Competing phases or interactions in complex many-particle systems can result in
free-energy barriers that strongly suppress thermal equilibration. Prominent examples
include frustrated magnets, glasses or proteins. In this talk I will review a line of
research developing adaptive Monte Carlo simulation techniques that systematically
explore and overcome the entropic barriers which cause the slow-down. In contrast to
flat-histogram sampling techniques these approaches do not suffer from critical slowing
down at second-order transitions -- despite using local updates. I will discuss some
recent applications to thermal order-by-disorder transitions in frustrated magnets,
folding transitions in proteins, and first-order transitions in quantum spin systems.
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