Physica A 272 (1999) 188–205 www.elsevier.com/locate/physa Brachistochrone approach to the relaxation dynamics of complex hierarchical systems Gustavo A. Appignanesi Departamento de Qu  mica e Ingenier  a Qu  mica, and Instituto de Matem atica de Bah  a Blanca (INMABB-UNS-CONICET), Universidad Nacional del Sur, Av., Alem 1253, 8000-Bah  a Blanca, Argentina Received 27 November 1998; received in revised form 3 May 1999 Abstract We investigate the structure and the relaxation dynamics of complex hierarchical systems from a variational point of view. First, an additional argument for the use of the ultrametric caricature to describe disordered systems is provided. Focusing on ultrametric models, we show that two relevant dynamical limit behaviors of such models, the limit of convergence of the dynamics and the transition from compact to noncompact exploration, are in fact realizations of brachistochrone relaxation pathways. In turn, by making use of a rugged model whose conformation space topology deviates from ultrametricity under selective controllable conditions, we show that while the exponent of the resulting relaxation law behaves as ruggedness-dependent, its functional form is robust with respect to the introduction of ruggedness. Finally, within this rugged context, the relaxation dynamics of the two above-mentioned limit behaviors are shown to correspond to characteristic relaxation laws: Debye–Kohlrausch and power decay, respectively. c  1999 Elsevier Science B.V. All rights reserved. Keywords: Ultrametric spaces; Relaxation; Kohlrausch law; Compact exploration; Percolation clusters; Rugged systems PACS: 75.40.-y; 61.40.Df; 05.50.+q; 46.30.Jv; 61.40.-a; 64.70.Ew 1. Introduction A variational principle valid in the context of disordered biopolymer folding has al- ready been derived by identifying the brachistochrone or least over-all-time relaxation pathway [1,2]. Nevertheless, the generality of the relaxation situation we considered suggests that the results should be relevant to a more generic context, namely that E-mail address: appignan@criba.edu.ar (G.A. Appignanesi) 0378-4371/99/$ - see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0378-4371(99)00241-1