Temperature-switch Monte Carlo (TSMMC)

Keywords:: MOVE_CTSMMC (9), MOVE_MULTICHAIN_TSMMC (10), MOVE_SYSTEM_TSMMC (12)
Status:: stable

TSMMC is one algorithm applied at three scopes, so the three move codes share this page. All three are governed by the same TSMMC_* keywords (see Temperature-switch Monte Carlo (TSMMC)).

How it works

A TSMMC move takes part of the system on a temperature excursion to hop over energy barriers that ordinary moves cannot cross. Starting from the simulation temperature \(T\), the temperature is ramped up a schedule to a high “jump” temperature TSMMC_JUMP_TEMP and back down again; at each temperature on the schedule a burst of ordinary Monte Carlo moves is performed. At the high temperature the system can climb out of a local energy minimum and explore; on the way back down it re-settles, hopefully into a different basin. The whole excursion is proposed as a single move and accepted or rejected as a unit.

The three variants differ only in what is heated:

MOVE_CTSMMC (code 9) - a single, randomly chosen chain.
MOVE_MULTICHAIN_TSMMC (code 10) - a random subset of chains.
MOVE_SYSTEM_TSMMC (code 12) - the entire system (most powerful, most expensive).

Why detailed balance holds

A naïve “heat, move, cool, then accept on the final energy” scheme would not be balanced, because the moves made at the elevated temperatures bias the proposal. TSMMC instead uses tempered transitions (Neal, 1996), which restore exact balance by accumulating the thermodynamic work done as the temperature is switched.

Let the inverse-temperature schedule be \(\beta_0 = \beta,\, \beta_1,\, \dots,\, \beta_{2n} = \beta\) (with \(\beta = 1/T\)), rising to the jump temperature at the midpoint and returning, and let \(x_k\) be the configuration just before the \(k\)-th temperature change. Each segment runs ordinary, balanced MC moves at its own temperature. The excursion is accepted with

\[A = \min\!\left(1,\; e^{W}\right), \qquad W = \sum_{k} \bigl(\beta_k - \beta_{k+1}\bigr)\, E(x_k),\]

where \(W\) is the accumulated log-weight (“work”) of the temperature switching. Because the schedule is symmetric (it returns to \(\beta\)) and the sub-moves at each temperature are themselves balanced, this acceptance makes the whole excursion satisfy detailed balance with respect to the target Boltzmann distribution at \(T\) - the elevated-temperature exploration is corrected for exactly, so it changes the dynamics but not the sampled distribution. The PIMMS implementation accumulates \(W\) as \(\sum(\beta_\text{before} - \beta_\text{after})\,E(x)\) over the schedule and is checked against a crankshaft-only reference by the detailed-balance test suite.

Configuration

The move fractions MOVE_CTSMMC / MOVE_MULTICHAIN_TSMMC / MOVE_SYSTEM_TSMMC (all MOVE_* must sum to 1.0) select the variants. The excursion itself is shaped by:

TSMMC_JUMP_TEMPfloat: Peak temperature of the excursion. Must exceed TEMPERATURE (unless TSMMC_FIXED_OFFSET is used).
TSMMC_NUMBER_OF_POINTSint: Number of temperature points on the ramp (more = smoother, more expensive).
TSMMC_STEP_MULTIPLIERint: MC sub-steps performed at each temperature point.
TSMMC_INTERPOLATION_MODEstr: How temperatures are spaced; currently only LINEAR.
TSMMC_FIXED_OFFSETfloat or False: If set, the jump temperature is TEMPERATURE + TSMMC_FIXED_OFFSET rather than the absolute TSMMC_JUMP_TEMP (handy inside quench runs).

TSMMC is most useful for strongly-interacting systems that get stuck; it is expensive (each move is many sub-moves across the schedule), so it is typically used at a small fraction alongside the crankshaft.

For the full treatment - the temperature schedule, a step-by-step account of the tempered-transitions work correction, a separate description of each of the three variants (what is heated, which sub-moves run, and how the cost scales), and cost / tuning guidance - see the dedicated Temperature-switch Monte Carlo (TSMMC) page.