5. Ergodicity, barriers, and convergence
Position in the course: Lesson 5 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Transition matrices; eigenvalues
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Equilibrium is not the same as reaching it
For a finite chain, irreducibility means every state communicates with every other state through a finite sequence of allowed moves. Aperiodicity excludes a compulsory return period larger than one. Together they ensure a unique stationary distribution and convergence from any initial distribution. Continuous spaces require analogous measure-theoretic conditions; do not apply a finite-matrix theorem without qualification. Detailed balance establishes the target as stationary but does not establish these other properties.
2. Solve a two-state relaxation exactly
Let \(P=\begin{pmatrix}1-a&a\\b&1-b\end{pmatrix}\) with \(a,b>0\). Solving \(\pi_AP_{AB}=\pi_BP_{BA}\) gives \(\pi_B=a/(a+b)\). If \(p_t\) is the probability of \(B\), then
The second eigenvalue \(\lambda=1-a-b\) controls relaxation. For \(a=b=0.001\), \(\lambda=0.998\) and a characteristic decay time is \(-1/\ln\lambda\approx499.5\) steps. A run of twenty steps can be numerically busy but physically uninformative about equilibrium. If \(a=b=1\), \(\lambda=-1\) and the chain oscillates forever. If both vanish, two disconnected stationary sectors remain.
3. Free-energy barriers rather than acceptance alone
In many-body systems, moving between phases may require crossing a rare region of configuration space. A local move can have moderate acceptance inside each basin while almost never crossing the bottleneck. The relevant barrier often includes entropy and is a free-energy barrier, not simply the highest individual potential energy. Near critical points large-scale fluctuations relax slowly; cluster moves or other nonlocal proposals may change the dynamic sampling behavior without changing equilibrium physics.
4. Practical evidence for convergence
Start independent chains in dispersed states, inspect slow order parameters as well as energy, compare early and late averages, and check whether chains visit competing regions. Discarding a fixed percentage as warm-up is not proof: determine whether the surviving distribution is stable under more warm-up and longer runs. Adapt proposals during a dedicated tuning stage; arbitrary adaptation during production can invalidate the ordinary stationary-chain derivation. Specialized adaptive algorithms need their own convergence conditions.
The slowest mode may be nearly invisible in energy. Two basins can have similar energy but different magnetization or molecular conformations. Monitoring energy alone can miss the mode controlling equilibration. Choose diagnostics related to the scientific question and suspected barriers. Agreement is more convincing when independent starting states deliberately span those barriers rather than merely perturbing the same initial structure.
5. Exercises and explained answers
Exercise: With \(a=0.02,b=0.08\), find the stationary distribution and relaxation eigenvalue.
Solution
\(\pi_B=0.2\) and \(\lambda=0.9\). A bias in \(p_0\) decays by \(0.9^t\). The target probabilities and relaxation time are distinct properties: multiplying both rates by a small common factor leaves \(\pi\) unchanged but slows mixing.
Exercise: Can increasing the number of chains overcome a forbidden transition?
Solution
Independent starts may expose disconnected sectors but cannot fix a reducible transition kernel. Correct sector weights must come from physics or a sampler that permits the missing transitions.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
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