Parallel Tempering

In the case of a system with a complex potential energy surface (e.g. with high energy barriers at a phase transition or many shallow minima in a glassy system), traditional Metropolis MC can be dramatically inefficient because its local moves do not allow the system to explore all of configuration space [4]. In practice, then, the MC simulation becomes non-ergodic and gives unreliable statistical information.

The parallel tempering (PT) algorithm solves this problem by supplementing local configurational Metropolis moves with global `swap' moves that update an entire set of configurations. Several MC simulations (`replicas') are run in parallel at a series of different temperatures ${T_i}$, with inverse temperature denoted $\beta_i$=$k_B T_i$. The simulations at higher temperatures will be able to explore configuration space more freely, crossing energy barriers at phase transitions and `hopping' among shallow energy minima. The PT algorithm takes advantage of this by exchanging these higher-temperature configurations with configurations at the low temperature of interest, allowing the low-temperature simulation to sample configurations much more efficiently than with local Metropolis updates only.

Figure 2: Schematic of the parallel tempering algorithm. Markov chains (replicas) at different temperatures are swapped. Figure taken from M. Falcioni and M.W. Deem, J. Chem. Phys 110, 1754 (1999) [4].

To understand the theory, consider a simulation with configuration $C_i$ at inverse temperature $\beta_i$ and one with $C_j$ at $\beta_j$. After a series of local updates on each replica, we consider a swap move, wherein the replica at $\beta_i$ assumes configuration $C_j$ and the replica at $\beta_j$ has configuration $C_i$. Given a hamiltonian $H(C)$, a system with configuration $C_i$ has energy $H(C_i)$. Then the proposed swap is accepted with probability

$$p  = \min [1,e^{-\Delta S}] with \Delta S = (\beta_j - \beta_i)(H(C_i) - H(C_j)).$$

This quantity is the change in action for the swap move.

Parallel tempering was developed in 1991 [5] and has since been used for a number of applications [6], including studying systems with large energy barriers, solving zeolite structure [4], and investigating water clusters [7,8].