arXiv subjects

A. S. Trushechkin

Publications and source records attributed to A. S. Trushechkin.

23 records · Page 2Linked to original sources

arXiv2011

Derivation of the Boltzmann equation and entropy production in functional mechanics

A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation of classical mechanics. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which are rational numbers. Hence, one can speak about the accuracy of the initial probability density function in the Liouville equation. We assume that the initial data for the microscopic density functions are assigned by the macroscopic one (so, one can say about a kind of hierarchy and subordination of the microscale to the macroscale) and derive the Boltzmann equation, which leads to the entropy production.

math-ph↗

arXiv2011

Hierarchy of space-time structures, Boltzmann equation, and functional mechanics

In this report we discuss the organization of different levels of nature and the corresponding space-time structures by the consideration of a particular problem of time irreversibility. The fundamental time irreversibility problem consists in the following: how to reconcile the time-reversible microscopic dynamics and the irreversible macroscopic one. The recently proposed functional formulation of mechanics is aimed to solve this problem. The basic concept of this formulation is not a material point and a trajectory, like in the traditional formulation of mechanics, but a probability density function. Even if we deal with a single particle (not with an ensemble of particles), we describe its state as a probability density function. We justify this approach using measurement theory. A particular problem in the framework of the irreversibility problem is the derivation of the Boltzmann kinetic equation from the equations of microscopic dynamics. We propose a procedure for obtaining the Boltzmann equation from the Liouville equation based on the BBGKY hierarchy, the recently proposed functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. The notion of a space-time structure is introduced. It takes into account not only the space-time itself (i.e., a pseudo-Riemannian manifold), but also a characteristic length and time. The space-time structures form a hierarchy in sense that the initial values for the processes on the microscopic space-time structure (interactions of the particles) are assigned from the processes on the macroscopic one (kinetic phenomena).

math-ph↗

arXiv2009

Functional Classical Mechanics and Rational Numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A "functional" formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a "beam" of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.

math-ph↗

arXiv2006

On Standards and Specifications in Quantum Cryptography

Quantum cryptography is going to find practically useful applications. Recently some first quantum cryptographic solutions became available on the market. For clients it is important to be able to compare the quality and properties of the proposed products. To this end one needs to elaborate on specifications and standards of solutions in quantum cryptography. We propose and discuss a list of characteristics for the specification, which includes numerical evaluations of the security of solution and can be considered as a standard for quantum key distribution solutions. The list is based on the average time of key generation depending on some parameters. In the simplest case for the user the list includes three characteristics: the security degree, the length of keys and the key refresh rate.

quant-ph↗

arXiv2005

General model of quantum key distribution

A general mathematical framework for quantum key distribution based on the concepts of quantum channel and Turing machine is suggested. The security for its special case is proved. The assumption is that the adversary can perform only individual (in essence, classical) attacks. For this case an advantage of quantum key distribution over classical one is shown.

quant-ph↗