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When liquids are cooled sufficiently rapidly below their melting temperature,they may bypass crystalization and, instead, enter a long-lived metastablesupercooled state that has long been the focus of intense research. Althoughthey exhibit strikingly different properties, both the (i) long-livedsupercooled liquid state and (ii) truly equilibrated (i.e., conventionalequilibrium fluid or crystalline) phases of the same material share anidentical Hamiltonian. This suggests a mapping between dynamical and otherobservables in these two different arenas. We formalize these notions via asimple theorem and illustrate that given a Hamiltonian defining the dynamics:(1) the reduced probability densities of all possible stationary states arelinear combinations of reduced probability densities associated with thermalequilibria at different temperatures, chemical potentials, etc. (2) Excusingspecial cases, amongst all of these stationary states, a clustering ofcorrelations is only consistent with conventional thermal equilibrium states(associated with a sharp distribution of the above state variables). (3) Otherstationary states may be modified so as to have local correlations. Thesedeformations concomitantly lead to metastable (yet possibly very long-lived)states. Since the lifetime of the supercooled state is exceptionally longrelative to the natural microscopic time scales, their reduced probabilitydensities may be close to those that we find for exact stationary states (i.e.,a weighted average of equilibrium probability densities at different statevariables). This form suggests several new predictions such as the existence ofdynamical heterogeneity stronger than probed for thus far and a relationbetween the specific heat peak and viscosities. Our theorem may further placeconstraints on the putative "ideal glass" phase.

Reduced probability densities of long-lived metastable states as those  of distributed thermal systems: possible experimental implications for  supercooled fluids