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.

Reasoning from two basic principles of molecular physics, P invariance of electromagnetic interaction and the second law of thermodynamics, one would conclude that mirror symmetry is retained in the world of chiral molecules. This inference is fully consistent with what is observed in inorganic nature. However, in the bioorganic world, the reverse is true. Mirror symmetry there is definitely broken. Is it possible to account for this phenomenon without going beyond conventional concepts of the kinetics of enantioselective processes? This study is an attempt to survey all existing hypotheses concerning this phenomenon.Operation of mirror reflection, or space inversion P, enables one to classify any molecular structure under either of two groups. One group involves molecules having neither symmetry planes nor symmetry centers, i.e., noninvariant with respect to P. These molecules occur in the form of two mirror antipodes (L and D enantiomers), possess optical activity, and are called chiral (from the Greek word XsLp, meaning hand). Among members of the other group are achiral molecules having either symmetry planes or symmetry centers. These molecules are invariant with respect to P and are optically inactive.The main characteristic of the chemistry of chiral compounds is associated with P invariance of electromagnetic interaction. This type of interaction as a rule dominates coupling of intramolecular electrons and nuclei, and therefore, the states of a chiral molecule are described by a symmetric double-well potential with minima corresponding to the L and D configurations (1).In compounds with an asymmetry center, e. However, in living nature, the situation changes dramatically. Broken mirror symmetry of bioorganic objects was first noticed by Louis Pasteur and led him to the conclusion that the molecular substrate of life was not only chiral but also asymmetric (4).What can be said about it now that relatively simple organisms have been studied in so much detail that apparently the only question that remains unanswered is how all these organisms could arise?It is common knowledge that polymer constituents of the double-stranded DNA structure may involve millions of nucleotide links, that similar RNA chains incorporate hundreds and even thousands of nucleotide monomers, and that polymer chains of enzymes usually consist of several hundred amino acid links. DNA and enzymes play essentially different roles; DNA macromolecules are informational carriers, whereas macromolecules of enzymes are functional carriers. RNA plays a part of an intermediary between DNA and enzymes and occasionally takes on the duties of either of the sides (5, 6).However, from the "chiral" viewpoint, all these biopolymers feature a remarkable trait, namely, nucleotide links of RNA and DNA incorporate exclusively D-ribose and D-deoxyribose, respectively, whereas enzymes involve solely L enantiomers of amino acids. In other words, the primary structures of DNA, RNA, and enzymes are homochiral. This property is inherent in all informat...

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.

Biological polymers have a preferred chirality ond can replicate themselves. Physical arguments provide insight into which of these unique and apparently related properties evolved first, and by what mechanism.

We consider the canonical ensemble of N -vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p −1 ] almost full subgraphs (cliques) above critical fugacity, µc, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of two-zonal support for µ > µc. Eigenvalue tunnelling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value N p. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.Investigation of critical and collective effects in graphs and networks has becoming a new rapidly developing interdisciplinary area, with diverse applications and variety of questions to be asked, see [1] for review. Ensembles of random Erdős-Rényi topological graphs (networks) provide an efficient laboratory for testing collective phenomena in statistical physics of complex systems, being also tightly linked to conventional random matrix theory. Besides investigating typical statistical properties of networks, like vertex degree distribution, clustering coefficients, "small world" structure etc, last two decades have been marked by rapidly growing interest in more refined graph characteristics, such as distribution of small subgraphs involving triads of vertices.Triadic interactions, being the simplest interactions beyond the free-field theory, play crucial role in the network statistics. Presence of such interactions is responsible for emergence of phase transitions in complex distributed systems. First example of a phase transition in random networks, known as Strauss clustering [2], has been treated by the Random Matrix Theory (RMT) in [3]. It was argued that, when the increasing fugacity, µ, the system develops two phases with essentially different triad concentrations. At large µ the system falls into the Strauss phase with the single clique of nodes. The condensation of triads is a non-perturbative phenomenon identified in [4] with the 1st order phase transition in the framework of mean-field cavity-like approach.Similar critical behavior was found in [5] for the vertexdegree-conserved ER graphs. It was demonstrated in the framework of the mean-field approach that the phase transition takes place in this case as well. The hysteresis for dependence of the triad concentration on the fugacity, µ also has been observed in [5]. For bi-color networks with conserved vertex degree a new phenomena of a wide plateau formation in concentration of black-white bonds as a function of the fugacity of unicolor triples of bonds has been found in...

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.

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95% of all finite subgraphs, and the distribution of linear chains is purely exponential. We analyze in detail the spectral density of ensembles of linear subgraphs, discuss its ultrametric nature and show that near the spectrum boundary, the tail of the spectral density exhibits a Lifshitz singularity typical for Anderson localization. We also discuss an intriguing connection of the spectral density to the Dedekind η-function. We conjecture that ultrametricity is inherit to complex systems with extremal sparse statistics and argue that a numbertheoretic ultrametricity emerges in any rare-event statistics. arXiv:1506.05037v1 [cond-mat.dis-nn]