arXiv subjects
Publications and source records attributed to Yuriy E. Kuzovlev.
It is shown that if kinetics of quantum transitions takes account of energy uncertainty of intermediate states, then it creates non-decaying correlations and non-averagable (flicker) fluctuations in the energy as well as in rates of transitions-induced irreversible processes, in particular, flicker noise or maybe suppression of mobility (rate of wandering) of particle interacting with thermally equilibrium medium
We suggest some principal ideas on origin, statistical properties and theoretical description of 1/f-noise exactly as they for the first time were expounded in our preprint published in Russian in 1982, and supplement them with short today's comments and selected references, with wish to support improvements of present generally poor ideologic and mathematical base of the 1/f-noise theory.
Influence of equilibrium thermal 1/f-type mobility fluctuations on time-of-flight measurements is considered. We show that it can explain experimental time dependencies of transient photocurrents.
It is shown that early suggested derivation of the Boltzmann kinetic equation for dilute hard sphere gas from the time-reversible BBGKY equations is incorrect since in fact a priori substitutes for them definite irreversible equations. Alternative approach to analysis of the hard sphere gas is formulated which conserves the reversibility and makes it clear that at any gas density one can reduce the BBGKY equations to the Boltzmann equation only in case of spatially uniform gas.
The BBGKY hierarchy of equations for a particle interacting with ideal gas is analyzed in terms of irreducible many-particle correlations between gas atoms and the particle's motion. The transition to the hard-sphere interaction is formulated from viewpoint of the recently discovered exact relations connecting the correlations with the particle's probability distribution. Then the Boltzmann-Grad limit is considered and shown not to lead to the Bolzmann hierarchy and the molecular chaos, since correlations of all orders keep significant.in this limit, merely taking a singular form.
Statistics of molecular random walks in a fluid is considered with the help of the Bogolyubov equation for generating functional of distribution functions. An invariance group of solutions to this equation as functions of the fluid density is discovered. It results in many exact relations between probability distribution of the path of a test particle and its irreducible correlations with the fluid. As the consequence, significant restrictions do arise on possible shapes of the path distribution. In particular, the hypothetical Gaussian form of its long-range asymptotic proves to be forbidden (even in the Boltzmann-Grad limit). Instead, a diffusive asymptotic is allowed which possesses power-law long tail (cut off by ballistic flight length).
The hypothesis of ``molecular chaos'' is shown to fail when applied to spatially inhomogeneous evolution of a low-density gas, because this hypothesis is incompatible with reduction of interactions of gas particles to ``collisions''. The failure of molecular chaos means existence of statistical correlations between colliding and closely spaced particles in configuration space. If this fact is taken into account, then in the collisional approximation (in the kinetic stage of gas evolution) in the limit of infinitely small gas parameter the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations yields an autonomous system of kinetic equations for the many-particle distribution functions of closely spaced particles. This system of equations can produce the Boltzmann equation only in the homogeneous case. It is used to analyze statistical properties of Brownian motion of a test gas particle. The analysis shows that there exist fluctuations with a 1/f spectrum in the diffusivity and mobility of any particle. The physical cause of these fluctuations is randomness of distribution of particles' encounters over the impact parameter values and, consequently, randomness of the rate and efficiency of collisions. In essence, this is {\bf reprint} of the like author's paper published in Russian in [ Zh. Eksp. Teor. Fiz. {\bf 94} (12), 140-156 (Dec. 1988)] and translated into English in [ Sov. Phys. JETP {\bf 67} (12), 469-2477 (Dec. 1988)] twenty years ago but seemingly still unknown to those to whom it might be very useful. The footnotes contain presently added comments.
An universal exact description of kinetics of open quantum systems in terms of random wave functions and stochastic Schrödinger equation is suggested. It is shown that evolution of random quantum states of an open system is unitary on average, and this implies validity of the optical theorem for any inelastic scattering.
The BBGKY hierarchy of equations for a particle interacting with an ideal gas is investigated. Principal properties of its solutions are disclosed, as exact identities which connect probability distribution of path of the particle, its derivatives in respect to gas density and irreducible many-particle correlations between gas molecules and the path. They show that all the correlations always give equally important contributions to evolution of the path distribution, and therefore the exact theory does not reduce to the classical kinetics even at arbitrary small gas density.
Statistics of molecular random walks in a fluid is considered with the help of Bogolyubov equation for generating functional of distribution functions. An invariance group of this equation is found. It results in many exact relations between path probability distribution of a test particle and its correlations with the fluid. As the consequence, significant restrictions on possible shape of the path distribution do arise. In particular, the hypothetical Gaussian form of long-range asymptotic proves to be forbidden, even (and first of all) under the Boltzmann-Grad limit. An allowed diffusive asymptotic possesses power-law long tail (cut off by free flight length).
An universal form of kinetic equation for open systems is considered which naturally unifies classical and quantum cases and allows to extend concept of wave function to open quantum systems. Corresponding stochastic Schrödinger equation is derived and illustrated by the example of inelastic scattering in quantum conduction channel.
Brownian motion of particle interacting with atoms of ideal gas is discussed as a key problem of kinetics lying at the border between ``dead'' systems like the Lorentz gas or formal constructs of conceptual Boltzmannian kinetics and actual ``alive'' systems like mere gas possessing scaleless (1/f) fluctuations in their kinetic characteristics (e.g. in diffusuvity and mobility of the ``Brownian particle'').
The problem of statistics of molecular random walks in a classical fluid is analyzed by means of the BBGKY hierarchy of equations reformulated in terms of the Bogolyubov evolution equation for generating functional of many-particle distribution functions. A proper equivalent set of correlation functions is introduced so that all they are integrable, vanish in statistical equilibrium, otherwise accumulate statistical information about history of collisions of a ``molecular Brownian particle'' (test molecule) with other molecules of the fluid. An exact evolution equation for generating functional of such correlation functions is derived. Then it is shown that time-dependent solution to this equation, as well as a properly defined generating functional of static thermodynamically equilibrium correlations, possesses invariance with respect to a definite group of transformations of independent variables of the functional, if density of the fluid (number of molecules per unit volume) is treated as one of the independent variables. Such invariance results in infinitely many exact relations between the correlation functions and probability distribution of path of the molecular Brownian particle. Even simplest of these relations suggest significant restrictions on a profile of the path probability distribution, even without literal solving the BBGKY hierarchy.
Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path never can be disjointed into statistically independent fragments, even in the Boltzmann-Grad gas, but behaves as if Brownian particle's diffusivity undergoes scaleless low-frequency fluctuations.
Real thermal motion of gas molecules, free electrons, etc., at long time intervals (much greater than mean free-flight time) possesses, contrary to its popular mathematical models, essentially non-Gaussian statistics. A simple proof of this statement is suggested basing on only the determinism and reversibility of microscopic dynamics and besides incidentally derived virial expansion of a path probability distribution of molecular Brownian particle.
A formally exact relation is derived which connects thermodynamically non-equilibrium evolution of gas density distribution after its arbitrary strong spatially non-uniform perturbation and evolution of many-particle correlations between path of some marked particle and its surroundings in equilibrium gas. This relation directly confirms significance of the many-particle correlations even under the Boltzmann-Grad limit and thus validates the earlier suggested revision of kinetics.
An attempt is made to compare statistical properties of self-diffusion of particles constituting gases in infinite volume and on torus. In this second part, derivation, from BBGKY equations, of roughened model of self-diffusion is revised as applied to finite $N$-particle gas under micro-canonical ensemble. The model confirms existence of characteristic time $\approx N$, in units of free flight time, for cross-over between non-Gaussian and Gaussian regimes of diffusion, but then loses its legacy.
An attempt is made to compare statistical properties of self-diffusion of particles constituting gases in infinite volume and on torus. In this first part, equations are derived which represent roughened but solvable variant of the collisional approximation to exact BBGKY equations. With their help, statistics of Brownian motion in infinite gas is considered, under the Boltzmann-Grad limit, and shown to be essentially non-Gaussian, involving 1/f fluctuations in diffusivity.