We demonstrate that p-adic analysis is a natural basis for the construction of a wide variety of the ultrametric diffusion models constrained by hierarchical energy landscapes. A general analytical description in terms of p-adic analysis is given for a class of models. Two exactly solvable examples, i.e. the ultrametric diffusion constraned by the linear energy landscape and the ultrametric diffusion with reaction sink, are considered. We show that such models can be applied to both the relaxation in complex systems and the rate processes coupled to rearrangenment of the complex surrounding.

Application of p-adic analysis to models of spontaneous breaking of the replica symmetry V.A.Avetisov, A.H.Bikulov, S.V.Kozyrev April 3, 2018 AbstractMethods of p-adic analysis are applied to the investigation of the spontaneous symmetry breaking in the models of spin glasses. A p-adic expression for the replica matrix is given and moreover the replica matrix in the models of spontaneous breaking of the replica symmetry in the simplest case is expressed in the form of the Vladimirov operator of p-adic fractional differentiation. Also the model of hierarchical diffusion (that was proposed to describe relaxation of spin glasses) investigated using p-adic analysis.

Abstract. This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p-adic analysis. We show that three types of relaxation fitted into the Kohlrausch-Williams-Watts law, the power decay law, or the logarithmic decay law, are similar random processes. Inherently, these processes are ultrametric and are described by the p-adic master equation. The physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape. We also discuss relations between the relaxation kinetics and the energy landscapes.

In this paper, we consider a homogeneous Markov process ξ(t; ω) on an ultrametric space Qp, with distribution density f(x, t), x ∊ Qp, t ∊ R+, satisfying the equation , usually called the ultrametric diffusion equation. We construct and examine a random variable that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0, t] and give its asymptotic estimates for large t.

Surprisingly accurate description of spectral diffusion in deeply frozen proteins is constructed on the basis of protein ultrametricity. We show that the ultrametric diffusion equation with self-similar hierarchy of the transition rates offers a simple description of protein dynamics in the spectral diffusion context. Earlier the same ultrametric diffusion equation has been used in the description of ligand-rebinding kinetics of myoglobin. Thus ultrametricity offers a universal background for the description of protein dynamics on very different scales of protein motions.

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