In the Metropolis algorithm, a proposed move increases the system's energy by ΔE > 0. The algorithm's response is to:
AAlways reject the move, since higher-energy states are less probable in the Boltzmann distribution
BAccept the move with probability e^{−ΔE/kT}, occasionally allowing uphill steps
CAccept the move only if the current state has been visited fewer times than average
DDefer the decision until the partition function Z has been computed for normalization
Rejecting all uphill moves would trap the algorithm in local energy minima and produce samples biased toward ground states, not the full thermal distribution. The Boltzmann distribution includes high-energy states, weighted by e^{−E/kT}. Accepting uphill moves with probability e^{−ΔE/kT} is precisely the ratio of Boltzmann weights P(s_new)/P(s_old) = e^{−ΔE/kT}, which is what detailed balance requires. This probabilistic uphill acceptance allows the Markov chain to explore the full configuration space with the correct long-run frequencies.
Question 2 Multiple Choice
What mathematical property guarantees that the Metropolis algorithm's Markov chain has the Boltzmann distribution as its stationary distribution?
AThe algorithm computes the partition function Z at each step to correctly normalize the probabilities
BErgodicity alone — any algorithm that visits all states will eventually sample from the correct distribution
CDetailed balance — the rate of transitions from state s to s′ is equal to the rate of reverse transitions, weighted by Boltzmann factors
DThe acceptance rule was empirically calibrated against exact solutions of the Ising model
Detailed balance — P(s)·A(s→s′) = P(s′)·A(s′→s) — is satisfied by the Metropolis acceptance rule by construction. Combined with ergodicity (the ability to reach any state from any other), detailed balance guarantees the Boltzmann distribution is the unique stationary distribution. Ergodicity alone is insufficient; many ergodic chains converge to wrong distributions. And crucially, Z never needs to be computed: it cancels in the ratio of Boltzmann weights that appears in the acceptance criterion.
Question 3 True / False
The Metropolis algorithm should compute the partition function Z to correctly normalize probabilities and sample from the Boltzmann distribution.
TTrue
FFalse
Answer: False
This is the key practical insight of the Metropolis algorithm: Z cancels. The acceptance probability involves only the ratio e^{−ΔE/kT} = e^{−E(s_new)/kT} / e^{−E(s_old)/kT}, from which Z (which would appear identically in numerator and denominator) cancels exactly. This is crucial because computing Z requires summing over all 2^N microstates — an intractable calculation for any large system. The algorithm samples correctly from the Boltzmann distribution without ever knowing the absolute probability of any state.
Question 4 True / False
After equilibration, consecutive samples generated by the Metropolis algorithm are statistically independent of one another.
TTrue
FFalse
Answer: False
Consecutive Metropolis samples are correlated — each proposed state is derived by a small perturbation of the previous one, so nearby samples are similar. The correlation decays over a characteristic time τ (the autocorrelation time). To obtain approximately independent samples, you must space them by several τ. This correlation reduces the effective sample size: M correlated samples contain less information than M independent samples. Equilibration removes the memory of the initial state, but it does not remove correlations between successive states.
Question 5 Short Answer
Why does the Metropolis algorithm sometimes accept moves that increase the system's energy, and why is this acceptance essential for sampling the correct thermal distribution?
Think about your answer, then reveal below.
Model answer: The thermal (Boltzmann) distribution is not concentrated at the energy minimum — it is a broad distribution that includes higher-energy configurations, weighted by e^{−E/kT}. If the algorithm only accepted energy-decreasing moves, it would converge to a local minimum and sample only near-ground-state configurations, completely missing the thermally relevant configurations at finite temperature. Accepting uphill moves with probability e^{−ΔE/kT} — exactly the ratio of Boltzmann weights — allows the chain to explore the full configuration space with frequencies proportional to their Boltzmann weights. This is what detailed balance enforces: the long-run time spent in each state matches its thermal probability.
The insight is that Monte Carlo sampling is not optimization. You don't want the lowest-energy state; you want a representative thermal ensemble. The Metropolis acceptance criterion is precisely calibrated to produce that ensemble, via detailed balance. Rejecting all uphill moves would be a zero-temperature simulation regardless of T — a minimization problem, not a sampling problem.