## Definition **The Metropolis acceptance criterion** is the rule that decides whether to accept a candidate solution $x'$ that is worse than the current solution $x$. Given an energy difference $\Delta = f(x') - f(x) > 0$ and a temperature parameter $T$, the move is accepted with probability $P(\text{accept}) = \exp\!\left(-\frac{\Delta}{T}\right)$ and rejected otherwise. Better moves ($\Delta \leq 0$) are always accepted. The criterion originates from the Metropolis–Hastings algorithm (Metropolis et al. 1953), which was designed to sample from the Boltzmann distribution in statistical physics. [[Simulated Annealing]] borrows it directly: at high $T$ the exponential is close to 1, so nearly all moves are accepted (broad exploration); as $T \to 0$ only downhill moves survive (greedy local search). ## Why It Matters Beyond SA The criterion is reusable in any context that requires sampling or escaping local optima: - **MCMC / Metropolis–Hastings sampling** — samples a target probability distribution $\pi(x)$ by accepting proposals with probability $\min\!\left(1,\frac{\pi(x')}{\pi(x)}\right)$, which reduces to the same exponential form under the Boltzmann distribution. - **[[Parallel Tempering]]** — each replica runs its own Metropolis chain; swaps between replicas are themselves accepted with a Metropolis-like criterion. - Any stochastic local search that needs a principled way to balance exploration and exploitation can plug in this rule. The key insight is that $\exp(-\Delta/T)$ is a smooth, temperature-controlled knob: it interpolates between pure random walk ($T \to \infty$) and pure greedy descent ($T \to 0$). ## Related - [[Simulated Annealing]] — uses this criterion at every iteration to decide whether to accept a worse neighbour - [[Cooling Schedule]] — the schedule that lowers $T$ over time, controlling how the acceptance probability shrinks - [[Parallel Tempering]] — runs multiple Metropolis chains at different temperatures - [[Local vs Global Optimum]] — the criterion's purpose is to allow escape from local optima