# Sergei V. KozyrevRussian Academy of Sciences | RAS · Steklov Mathematical Institute

Sergei V. Kozyrev

doctor of science

## About

115

Publications

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2,369

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Introduction

Additional affiliations

January 2004 - present

## Publications

Publications (115)

p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas...

We present a brief review of some parts of p-adic mathematical physics related to the scientific work of Branko Dragovich on the occasion of his 70th birthday.

A three-level quantum system interacting with non-equilibrium environment is investigated. The stationary state of the system is found (both for non-coherent and coherent environment) and relaxation and decoherence to the stationary state is described. The stationary state of the system will be non-equilibrium and will generate flows. We describe t...

We consider the dynamics of an open three-level quantum degenerate system. One of the levels in this system is degenerate. The system interacts with three reservoirs (quantum fields) and a classical external field. We show that nondecaying so-called dark states are generated in this system. Since the interactions of the degenerate level with two di...

We discuss a model of quantum photosynthesis with degeneracy in the light-harvesting system. We consider interaction of excitons in chromophores with light and phonons (vibrations of environment). These interactions have dipole form but are different (are related to non-parallel vectors of "bright" states). We show that this leads to excitation of...

We present a brief biography and a brief review of the scientific work of Branko Dragovich on the occasion of his 70th birthday on March 11, 2015.

We review applications of p-adic and ultrametric methods in the theory of complex systems. We consider the following examples: the p-adic parameterization of the Parisi matrix in the replica method; the method of hierarchical (interbasin) kinetics, which allows describing macromolecular dynamics by models of ultrametric diffusion; the two-dimension...

We consider the problem of excitation energy transfer in quantum many-particle systems with a dipole interaction. The considered exciton transfer mechanism is based on quantum interference. We show that by a special choice of interaction parameters, an enhancement of the exciton transfer to a sink and suppression of the transfer to alternative sink...

We discuss a model of protein conformations where conformations are
combinations of fragments from some small set. For these fragments we consider
a distribution of frequencies of occurrence of pairs (sequence of amino acids,
conformation), averaged over some balls in the spaces of sequences and
conformations. These frequencies can be estimated due...

The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first time, bases of compactly supported eigenvectors for p-adic pseudodifferential operators were considered by V.S. Vladimirov. In contrast to real wavelets...

Clustering procedure for the case where instead of a fixed metric one applies
a family of metrics is considered. In this case instead of a classification
tree one obtains a classification network (a directed acyclic graph with non
directed cycles).
Relation to Bruhat-Tits buildings is discussed. Dimension of a general
cluster system is considered.

We discuss the approach to investigation of molecular machines using systems
of integro--differential ultrametric (p-adic) reaction--diffusion equations
with drift. This approach combines the features of continuous and discrete
dynamic models. We apply this model to investigation of actomyosin molecular
motor.
The introduced system of equations is...

Time series defined by a p-adic pseudo-differential equation is investigated
using the expansion of the time series over p-adic wavelets. Quadratic
correlation function is computed. This correlation function shows a
degree--like behavior and is locally constant for some time periods. It is
natural to apply this kind of models for the investigation...

We present a brief biographical review of the scientific work and achievements of Vladimir M. Shelkovich on the occasion of his sudden death in February 2013.

We present a brief information on “The Workshop on p-Adic Methods for Modeling of Complex Systems”, which was held in the Center for Interdisciplinary Research (Zentrum für interdisziplinäre Forshung — ZiF), Bielefeld University, Bielefeld, Germany, April 15–19, 2013.

We present a brief biographical review of the scientific work and achievements of Vasiliy Sergeevich Vladimirov on the occasion of his death on November 3, 2012.

We introduce a lattice model of protein conformations which is able to
reproduce second structures of proteins (alpha--helices and beta--sheets). This
model is based on the following two main ideas. First, we model backbone parts
of amino acid residues in a peptide chain by edges in the cubic lattice which
are not parallel to the coordinate axes. S...

We construct a model of a lattice polymer which describes secondary
structures of proteins. In this model the energy of a conformation of a polymer
is equal to a sum of energies of conformations of segments of the polymer chain
of the length five.
We show that for this model with cooperative interaction all conformations
with minimal energy are com...

In the present paper we discuss the clustering procedure in the case where
instead of a single metric we have a family of metrics. In this case we can
obtain a partially ordered graph of clusters which is not necessarily a tree.
We discuss a structure of a hypergraph above this graph. We propose two
definitions of dimension for hyperedges of this h...

We introduce a new procedure for training of artificial neural networks by
using the approximation of an objective function by arithmetic mean of an
ensemble of selected randomly generated neural networks, and apply this
procedure to the classification (or pattern recognition) problem. This approach
differs from the standard one based on the optimi...

We present a brief review of the scientific work and achievements of Igor V. Volovich on the occasion of his 65th birthday.

In the present paper we propose to describe gene networks in biological
systems using probabilistic algorithms. We describe gene duplication in the
process of biological evolution using introduction of the replica procedure for
probabilistic algorithms. We construct the examples of such a replica procedure
for hidden Markov models. We introduce the...

A wide class of p-adic integral operators in bases of p-adic wavelets is considered and matrix elements of the corresponding matrices of these operators are shown to be nonzero only on a finite number of main diagonals. A method is described for approximating real integral operators by p-adic ones in wavelet bases. This approach is based on the exi...

In the p-adic approach to description of the genetic code the degeneracy of the genetic code is described by some p-adic metric (in particular, 2-dimensional 2-adic metric). In this paper we consider some deformation of the standard multidimensional
p-adicmetric. This kind of deformed metric possesses a different set of balls (compared to the stand...

The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These groups are automorphisms of the tree of balls in the p-adic space. In the present paper we consider deformations of the standard p-adic metric in many...

We discuss a multidimensional generalization of the clustering method. In our approach, the clustering is realized by partially
ordered hypergraphs belonging to some family. The suggested procedure is applicable in the case where the original metric
depends on a set of parameters. The clustering hypergraph studied here can be regarded as an object...

A new approach to the proof of the Arrhenius formula of kinetic theory is
proposed. We prove this formula starting from the equation of diffusion in a
potential. We put this diffusion equation in the form of evolutionary equation
generated by some Schroedinger operator. We show that the Arrhenius formula for
the rate of over the barrier transitions...

We discuss the interbasin kinetics approximation for random walk on a complex (rugged) landscape of energy. In this approximation
the random walk is described by the system of kinetic equations corresponding to transitions between the local minima of energy.
If we approximate the transition rates between the local minima by the Arrhenius formula th...

A team of researchers conducted a study to propose an alternative approach to investigating genetics in mathematical physics. Their alternative approach consisted in considering a p-adic parameterization of the genetic code where the code was a locally constant mapping of a p-adic argument. The p-adic approach investigated the local constancy of ma...

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find
the...

Recently systems of p-adic numbers were applied to genetics. These systems are fundamentally hierarchical. Geometrically they are represented by homogeneous trees (here p is the number of branches leaving each vertex). It was shown that p-adics provides a natural number theoretic representation of the genetic code. In our previous paper we construc...

We discuss the relation between ultrametric analysis, wavelet theory, and cascade models of turbulence. We construct explicit
solutions of the nonlinear ultrametric integral equation with quadratic nonlinearity, using a recursive hierarchical procedure
analogous to the procedure used for the cascade models of turbulence.

A multidimensional basis of p-adic wavelets is constructed. The relation of the constructed basis to a system of coherent states i.e., orbit of action)
for some p-adic group of linear transformations is discussed. We show that the set of products of the vectors from the constructed basis
and p-roots of unity is the orbit of the corresponding p-adic...

In this paper we demonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, degeneracy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic distance is an ultrametric and applications of ultrametric in bioinformatics are not surprising. However, by us...

A brief review of some selected topics in p-adic mathematical physics is presented. Comment: 36 pages

In this paper we denonstrate that the use of the system of 2-adic numbers
provides a new insight to some problems of genetics, in particular, generacy of
the genetic code and the structure of the PAM matrix in bioinformatics. The
2-adic distance is an ultrametric and applications of ultrametrics in
bioinformatics are not surprising. However, by usi...

The general construction of frames of p-adic wavelets is described. We
consider the orbit of a mean zero generic locally constant function with
compact support (mean zero test function) with respect to the action of the
p-adic affine group and show that this orbit is a uniform tight frame. We
discuss relation of this result to the multiresolution w...

We discuss the interbasin kinetics approximation for random walk on a complex landscape. We show that for a generic landscape the corresponding model of interbasin kinetics is equivalent to an ultrametric diffusion, generated by an ultrametric pseudodifferential operator on the ultrametric space related to the tree of basins. The simplest example o...

The spectral theory of pseudodifferential operators on ultrametric spaces of general form is investigated with the use of the analysis of ultrametric wavelets. Bases of ultrametric wavelets are constructed on ultrametric spaces of analytic type; it is proved that bases of ultrametric wavelets are bases of eigenvectors for the introduced pseudodiffe...

We construct a new orthonormal basis of eigenfunctions of the Vladimirov -adic fractional differentiation operator. We construct a map of the -adic numbers onto the real numbers (the -adic change of variables), which transforms the Haar measure on the -adic numbers to the Lebesgue measure on the positive semi-axis. The -adic change of variables (fo...

We develop an analysis of wavelets and pseudodifferential operators on
multidimensional ultrametric spaces which are defined as products of locally
compact ultrametric spaces. We introduce bases of wavelets, spaces of
generalized functions and Lizorkin generalized functions on multidimensional
ultrametric spaces.
We also consider some family of pse...

We show that translations and dilations of a p-adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p-adic wavelets. In this sense the continuous p-adic wavelet transform coincides with the discrete p-adic wavelet transform. The p-adic multiresolution approximation is introduced and re...

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8\times 8 table. This table (which we call the dyadic plane) possesses the natural 2-adic ultrametric. We show that after this parametrization the genetic code will be a locally constant map of the simple form. The local constancy of this map will describe d...

Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball. © 2006 American Institute of Physics

We use the stochastic limit technique to predict a new phenomenon concerning a two-level atom with degenerate ground state interacting with a quantum field. We show, that the field drives the state of the atom to a stationary state, which is non-unique, but depends on the initial state of the system through some conserved quantities. This non uniqu...

Family of replica matrices, related to general ultrametric spaces with
general measures, is introduced. These matrices generalize the known Parisi
matrices. Some functionals of replica approach are computed. Replica symmetry
breaking solution is found.

Family of replica matrices, related to general ultrametric spaces, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed. Comment: 19 pages

Replica symmetry breaking solutions for the new replica anzats, related to general ultrametric spaces, are investigated. A variant of analysis on trees is developed and applied to the computation of the n\to0 limit in the new replica anzats. Comment: 22 pages

Gaussian random field on general ultrametric space is introduced as a
solution of pseudodifferential stochastic equation. Covariation of the
introduced random field is computed with the help of wavelet analysis on
ultrametric spaces.
Notion of ultrametric Markovianity, which describes independence of
contributions to random field from different ult...

We discuss ultrametric pseudodifferential operators and wavelets and applications to models of interbasin kinetics. We show, that, using the language of ultrametric pseudodifferential operators, it is possible to describe interbasin kinetics for general complex landscape.

A study was conducted to construct models of quantum dynamics on general ultrametric spaces and prove the localization property for an ultrametric free quantum particle. An ultrametric quantum particle without any group structure and propagating in a generic ultrametric space was considered for the study. The dynamics of the particle was described...

Free evolution for quantum particle in general ultrametric space is
considered. We find that if mean zero wave packet is localized in some ball in
the ultrametric space then its evolution remains localized in the same ball.

We consider the procedure for analytic continuation of the replica matrix. We formulate a particular form of this procedure
in which the analytic continuation is defined by a sequence of maps. Using this definition, we construct a solution that breaks
the Parisi replica symmetry and find the corresponding p-adic pseudodifferential operator.

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators on this ultrametric spaces is introduced. We show that these operators are diagonal in these ultrametric wavelet bases. A map of considered...

The general non-degenerate p-adic operators of ultrametric diffusion are introduced. Bases of eigenvectors for the introduced operators are constructed and the corresponding eigenvalues are computed. The long-time relaxation behavior of the ultrametric diffusion generated by the introduced operators are investigated.

A family of orthonormal bases of ultrametric wavelets in the space of
quadratically integrable with respect to arbitrary measure functions on general
(up to some topological restrictions) ultrametric space is introduced.
Pseudodifferential operators (PDO) on the ultrametric space are investigated.
We prove that these operators are diagonal in the i...

A family of orthonormal bases, the ultrametric wavelet bases, is introduced
in quadratically integrable complex valued functions spaces for a wide family
of ultrametric spaces.
A general family of pseudodifferential operators, acting on complex valued
functions on these ultrametric spaces is introduced. We show that these
operators are diagonal in...

We discuss the stochastic limit approach to superfluidity.
Starting from the usual Hamiltonian describing the interaction
between the Bose condensate and the normal phase, we prove the
existence of superfluidity in the stochastic limit and deduce a
nonlinear (quadratic) kinetic equation describing the evolution
of the density of superfluid liquid.

We investigate, using the stochastic limit method, the coherent quantum control of a 3-level atom in $\Lambda$-configuration interacting with two laser fields. We prove that, in the generic situation, this interaction entangles the two lower energy levels of the atom into a single qubit, i.e. it drives at an exponentially fast rate the atom to a st...

We introduce a new wide class of p-adic pseudodifferential operators. We show that the basis of p-adic wavelets is the basis of eigenvectors for the introduced operators.

The contextual probabilistic quantization procedure is formulated. This approach to quantization has much broader field of applications, compared with the canonical quantization. The contextual probabilistic quantization procedure is based on the notions of probability context and the principle of complementarity of probabilities. The general defin...

Irreversibility appears at the microscopic level in radiative phenomena, decoherence, CP violation. Radiation interactions of single atoms are discussed by Walther. A scheme providing optical evidence of macroscopic quantum interference and the associated decoherence are discussed by Arecchi and Montina. Accardi and Kozyrev discuss decoherence of s...

Two examples of the situation when the classical observables should be described by a noncommutative probability space are investigated. A possible experimental approach to find quantum-like correlations for classical disordered systems is discussed. The interpretation of noncommutative probability in experiments with classical systems as a result...

A 2-level atom with degenerate ground state interacting with a quantum field
is investigated. We show, that the field drives the state of the atom to a
stationary state, which is non-unique, but depends on the initial state of the
system through some conserved quantities. This non-uniqueness follows from the
degeneracy of the ground state of the at...

We define dynamical systems where time is a quantum group. We give the definition of quantum ergodicity for the introduced dynamical system with noncommutative (or quantum) time, and discuss the examples.

The alternative to the replica procedure, which we call the noncommutative replica procedure, is discussed. The detailed comparison with the standard replica procedure is performed.

Representation of the Cuntz algebra in the space of (complex valued) functions on p-adic disk is introduced. The relation of this representation and the free coherent states is investigated.

Rigged Hilbert space of the free coherent states is investigated. We prove that this rigged Hilbert space is isomorphous to the space of generalized functions on p-adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show, that the considered example realises the isomorphism of the...

p-Adic and noncommutative analysis are applied to describe phase transitions in disordered systems. In the noncommutative replica approach we replicate the disorder instead of the system degrees of freedom. The noncommutatibe replica symmetry breaking is formulated using the language of noncommutative analysis. This allows to derive the ultrametric...

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...

In the present paper we outline the stochastic limit approach to superfluidity. The Hamiltonian describing the interaction between the Bose condensate and the normal phase is introduced. Sufficient in the stochastic limit condition of superfluidity is proposed. Existence of superfluidity in the stochastic limit of this system is proved and the non-...

The interaction of 3-level system with a quantum field in a non-equilibrium state is considered. We describe a class of states of the quantum field for wich a stationary state drives the system to inverse populated state. We find that the quotient of the population of the energy levels in the simplest case is described by the double Einstein formul...

The stimulated emission from an atom interacting with radiation in non-equilibrium state is considered. The stochastic limit, applied to the non-relativistic Hamiltonian describing the interaction, shows that the state of atoms, driven by some non-equilibrium state of the field approaches a stationary state which can continuously emit photon, unlik...

Collective operators that describe interaction of generic quantum system with
discrete spectrum with a quantum field are investigated. These operators,
considered as operators in the entangled Fock space (space generated by action
of collective creations on the vacuum) in the stochastic limit satisfy a
particular kind of Quantum Boltzmann (or free)...

New orthonormal basis of eigenfunctions for the Vladimirov operator of p-adic
fractional derivation is constructed. The map of p-adic numbers onto real
numbers (p-adic change of variables) is considered. This map (for p=2) provides
an equivalence between the constructed basis of eigenfunctions of the
Vladimirov operator and the wavelet basis in L^2...

The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In...

A new infinitesimal characterization of completely positive but not necessarily homomorphic Markov flows from a C*-algebra to bounded operators on the boson Fock space over L2(R) is given. Contrarily to previous characterizations, based on stochastic differential equations, this characterization is universal, i.e., valid for arbitrary Markov flows....

Methods of p-adic analysis are applied to the investigation of spontaneous symmetry breaking in the models of spin glasses. A p-adic expression for the Parisi replica matrix is given and, moreover, the Parisi replica matrix in models of spontaneous breaking of the replica symmetry in the simplest case is expressed in the form of the Vladimirov oper...

The stochastic limit for the system of spins interacting with a boson field is investigated. In the finite volume an application of the stochastic golden rule shows that in the limit the dynamics of a quantum system is described by a quantum white noise equation that after taking of normal order is equivalent to quantum stochastic differential equa...

A model of particle interacting with quantum field is considered. The model includes as particular cases the polaron model and non-relativistic quantum electrodynamics. We compute matrix elements of the evolution operator in the stochastic approximation and show that depending on the state of the particle one can get the non-exponential decay with...

When the stochastic approximation is used to calculate correlation functions in the model of a particle interacting with a
quantum field, a new algebra with temperature-dependent commutation relations appears. This algebra generalizes the free (Boltzmann)
algebra.

A model of particle interacting with quantum field is considered. The model includes as particular cases the polaron model and non-relativistic quantum electrodynamics. We show that the field operators obey q-commutation relations with q depending on time. After the stochastic (or van Hove) limit, due to the nonlinearity, the atomic and field degre...

## Projects

Project (1)

Mathematical study of systems and their dynamics with ultrametric space of states.