thermalized

Thermalized describes a system that has reached thermal equilibrium with its surroundings or a heat bath. In this state, energy exchange has occurred so that all accessible degrees of freedom share a common temperature, and the distribution over microstates corresponds to a Boltzmann distribution at that temperature in classical contexts, or a Gibbs state in quantum contexts.

In practice, thermalization arises from interactions such as collisions, radiation, or coupling to an environment. In classical gases, frequent collisions redistribute energy until the system exhibits a Maxwell-Boltzmann energy distribution and equipartition of energy among degrees of freedom. In quantum systems, the thermal state is described by ρ proportional to e^{-H/kT}, with observables reflecting thermodynamic expectations. Even isolated quantum systems can show thermalized behavior in observables due to concepts like the eigenstate thermalization hypothesis, even though the global state remains pure.

Timescales for thermalization vary with interaction strength, density, dimensionality, and coupling to the environment, ranging from very fast (picoseconds to nanoseconds in some solids) to much longer in dilute gases. Strongly interacting or chaotic systems tend to thermalize efficiently, whereas integrable or weakly interacting systems may relax slowly or exhibit non-thermal steady states.

Applications and implications: thermalization underpins thermodynamics and statistical mechanics, guiding how systems approach standard temperature-based descriptions after perturbations or drives. See also thermal equilibrium, Boltzmann distribution, Gibbs state, and eigenstate thermalization hypothesis.