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I. V. Volovich

Publications and source records attributed to I. V. Volovich.

At least 19 recordsLinked to original sources
arXiv2024

On the theory of relativistic Brownian motion

The approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into account the boundedness of speed. An attempt was made to build a relativistic analogue of the Wiener measure as a weak limit of finite-difference approximations. A formula has been proposed for calculating the probability particle transition during relativistic Brownian motion. Calculations were carried out by three different methods with identical results. Along the way, exact and asymptotic formulas for the volume of some parts and sections of an N-1-dimensional unit cube were obtained. They can have independent value.

gr-qc↗
arXiv2023

Analogues of Jacobi and Weyl Theorems for Infinite-Dimensional Tori

Generalizations of the Jacobi and Weyl theorems on finite-dimensional linear flows to the case of linear flows on infinite-dimensional tori are presented. Conditions for periodicity, non-wandering, ergodicity and transitivity of trajectories of an infinite-dimensional linear flow are obtained. It is shown that for infinite-dimensional linear flows there is a new type of trajectories that is absent in the finite-dimensional case.

math.DS↗
arXiv2023

Bose Gas Modeling of the Schwarzschild Black Hole Thermodynamics

Black holes violate the third law of thermodynamics, and this gives rise to difficulties with the microscopic description of the entropy of black holes. Recently, it has been shown that the microscopic description of the Schwarzschild black hole thermodynamics in $D = 4$ spacetime dimensions is provided by the analytical continuation of the entropy of Bose gas with non-relativistic one particle energy to d =-4 negative spatial dimension. In this paper, we show that the D=5 and D=6 Schwarzschild black holes thermodynamics can be modeled by the d-dimensional Bose gas, d=1,2,3..., with the one particle energy $\varepsilon(k)=k^α$ under conditions $α=-d/3$ and $α=-d/4$, respectively. In these cases the free energy of the Bose gas has divergences and we introduce a cut-off and perform the minimal renormalizations. We also perform renormalizations using analytical regularization and prove that the minimal cut-off renormalization gives the same answer as the analytical regularization by the Riemann zeta-function.

hep-th↗
arXiv2017

$p$-Adic Mathematical Physics: The First 30 Years

$p$-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 including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

math-ph↗
arXiv2017

Self-adjoint approximations of degenerate Schrodinger operator

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

math-ph↗
arXiv2016

Flows in nonequilibrium quantum systems and quantum photosynthesis

A three level quantum system interacting with nonequilibrium 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 the dependence of the flows on the state of the environment. We also discuss application of this model to the problem of quantum photosynthesis, in particular, to description of flows of excitons and generation of excitonic coherences.

quant-ph↗
arXiv2016

Dark states in quantum photosynthesis

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 non-decaying "dark" states. We discuss relation of this model to the known from spectroscopical experiments phenomenon of existence of photonic echo in quantum photosynthesis.

physics.bio-ph↗
arXiv2015

Perturbative treatment of inter-site couplings in the local description of open quantum networks

The problem of construction of a quantum master equation for a system of sites weakly coupled to each other and to one or more reservoirs (open quantum network) is considered. Microscopic derivation of a quantum master equation requires a diagonalization of the Hamiltonian of the network, which can be a difficult task. When the inter-site couplings are weak, the local approach, which neglects the influence of the inter-site couplings on the system-reservoir couplings, is often used. Recently, some doubts were cast to the consistency of the local approach. We develop a systematic perturbation expansion to derive corrections to the local approach and establish its range of validity. Using this extension of the local approach, we derive an expression for the heat flux for a particular model and show that it does not violate the second law of thermodynamics.

quant-ph↗
arXiv2014

Fermionic Meixner probability distributions, Lie algebras and quadratic Hamiltonians

We introduce the quadratic Fermi algebra, which is a Lie algebra, and show that the vacuum distributions of the associated Hamiltonians define the fermionic Meixner probability distributions. In order to emphasize the difference with the Bose case, we apply a modification of the method used in the above calculation to obtain a simple and straightforward classification of the 1--dimensional Meixner laws in terms of homogeneous quadratic expressions in the Bose creation and annihilation operators. There is a huge literature of the Meixner laws but this, purely quantum probabilistic, derivation seems to be new. Finally we briefly discuss the possible multi-dimensional extensions of the above results.

math-ph↗
arXiv2013

Quinary lattice model of secondary structures of polymers

In the standard approach to lattice proteins the models based on nearest neighbor interaction are used. In this kind of models it is difficult to explain the existence of secondary structures --- special preferred conformations of protein chains. In the present paper a new lattice model of proteins is proposed which is based on non-local cooperative interactions. In this model the energy of a conformation of a polymer is equal to the sum of energies of conformations of fragments of the polymer chain of the length five. It is shown that this quinary lattice model is able to describe at qualitative level secondary structures of proteins: for this model all conformations with minimal energy are combinations of lattice models of alpha--helix and beta--strand. Moreover for lattice polymers of the length not longer that 38 monomers we can describe all conformations with minimal energy.

cond-mat.soft↗
arXiv2013

A universal boundary value problem for partial differential equations

A new boundary value problem for partial differential equations is discussed. We consider an arbitrary solution of an elliptic or parabolic equation in a given domain and no boundary conditions are assumed. We study which restrictions the boundary values of the solution and its normal derivatives must satisfy. Linear integral equations for the boundary values of the solution and its normal derivatives are obtained, which we call the universal boundary value equations. A universal boundary value problem is defined as a partial differential equation together with the boundary data which specify the values of the solution on the boundary and its normal derivatives and satisfy to the universal boundary value equations. For the equations of mathematical physics such as Laplace's and the heat equation the solution of the universal boundary value problem is presented. Applications to cosmology and quantum mechanics are mentioned.

math.AP↗
arXiv2013

Localization properties of squeezed quantum states in nanoscale space domains

We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.

quant-ph↗
arXiv2013

Semiclassical description of collapses and revivals of quantum wave packets in bounded domains

We study a special kind of semiclassical limit of quantum dynamics on a circle and in a box (infinite potential well with hard walls) as the Planck constant tends to zero and time tends to infinity. The results give detailed information about all stages of evolution of quantum wave packets: semiclassical motion, collapses, revivals, as well as intermediate stages. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most of the time close to uniform. This fact was previously known only from numerical calculations. We apply the obtained results to a problem of classical mechanics: deciding whether recently suggested functional classical mechanics is preferable to traditional Newtonian one from the quantum-mechanical point of view. To do this, we study the semiclassical limit of the Husimi functions of quantum states. We show that functional mechanics remains valid at larger time scales than Newtonian one and, therefore, is preferable. Finally, we analyse the quantum dynamics in a box in case when the size of the box is known with a random error. We show that, in this case, the probability distribution of the position of a quantum particle is not almost periodic, but tends to a limit distribution as time indefinitely increases.

quant-ph↗
arXiv2013

Note on the Unruh Effect

It was suggested by Unruh that a uniformly accelerated detector in vacuum would perceive a noise with a thermal distribution. We obtain a representation of solutions of the wave equation in two dimensions suitable for the Rindler regions. The representation includes the dependence on a parameter. The Unruh field corresponds to a singular limit of the representation.

hep-th↗
arXiv2012

On Holographic Thermalization and Dethermalization of Quark-Gluon Plasma

We estimate the ratio of the thermalization time over the freeze-out time using a holographic AdS-Vaidya model for quark-gluon plasma formed in the heavy ion collisions. In the model the process of thermalization is described as formation of the black hole in AdS space while dethermalization process, related with the freeze-out, as the black hole evaporation due to the Hawking radiation that is modeled by the Vaidya metric with a negative mass. In this model the thermalization takes place only at small scales and absent in the infrared region. At small scales the system tends towards a state of the thermal equilibrium only for the short time after which the processes of dethermalization starts. In this simple model the dethermalization time has a low bound about 7 fm/c which is consistent with experimental data.

hep-th↗
arXiv2012

Rolling in the Higgs Model and Elliptic Functions

Asymptotic methods in nonlinear dynamics are used to improve perturbation theory results in the oscillations regime. However, for some problems of nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and the Friedmann cosmological equations, not only small oscillations regime is of interest but also the regime of rolling (climbing), more precisely the rolling from a top (climbing to a top). In the Friedman cosmology, where the slow rolling regime is often used, the rolling from a top (not necessary slow) is of interest too. In the present work a method for approximate solution to the Higgs equation in the rolling regime is presented. It is shown that in order to improve perturbation theory in the rolling regime turns out to be effective not to use an expansion in trigonometric functions as it is done in case of small oscillations but use expansions in hyperbolic functions instead. This regime is investigated using the representation of the solution in terms of elliptic functions. An accuracy of the corresponding approximation is estimated.

hep-th↗
arXiv2011

Mathematical Physics : Problems and Solutions of The Students Training Contest Olympiad in Mathematical and Theoretical Physics (May 21st - 24th, 2010)

The present issue of the series < > represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and the solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia's Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation. The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics.

math-ph↗