8. Cluster updates, replica exchange, and Hamiltonian proposals
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Balance; ergodicity; ensembles
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Change the proposal, preserve the target
Slow exploration invites a different transition kernel rather than a different physical answer. For a mixture of individually stationary kernels, choosing each with a state-independent probability preserves stationarity. Composing stationary kernels also preserves the target, although the complete composition need not be reversible. State-dependent selection requires more care because it can introduce an omitted proposal ratio.
2. Why an Ising bond probability appears
At zero field with ferromagnetic \(J>0\), a bond weight is
Check aligned spins: the bracket is one. For opposite spins it is \(1-p=e^{-2\beta J}\), giving \(e^{-\beta J}\). This joint spin–bond representation permits bonds between aligned spins, followed by collective cluster flips. Large correlated regions can change together, reducing critical slowing for this model. Frustration, antiferromagnetic interactions, external fields, and other Hamiltonians require modified constructions; blindly using the same rule is incorrect.
3. Derive replica-exchange acceptance
Two replicas at inverse temperatures \(\beta_i,\beta_j\) have joint weight \(e^{-\beta_iU(R_i)-\beta_jU(R_j)}\). Swapping configurations changes the exponent by \((\beta_i-\beta_j)[U(R_i)-U(R_j)]\), so a symmetric swap accepts with
With \(\beta_i=2,\beta_j=1,U_i=0,U_j=3\) in consistent units, the ratio is \(e^{-3}\). Sending a high-energy configuration into the colder replica is disfavored. Each temperature retains its own target; do not average all temperatures into one result. Temperature spacing needs overlapping energy distributions and adequate round trips. Many successful nearest-neighbor swaps do not alone prove global exploration.
4. Hamiltonian Monte Carlo as a proposal construction
For differentiable \(\pi(R)\propto e^{-U_*(R)}\), introduce auxiliary Gaussian momenta and Hamiltonian \(H_*=U_*+p^TM^{-1}p/2\). Reversible volume-preserving leapfrog integration followed by momentum reversal defines a proposal accepted with \(\min(1,e^{-\Delta H_*})\). Momentum refresh and Metropolis correction are essential parts of the sampling method. The trajectory integration time is an algorithm parameter, not necessarily physical time. Poor integration, constrained coordinates, nondifferentiable targets, or disconnected regions require special treatment.
Measure efficiency against uncertainty in the desired observable at a given cost. A cluster update and a particle move require different work, so raw acceptance percentages are not comparable. Replica exchange spends some effort at temperatures that aid exploration without contributing directly to the final cold estimate. A method can improve one slow coordinate while leaving another poorly mixed; retain observable-specific diagnostics.
5. Exercises and interpretation
Exercise: At \(\beta J=\ln2/2\), what is the ferromagnetic cluster bond probability?
Solution
\(p=1-e^{-\ln2}=1/2\). Bonds are considered only between aligned neighbors in this construction; it is not the probability to accept a single-spin flip.
Exercise: Does a perfect energy-conserving Hamiltonian proposal guarantee ergodicity?
Solution
No. It may follow confined trajectories or fail to reach disconnected support. Momentum refresh and trajectory choices help, but reachability and mixing remain separate questions.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
- Swendsen and Wang (1987), Nonuniversal critical dynamics in Monte Carlo simulations
- Hukushima and Nemoto (1996), Exchange Monte Carlo Method
- Duane et al. (1987), Hybrid Monte Carlo
Related theory: Molecular dynamics · Quantum Monte Carlo
Software connection: CP2K · Quantum ESPRESSO
These software courses provide related background on energies, orbitals, or convergence management; they do not imply that the Monte Carlo or QMC examples on this page were executed there.
7. Related theory and practice
Molecular dynamics · Quantum Monte Carlo · RASPA3