\beta_k\), so it needs no correction and does not appear in (1). All of the bookkeeping lives in the discrete temperature changes, each weighted by the instantaneous energy.
A no-op is always accepted. If no sub-move is ever accepted, the energy never changes, \(U(x^{(k)}) = U_0\) for all \(k\), and because the schedule returns to \(\beta_0\) the coefficients telescope:

\[W \;=\; U_0\!\left[\sum_{k=1}^{M}(\beta_{k-1}-\beta_k) + (\beta_M-\beta_0)\right] \;=\; U_0\,(\beta_0-\beta_0) \;=\; 0,\]

giving \(A = 1\). A move that does nothing is never spuriously rejected - as it must not be.

More generally, because the schedule is a closed loop and every intra-rung kernel is balanced, the correction (1) exactly cancels the bias introduced by sampling at the elevated temperatures, so the composite excursion satisfies detailed balance with respect to the target Boltzmann distribution at \(T\). Intuitively, \(W\) measures the reversible work of carrying the system around the temperature loop: excursions that end up in configurations which are favourable at \(T\) relative to where they started are accepted readily, while those that end “expensively” are penalised by exactly the amount needed to keep the ensemble correct. PIMMS accumulates \(W\) incrementally as \(\sum(\beta_\text{before}-\beta_\text{after})\,U(x)\) over the schedule, and the implementation is checked against a crankshaft-only reference in the detailed-balance test suite.

The three variants

All three variants share the schedule and the acceptance rule above; they differ only in what is taken on the excursion and, consequently, which moves run at each rung and how expensive the excursion is.

Code	Keyword	Scope of excursion	Sub-moves used at each rung
9	`MOVE_CTSMMC`	One randomly chosen chain	Local (crankshaft) moves on that chain only
10	`MOVE_MULTICHAIN_TSMMC`	A random subset of chains	Local (crankshaft) moves on the selected chains only
12	`MOVE_SYSTEM_TSMMC`	The entire system	The full move set (any move except a nested TSMMC)

Chain TSMMC (`MOVE_CTSMMC`)

A single chain is chosen at random and taken on the temperature excursion by itself. Only local crankshaft perturbations of that one chain are performed at each rung - the chain “wiggles” more and more freely as it heats, then re-tightens as it cools. Restricting the excursion to local wiggling is deliberate: it lets the chain thoroughly rearrange its own conformation without decorrelating so violently that the final state is almost always rejected.

The amount of work per rung scales with the chain length: the number of sub-moves at each temperature is

\[\texttt{chain length}\;\times\;\texttt{TSMMC\_STEP\_MULTIPLIER},\]

so a single chain-TSMMC move costs roughly \(M \times \texttt{chain length} \times \texttt{TSMMC\_STEP\_MULTIPLIER}\) crankshaft sub-moves. Use it to shake a single stubborn chain out of a bad conformation.

Multichain TSMMC (`MOVE_MULTICHAIN_TSMMC`)

Exactly the chain excursion, but a random subset of chains is heated together, with local moves performed across all of the selected chains. This is the tool for chains that are stuck in a cooperative minimum - tangled or mutually frustrated in a way no single-chain move can undo - without paying for a full-system excursion.

Important

The subset is chosen as a plain random selection of chains (a random number of them, up to a fraction of the system), not as a connected cluster. Selecting a cluster would break detailed balance: after the excursion two clusters might have merged (or one split), and you could not propose the reverse move with the same probability, because the cluster structure has changed. Drawing the subset at random - independently of the configuration - makes its selection probability the same before and after the excursion, so the reverse move is proposable with equal probability and the acceptance (2) stays valid.

The cost per rung scales with the total number of beads in the selected subset times TSMMC_STEP_MULTIPLIER. In testing this variant is a particularly effective way to work down a rough energy landscape: it supplies the concerted, multi-chain motion that ordinary single-move MC lacks, without having to guess in advance which chains need to move together.

System TSMMC (`MOVE_SYSTEM_TSMMC`)

The system-wide excursion is different in kind. The entire lattice is backed up, and the main simulation is temporarily converted into a chain of auxiliary simulations: the main-loop temperature is stepped along the schedule, and at each rung a burst of ordinary full-system Monte Carlo moves is run - any move is available except a nested TSMMC (TSMMC excursions are not allowed to recurse). During the excursion the main step counter is not advanced and no analysis or trajectory output is produced; it is a self-contained super-move bolted into the main chain.

At the end the whole new system configuration is accepted or rejected with (2); on rejection the lattice is restored exactly from the backup. The number of sub-moves per rung is TSMMC_STEP_MULTIPLIER (not scaled by size, since each sub-move is already a full-system move). This is the most powerful variant - it can rearrange the entire configuration cooperatively - and the most expensive, both in time and in the memory needed to hold the backup.

Configuring the excursion

MOVE_CRANKSHAFT          : 0.9
MOVE_SYSTEM_TSMMC        : 0.1
TSMMC_JUMP_TEMP          : 120     # peak temperature (must exceed TEMPERATURE)
TSMMC_NUMBER_OF_POINTS   : 20      # points on each ramp (smoother = more)
TSMMC_STEP_MULTIPLIER    : 50      # sub-steps per temperature rung
TSMMC_INTERPOLATION_MODE : LINEAR

Keyword	Type	Meaning
`TSMMC_JUMP_TEMP`	float	Peak temperature of the excursion. Must be greater than `TEMPERATURE`.
`TSMMC_NUMBER_OF_POINTS`	int	Number of temperature points on each ramp between \(T\) and the jump temperature. More points = a smoother, more gradual (and more expensive) excursion.
`TSMMC_STEP_MULTIPLIER`	int	Monte Carlo sub-steps performed at each rung of the schedule (further multiplied by the number of beads for the chain/multichain variants). More = more thorough equilibration at each temperature, but slower.
`TSMMC_INTERPOLATION_MODE`	str	How the temperature is spaced across the schedule. Currently only `LINEAR` (equal increments) is supported.
`TSMMC_FIXED_OFFSET`	float / False	If set to a number, the jump temperature is defined relative to the current temperature as \(T + \texttt{TSMMC\_FIXED\_OFFSET}\) instead of the absolute `TSMMC_JUMP_TEMP`. Handy inside a quench (see below).

The move fractions MOVE_CTSMMC / MOVE_MULTICHAIN_TSMMC / MOVE_SYSTEM_TSMMC select the variants and, like all MOVE_* keywords, must sum to 1.0 across the whole move set.

Cost and tuning

A single TSMMC move is worth roughly

\[\underbrace{\bigl(2\times\texttt{TSMMC\_NUMBER\_OF\_POINTS} + \text{hold}\bigr)}_{\text{rungs, } M} \;\times\;\text{(sub-moves per rung)}\]

ordinary sub-moves, so even a small MOVE_*TSMMC probability represents a large amount of work. Guidance:

Jump temperature. Hot enough to melt the barrier you are trying to cross, but not so hot that the system fully evaporates and forgets everything useful. TSMMC_JUMP_TEMP must always exceed the base TEMPERATURE.
Points and step multiplier. More points and more sub-steps make each excursion gentler and more thorough, and therefore more likely to be accepted, at a directly proportional cost. If excursions are almost always rejected, make the schedule smoother (more points) before making it hotter.
Fraction. Because each excursion is expensive, TSMMC is normally mixed in at a small fraction, with cheap local moves (crankshaft, slither) doing the routine sampling in between.
Pick the smallest scope that works. Reach for MOVE_CTSMMC for a single stuck chain, MOVE_MULTICHAIN_TSMMC for a cooperatively-trapped handful, and MOVE_SYSTEM_TSMMC only when the whole configuration must move at once.

Inside a quench

If you combine TSMMC with a quench, the excursion coordinator is rebuilt at the new base temperature every time the quench updates, so the jumps always heat relative to wherever the ramp currently sits. TSMMC_FIXED_OFFSET is the natural choice there: it pins the jump temperature a fixed amount above the current temperature, so you never risk an absolute TSMMC_JUMP_TEMP accidentally falling below the (moving) base temperature during the ramp.

References

R. M. Neal, “Sampling from multimodal distributions using tempered transitions”, Statistics and Computing 6, 353-366 (1996).
J. P. Nilmeier, G. E. Crooks, D. D. L. Minh, J. D. Chodera, “Nonequilibrium candidate Monte Carlo is an efficient tool for equilibrium simulation”, PNAS 108, E1009-E1018 (2011).