arXiv subjects

Sergey K. Nemirovskii

Publications and source records attributed to Sergey K. Nemirovskii.

32 records · Page 2Linked to original sources

arXiv2008

Kinetics of a Network of Vortex Loops in He II and a Theory of Superfluid Turbulence

A theory is developed to describe the superfluid turbulence on the base of kinetics of the merging and splitting vortex loops. Because of very frequent reconnections the vortex loops (as a whole) do not live long enough to perform any essential evolution due to the deterministic motion. On the contrary, they rapidly merge and split, and these random recombination processes prevail over other slower dynamic processes. To develop quantitative description we take the vortex loops to have a Brownian structure with the only degree of freedom, which is the length $l$ of the loop. We perform investigation on the base of the Boltzmann type kinetic equation for the distribution function $n(l)$ of number of loops with length $l$. By use of the special ansatz in the collision integral we have found the exact power-like solution to kinetic equation in the stationary case. This solution is not (thermodynamically) equilibrium, but on the contrary, it describes the state with two mutual fluxes of the length (or energy) in space of sizes of the vortex loops. The term flux means just redistribution of length (or energy) among the loops of different sizes due to reconnections. Analyzing this solution we drew several results on the structure and dynamics of the vortex tangle in the turbulent superfluid helium. In particular, we evaluated the mean radius of the curvature and the full rate of the reconnection events. We also studied the evolution of the full length of vortex loops per unit volume-the so-called vortex line density. It is shown this evolution to obey the famous Vinen equation. The properties of the Vinen equation from the point of view of the developed approach had been discussed.

cond-mat.stat-mech↗

arXiv2006

Evolution of a Network of Vortex Loops in the Turbulent Superfluid Helium; Derivation of the Vinen Equation

The evolution a network of vortex loops due to the fusion and breakdown in the turbulent superfluid helium is studied. We perform investigation on the base of the "rate equation" for the distribution function $n(l)$ of number of loops in space of their length $l$. There are two mechanisms for change of quantity $n(l)$. Firstly, the function changes due to deterministic process of mutual friction, when the length grows or decreases depending on orientation. Secondly, the change of $n(l)$ occurs due to random events when the loop crosses itself breaking down into two daughter or two loops collide merging into one larger loop. Accordingly the "rate equation" includes the "collision" term collecting random processes of fusion and breakdown and the deterministic term. Assuming, further, that processes of random colliding are fastest we are in position to study more slow processes related to deterministic term. In this way we study the evolution of full length of vortex loops per unit volume-so called vortex line density ${\cal L}(t)$. It is shown this evolution to obey the famous Vinen equation. In conclusion we discuss properties of the Vinen equation from the point of view of the developed approach.

cond-mat.soft↗

arXiv2005

Evolution of a Network of Vortex Loops in HeII. Exact Solution of the "Rate Equation"

Evolution of a network of vortex loops in HeII due to the fusion and breakdown of vortex loops is studied. We perform investigation on the base of the ''rate equation'' for the distribution function $n(l)$ of number of loops of length $l$ proposed by Copeland with coauthors. By using the special ansatz in the ''collision'' integral we have found the exact power-like solution of ''kinetic equation'' in stationary case. That solution is the famous equilibrium distribution $n(l)\varpropto l^{-5/2}$ obtained earlier in numerical calculations. Our result, however, is not equilibrium, but on the contrary, it describes the state with two mutual fluxes of the length (or energy) in space of the vortex loop sizes. Analyzing this solution we drew several results on the structure and dynamics of the vortex tangle in the superfluid turbulent helium. In particular, we obtained that the mean radius of the curvature is of order of interline space. We also obtain that the decay of the vortex tangle obeys the Vinen equation, obtained earlier phenomenologically. We evaluate also the full rate of reconnection events. PACS-number 67.40

cond-mat.stat-mech↗

arXiv2005

Reconnections of Vortex Loops in the Superfluid Turbulent HeII. Rates of the Breakdown and Fusion processes

Kinetics of merging and breaking down vortex loops is the important part of the whole vortex tangle dynamics. Another part is the motion of individual lines, which obeys the Biot-Savart law in presence of friction force and of applied external velocity fields if any. In the present work we evaluate the coefficients of the reconnection rates $A(l_{1},l_{2},l)$ and $B(l,l_{1},l_{2})$. Quantity $A$ is a number (per unit of time and per unit of volume) of events, when two loops with lengths $l_{1}$and $l_{2}$ collide and form the single loop of length $ l=l_{1}+l_{2}$. Quantity $% B(l,l_{1},l_{2}) $ describes the rate of events, when the single loop of the length $l$ breaks down into two the daughter loops of lengths $ l_{1}$ and $l_{2}$. These quantities ave evaluated as the averaged numbers of zeroes of vector $\mathbf{S}%_{s}(ξ_{2},ξ_{1},t)$ connecting two points on the loops of $ξ_{2}$ and $ ξ_{1}$ at moment of time $t$. Statistics of the individual loops is taken from the Gaussian model of vortex tangle. PACS-number 67.40

cond-mat.supr-con↗

arXiv2004

Slow Transient Processes in the Second Sound Resonator

The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive the evolution equation for the slow processes only. We then demonstrate that the long-term evolution of a transient heat load of moderate intensity obeys the nonlinear heat conductivity equation. The second example of the methods developed is investigation of unsteady processes in the second sound resonator. The latter is frequently used for study of nonstationary behavior of vortex tangle, just by monitoring of the quality factor behavior. This procedure however is wrong when characteristic times of processes are comparable (or smaller) than the time constant of resonator. We show how to extract the correct information on the vortex line density (VLD) dynamics with use of procedure we developed. PACS numbers: 47.32.Cc, 47.37.+q, 67.40.Vs., 05.10.Gg.

cond-mat.other↗

arXiv2004

Multi-Scale Perturbation Analysis in Hydrodynamics of the Superfluid Turbulence. Derivation of the Dresner Equation

The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments, so called superfluid turbulence. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents an effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive the evolution equation for the slow processes only. We then demonstrate that the long-term evolution of a transient heat load of moderate intensity obeys the nonlinear heat conductivity equation, often referred to as the Dresner equation. We also compare our approach against the Dresner phenomenological derivation and establish a range of validity of the latter.

cond-mat.other↗

arXiv2003

Chaotic Quantum Vortexes In A Weakly Non Ideal Bose Gas. Thermodynamic Equilibrium And Turbulence

We study the stochastic behavior of a set of chaotic vortex loops appeared in imperfect Bose gas. Dynamics of Bose-gas is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has solution ${\cal P}(\{ψ({\bf r})\})={\cal N}\exp (-H\{ψ({\bf r)}\} /T),$ where $H\{ψ({\bf r})\} $ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field $ψ({\bf r})$ we derive a Langevin-type equation of motion of the line with the correspondingly transformed stirring force. The respective Fokker-Planck equation for probability functional ${\cal P}(\{{\bf s}(ξ)\})$ in vortex loop configuration space is shown to have a solution in the form of ${\cal P}(\{{\bf s}(ξ)\})={\cal N}\exp (-H\{{\bf s}\} /T),$ where ${\cal N}$ is the normalizing factor and $H\{{\bf s}\} $ is energy of vortex line configurations. Analyzing this result we discuss possible reasons for destruction of the thermodynamic equilibrium and follow the mechanisms of transition to the turbulent state

cond-mat.soft↗

arXiv2002

Energy Spectrum of Superfluid Turbulence without Normal Fluid

The energy spectrum of the superfluid turbulence without the normal fluid is studied numerically under the vortex filament model. Time evolution of the Taylor-Green vortex is calculated under the full nonlocal Biot-Savart law. It is shown that for k<2pi/l, the energy spectrum is very similar to the Kolmogorov's -5/3 law which is the most important statistical property of the conventional turbulence, where k is the wave number of the Fourier component of the velocity field and 1 the average intervortex spacing. The vortex length distribution becomes to obey a scaling property reflecting the self-similarity of the tangle.

cond-mat.soft↗

arXiv2001

Energy Spectrum of the Random Velocity Field Induced by a Gaussian Vortex Tangle in HeII

Using the Gaussian model of the vortex tangle (VT) arising in the turbulent superfluid HeII, we calculate the energy spectrum $E(k)$ of the 3D random velocity field induced by that VT. If the VT is assumed to be a purely fractal object with Haussdorf dimension $H_D$, the $E(k)$ is a power-like function $E(k)\propto k^{-2+H_D}$. A more realistic VT in HeII is a semi-fractal object, behaving as smooth line for small separations $Δξ\ll R$ ($ξ$ is the label coordinate, $R$ is mean curvature)and having a random walk structure for large $Δξ$ with $H_D=2$. For that case calculations give a spectrum $E(k)$ that is $k$-independent for $k$ smaller than 1/R (but larger than the inverse size of the system) and that scales as $k^{-1}$ for larger $k$. The latter reflects the fact that for small scales a vortex filament behaves as a smooth line. Our results agree with recent numerical simulations. PACS numbers: 47.32.Cc, 47.37.+q, 67.40.Vs., 05.10.Gg.

cond-mat.soft↗

arXiv2001

Energy Spectrum of Vortex Tangle

The energy spectrum of superfluid turbulence in the absence of the normal fluid is studied numerically. In order to discuss the statistical properties, we calculated the energy spectra of the 3D velocity field induced by dilute and dense vortex tangles respectively, whose dynamics is calculated by the Biot-Savart law. In the case of a dense tangle, the slope of the energy spectrum is changed at $k=2π/l$, where $l$ is the intervortex spacing. For $k>2π/l$, the energy spectrum has $k^{-1}$ behavior in the same manner as the dilute vortex tangle, while otherwise the slope of the energy spectrum deviates from $k^{-1}$ behavior. We compare the behavior for $k<2π/l$ with the Kolmogorov law.

cond-mat.soft↗

arXiv2000

Stochastic Dynamics of a Vortex Loop. Large Scale Stirring Force

Stochastic dynamics of a vortex filament obeying local induced approximation equation plus random agitation is investigated by analytical and numerical methods. The character of a stirring force is supposed to be a white noise with spatial correlator concentrated at large distances comparable with size of the loop. Dependence of the spectral function $<\mathbf{s}_κ^α\mathbf{s}_κ^β>$ of the vortex line on both the one-dimensional wave vector $κ$ and intensity of the external force correlator $<\mathbfζ_κ^α\mathbfζ_κ^α>$ was studied. Here $\mathbf{s}_κ^α$ is the Fourier transform of the line element position $\mathbf{s}^α(ξ, t)$. It is shown that under the influence of an external random force a vortex ring becomes a small tangle whose mean size depends on external force intensity. The theoretical predictions and the numerical results are in reasonable agreement.

cond-mat.stat-mech↗

arXiv2000

Stochastic Dynamics of a Vortex Loop. Thermal Equilibrium

We study stochastic behavior of a single vortex loop appeared in imperfect Bose gas. Dynamics of Bose-condensate is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has a solution $\mathcal{P}(\{ψ(\mathbf{r})\})=\mathcal{N}\exp (-Hψ(\mathbf{r)} /T),$ where $Hψ(\mathbf{r})$ is a Ginzburg-Landau free energy. Considering a vortex filaments as a topological defects of the field $ψ(\mathbf{r})$ we derive a Langevin-type equation of motion of the line with correspondingly transformed stirring force. The respective Fokker-Planck equation for probability functional $\mathcal{P}(\{\mathbf{s}(ξ)\})$ in vortex loop configuration space is shown to have a solution of the form $\mathcal{P}(\{\mathbf{s}(ξ)\})=\mathcal{N}\exp (-H{\mathbf{s}} /T),$ where $\mathcal{N}$ is a normalizing factor and $H{\mathbf{s}}$ is energy of vortex line configurations. In other words a thermal equilibrium of Bose-condensate results in a thermal equilibrium of vortex loops appeared in Bose-condensate. Some consequences of that fact and possible violations are discussed.

cond-mat.stat-mech↗

arXiv2000

Applications of Gaussian model of the vortex tangle in the superfluid turbulent HeII

In spite of an appearance of some impressive recent results in understanding of the superfluid turbulence in HeII they fail to evaluate many characteristics of vortex tangle needed for both applications and fundamental study. Early we reported the Gaussian model of the vortex tangle in superfluid turbulent HeII. That model is just trial distribution functional in space of vortex loop configurations constructed on the basis of well established properties of vortex tangle. It is designed to calculate various averages taken over stochastic vortex loop configurations. In this paper we use this model to calculate some important characteristics of the vortex tangle. In particular we evaluate the average superfluid mass current J induced by vortices and the average energy E associated with the chaotic vortex filament.

cond-mat.stat-mech↗

arXiv2000

Dynamics of vortex tangle without mutual friction in superfluid $^4$He

A recent experiment has shown that a tangle of quantized vortices in superfluid $^4$He decayed even at mK temperatures where the normal fluid was negligible and no mutual friction worked. Motivated by this experiment, this work studies numerically the dynamics of the vortex tangle without the mutual friction, thus showing that a self-similar cascade process, whereby large vortex loops break up to smaller ones, proceeds in the vortex tangle and is closely related with its free decay. This cascade process which may be covered with the mutual friction at higher temperatures is just the one at zero temperature Feynman proposed long ago. The full Biot-Savart calculation is made for dilute vortices, while the localized induction approximation is used for a dense tangle. The former finds the elementary scenario: the reconnection of the vortices excites vortex waves along them and makes them kinked, which could be suppressed if the mutual friction worked. The kinked parts reconnect with the vortex they belong to, dividing into small loops. The latter simulation under the localized induction approximation shows that such cascade process actually proceeds self-similarly in a dense tangle and continues to make small vortices. Considering that the vortices of the interatomic size no longer keep the picture of vortex, the cascade process leads to the decay of the vortex line density. The presence of the cascade process is supported also by investigating the classification of the reconnection type and the size distribution of vortices. The decay of the vortex line density is consistent with the solution of the Vinen's equation which was originally derived on the basis of the idea of homogeneous turbulence with the cascade process. The obtained result is compared with the recent Vinen's theory.

cond-mat.soft↗