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Ergodicity-breaking transition and high-frequency response in a simple free-energy landscape

Here {/i„} is the set of functions specifying the range of each subwell, i.e., μ η= 1 if φ°η — Δ/2< ф°п + Δ/2 and zero oth­ erwise. The curvature of the megavalley is denoted by R.

The dynamics is modeled by relaxation in this free-energy surface and is defined by the Langevin equation

where 77 is a Gaussian noise with zero average and variance (η(ί)η(ί')) = Гô(t-1'). The temperature scale is set by β = Δ2/Γ. In the absence of any subvalley structure, (all rn = 0), Eq. (2) results in a Debye relaxation spectrum with a

relaxation time of HR. If R is taken to be of the form as­ sumed in Landau theory, such that it vanishes linearly at the critical temperature, then Eq. (2) provides a mean-field de­ scription of critical slowing down [5]. The effect of the sub­ valley structure on the relaxation spectrum, and the nature and existence of phase transitions in the two-dimensional space spanned by R and ß are the subjects of this paper.