Spectral signatures of hierarchical relaxation

The concept of phase transition is most precise in the context of equilibrium statistical mechanics in the thermodynamic limit. Unfortunately many systems of interest are excluded by the narrowness of this definition. In this paper we continue our master equation approach [1,2] to this problem and extend that method to structures having strong resemblance to those believed to exist in spin glasses. We remark that for spin glasses the perceived need to take a thermodynamic limit has vastly complicated the enterprise and that a dynamical approach, which is the essence of our method, is being pursued by other workers in this field as well [3]. The master equation approach is applicable to both traditional physical systems and to others of a more general nature [4,5]. In previous publications [1] we used this framework to provide concepts of entropy, dissipation, currents, and fluctuation-dissipation theorems, with or without detailed balance. In [2] we presented a concept of first order phase transition from this perspective. In the present paper we accommodate the hierarchical structure often attributed to spin glasses within the master equation approach, and show how spectral properties (especially left eigenfunctions) reflect the physically significant structure. We comment that our treatment of phase transitions has two principal differences from other approaches. First, it is dynamical, letting the system define its own metastability. Second, it is not infinitely sharp, in the sense that only asymptotic statements are made (no thermodynamic limit is taken). Arguments for these differences have already been made in the relatively simpler case of ordinary (macroscopic) metastability [6]. In Sec. 2 we outline our approach, with emphasis on phase transitions. Following that we comment on systems with slower than exponential relaxation and the implications for the transition matrix spectrum. In Sec. 3 a dynamical distance function is introduced and used to define a coarse graining. Finally in Sec. 4 the uses of our approach for spin-glasses are presented.