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We study the correlations of vorticity fluctuations inside a coherent vortex resulting from the inverse energy cascade in two-dimensional turbulence. The presence of a coherent flow, which is a differential rotation, suppresses small-scale fluctuations of the flow, which are created by an external force, and lead to the fact that these fluctuations can be considered as non-interacting and, therefore, examined in a linear approximation. We calculate the pair correlation function of vorticity and demonstrate that it has a power-law behavior both in space and in time. The obtained results allow us to start a systematic study of the effects associated with the nonlinear interaction of fluctuations, which play an essential role on the periphery of a coherent vortex. Our results are also applicable to the statistics of a passive scalar in a strong shear flow.

We analytically examine fluctuations of vorticity excited by an external random force in two-dimensional fluid. We develop the perturbation theory enabling one to calculate nonlinear corrections to correlation functions of the flow fluctuations found in the linear approximation. We calculate the correction to the pair correlation function and the triple correlation function. It enables us to establish the criterion of validity of the perturbation theory for different ratios of viscosity and bottom friction. We find that the corrections to the second moment are anomalously weak in the cases of small bottom friction and small viscosity and relate the weakness to the energy and enstrophy balances. We demonstrate that at small bottom friction the triple correlation function is characterized by universal scaling behavior in some region of lengths. The developed perturbation method was verified and confirmed by direct numerical simulations.

We examine the velocity profile of coherent vortices appearing as a consequence of the inverse cascade of two-dimensional turbulence in a finite box in the case of static pumping. We demonstrate that in the passive regime the flat velocity profile is realized, as in the case of pumping short correlated in time. However, in the static case the energy flux to large scales is dependent on the system parameters. We demonstrate that it is proportional to $f^{4/3}$ where $f$ is the characteristic force exciting turbulence., Comment: 5 pages

We analyze velocity fluctuations inside coherent vortices generated as a result of the inverse cascade in the two-dimensional ($2d$) turbulence in a finite box. As we demonstrated in \cite{16KL}, the universal velocity profile, established in \cite{14LBFKL}, corresponds to the passive regime of flow fluctuations. The property enables one to calculate correlation functions of the velocity fluctuations in the universal region. We present results of the calculations that demonstrate a non-trivial scaling of the structure function. In addition the calculations reveal strong anisotropy of the structure function., Comment: arXiv admin note: substantial text overlap with arXiv:1511.03113

We analyze magnetic kinematic dynamo in a conducting fluid where the stationary shear flow is accompanied by relatively weak random velocity fluctuations. The diffusionless and diffusion regimes are described. The growth rates of the magnetic field moments are related to the statistical characteristics of the flow describing divergence of the Lagrangian trajectories. The magnetic field correlation functions are examined, we establish their growth rates and scaling behavior. General assertions are illustrated by explicit solution of the model where the velocity field is short-correlated in time.

Two-dimensional turbulence generated in a finite box produces large-scale coherent vortices coexisting with small-scale fluctuations. We present a rigorous theory explaining the $\eta=1/4$ scaling in the $V\propto r^{-\eta}$ law of the velocity spatial profile within a vortex, where $r$ is the distance from the vortex center. This scaling, consistent with earlier numerical and laboratory measurements, is universal in its independence of details of the small-scale injection of turbulent fluctuations and details of the shape of the box., Comment: 4 pages, 1 figure

The free-energy distribution function of an elastic string in a quenched random potential, P(F), is investigated with the help of the optimal-fluctuation approach. The form of the far-right tail of P(F) is found by constructing the exact solution of the non-linear saddle-point equations describing the asymptotic form of the optimal fluctuation. The solution of the problem is obtained for two different types of boundary conditions and for an arbitrary dimension of the imbeding space 1+d with d from the interval 0

Eukaryotic cells possess a sensible chemical compass allowing them to orient toward sources of soluble chemicals. The extracellular chemical signal triggers separation of the cell membrane into two domains populated by different phospholipid molecules and oriented along the signal anisotropy. We propose a theory of this polarization process, which is articulated into subsequent stages of germ nucleation, patch coarsening and merging into a single domain. We find that the polarization time, $t_\epsilon$, depends on the anisotropy degree $\epsilon$ through the power law $t_\epsilon\propto \epsilon^{-2}$, and that in a cell of radius $R$ there should exist a threshold value $\epsilon_\mathrm{th}\propto R^{-1}$ for the smallest detectable anisotropy.

Cell polarization plays a central role in the development of complex organisms. It has been recently shown that cell polarization may follow from the proximity to a phase separation instability in a bistable network of chemical reactions. An example which has been thoroughly studied is the formation of signaling domains during eukaryotic chemotaxis. In this case, the process of domain growth may be described by the use of a constrained time-dependent Landau-Ginzburg equation, admitting scale-invariant solutions {\textit{\`a la}} Lifshitz and Slyozov. The constraint results here from a mechanism of fast cycling of molecules between a cytosolic, inactive state and a membrane-bound, active state, which dynamically tunes the chemical potential for membrane binding to a value corresponding to the coexistence of different phases on the cell membrane. We provide here a universal description of this process both in the presence and absence of a gradient in the external activation field. Universal power laws are derived for the time needed for the cell to polarize in a chemotactic gradient, and for the value of the smallest detectable gradient. We also describe a concrete realization of our scheme based on the analysis of available biochemical and biophysical data., Comment: Submitted to Journal of Statistical Mechanics -Theory and Experiments

The optimal fluctuation approach is applied to study the most distant (non-universal) tails of the free-energy distribution function P(F) for an elastic string (of a large but finite length L) interacting with a quenched random potential. A further modification of this approach is proposed which takes into account the renormalization effects and allows one to study the most close (universal) parts of the tails. The problem is analyzed for different dimensions of a space in which the polymer is imbedded. In terms the stochastic growth problem, the same distribution function describes the distribution of heights in the regime of a non-stationary growth in a situation when an interface starts to grow from a flat configuration., Comment: 17 pages, 2 figures, the final version, two paragraphs added to the conclusion

Passive scalar turbulence forced steadily is characterized by the velocity correlation scale, $L$, injection scale, $l$, and diffusive scale, $r_d$. The scales are well separated if the diffusivity is small, $r_d\ll l,L$, and one normally says that effects of diffusion are confined to smaller scales, $r\ll r_d$. However, if the velocity is single scale one finds that a weak dependence of the scalar correlations on the molecular diffusivity persists to even larger scales, e.g. $l\gg r\gg r_d$ \cite{95BCKL}. We consider the case of $L\gg l$ and report a counter-intuitive result -- the emergence of a new range of large scales, $L\gg r\gg l^2/r_d$, where the diffusivity shows a strong effect on scalar correlations., Comment: 4 pages, 1 figure, submitted to Physics of Fluids

A modification of the optimal fluctuation approach is applied to study the tails of the free energy distribution function P(F) for an elastic string in quenched disorder both in the regions of the universal behavior of P(F) and in the regions of large fluctuations, where the behavior of P(F) is non-universal. The difference between the two regimes is shown to consist in whether it is necessary or not to take into account the renormalization of parameters by the fluctuations of disorder in the vicinity of the optimal fluctuation., Comment: 4 pages, no figures

We report a numerical study, supplemented by phenomenological explanations, of ``energy condensation'' in forced 2D turbulence in a biperiodic box. Condensation is a finite size effect which occurs after the standard inverse cascade reaches the size of the system. It leads to emergence of a coherent vortex dipole. We show that the time growth of the dipole is self-similar, and it contains most of the injected energy, thus resulting in an energy spectrum which is markedly steeper than the standard $k^{-5/3}$ one. Once the coherent component is subtracted, however, the remaining fluctuations have a spectrum close to $k^{-1}$. The fluctuations decay slowly as the coherent part grows., Comment: 4 pages, 4 figures. This version includes some additional phenomenological explanations of the results, additional references and improved figures

Considering the dynamics of a polymer with finite extensibility placed in a chaotic flow with large mean shear, we explain how the statistics of polymer extension changes with Weissenberg number, ${\it Wi}$, defined as the product of the polymer relaxation time and the Lyapunov exponent of the flow. Four regimes, of the ${\it Wi}$ number, are identified. One below the coil-stretched transition and three above the coil-stretched transition. Specific emphasis is given to explaining these regimes in terms of the polymer dynamics., Comment: submitted to Journal of Fluid Mechanics

A polymer placed in chaotic flow with large mean shear tumbles, making a-periodic flips. We describe the statistics of angular orientation, as well as of tumbling time (separating two subsequent flips) of polymers in this flow. The probability distribution function (PDF) of the polymer orientation is peaked around a shear-preferred direction. The tails of this angular PDF are algebraic. The PDF of the tumbling time, $\tau$, has a maximum at the value estimated as inverse Lyapunov exponent of the flow. This PDF shows an exponential tail for large $\tau$ and a small-$\tau$ tail determined by the simultaneous statistics of velocity PDF., Comment: submitted to Journal of Fluid Mechanics

We consider asymptotics of the velocity derivatives probability distribution functions (PDFs) in Burgers' turbulence. We argue that in the forced case the same power laws as in the decaying case are realized for an infinite system., Comment: 2 pages

We present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise. Even though a weak noise is considered, we are interested in probabilities of large fluctuations (generally non-Gaussian) which are beyond perturbation theory. Our method is a further development of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve a fundamental problem of soliton statistics governing by noisy Nonlinear Schr\"odinger Equation (NSE). We then apply our method to optical soliton transmission systems using signal control elements (filters, amplitude and phase modulators)., Comment: 4 pages. Submitted to PRL

For the Burgers equation driven by thermal noise leading asymptotics of pair and high-order correlators of the velocity field are found for finite times and large distances. It is shown that the intermittency takes place: some correlators are much larger than their reducible parts., Comment: 7 pages, LaTeX2e, no figures

Numerous and all unsuccessful attempts of experimental search for monopole in cosmic rays and on accelerators in high energy particle collisions have been done since the possibility of existence of a magnetic monopole has been surveyed in 1931. Also the searches have been carried out in mica for monopole tracks as well as for relict monopoles, entrapped by ferromagnetic inclusions in iron-ores, moon rock and meteorites. A new method of search for supermassive cosmic and relict monopoles by magnetically ordered film is considered. This approach resembles the traditional method of nuclear emulsion chamber. Apparently the proposed method is particularly attractive for detection of relict monopoles, released from melting iron ore., Comment: 9 pages, LaTeX, no figures. This talk was given at 10th Baksan School "Particles & Cosmology", April '99 Baksan Valley

Kinematic dynamo theory is presented here for turbulent conductive fluids. We describe how inhomogeneous magnetic fluctuations are generated below the viscous scale of turbulence where the spatial smoothness of the velocity permits a systematic analysis of the Lagrangian path dynamics. We find analytically the moments and multi-point correlation functions of the magnetic field at small yet finite magnetic diffusivity. We show that the field is concentrated in long narrow strips and describe anomalous scalings and angular singularities of the multi-point correlation functions which are manifestations of field's intermittency. The growth rate of the magnetic field in a typical realization is found to be half the difference of two Lyapunov exponents of the same sign., Comment: 4 pages, no figures

We consider advection of a passive scalar theta(t,r) by an incompressible large-scale turbulent flow. In the framework of the Kraichnan model the whole PDF's (probability distribution functions) for the single-point statistics of theta and for the passive scalar difference theta(r_1)-theta(r_2) (for separations r_1-r_2 lying in the convective interval) are found., Comment: 19 pages, RevTeX

The possibility of existence of a magnetic monopole has been surveyed by P. Dirac even in 1931, and then from the point of view of the modern theory by A.M. Polyakov and G.~ 'tHooft in 1974. Numerous and unsuccessful attempts of experimental search for monopole in cosmic rays and on accelerators in high energy particle collisions have been done. Also the searches have been carried out in mica for monopole tracks as well as for relict monopoles, entrapped by ferromagnetic inclusions in iron-ores, moon rock and meteorites. These entrapped monopoles, when released, would have the lowest velocities $\beta<10^{-6}$ and do not yield ionization at all, and are hard to detect. Therefore it is necessary to examine thoroughly the mechanisms of slow heavy monopole interaction with matter and their scale of energy loss. We discuss here the interaction of a massive slow magnetic monopole with magnetically ordered matter, with conductors, superconductors and with condensed matter in general. Our results indicate that the energy loss of a slow supermassive monopole reach $10^{8} eV/cm$ and more if we take into consideration the Cherenkov radiation of magnons or phonons and conductivity of the media. A new method of search for cosmic and relict monopoles by magnetically ordered film is considered too. This approach resembles the traditional method of nuclear emulsion chamber. Apparently the proposed method is particularly attractive for detection of relict monopoles, released from melting iron ore., Comment: LaTeX 23 pages, no figures. This revised version subm to Phys. Rev. D

We compute analytically the probability distribution function ${\cal P}(\epsilon)$ of the dissipation field $\epsilon =(\nabla \theta)^{2}$ of a passive scalar $\theta$ advected by a $d$-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime). The tail of the distribution is a stretched exponential: for $\epsilon \to \infty$, $\ln {\cal P}(\epsilon)\sim -(d^2\epsilon)^{1/3}$., Comment: Latex, 20 pages, submitted to J. Stat. Phys

The Cherenkov radiation of magnons at passage of the heavy slow magnetic monopole through an ordered magnetic matter is considered. Also the Cherenkov radiation of phonons at monopole movement in medium is discussed., Comment: 7 pages, LaTeX, no figures

Probability density function (PDF) of passive scalar dissipation ${\cal P} (\epsilon)$ is found analytically in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. The tail of PDF at $\epsilon\gg<\epsilon>$ is shown to be stretched exponent $\ln{\cal P}(\epsilon)\propto\epsilon^{1/3}$. At $\epsilon\ll<\epsilon>$, ${\cal P}\propto 1/\sqrt{\epsilon}$., Comment: 4 pages, RevTeX 3.0, Submitted to Phys. Rev. Lett

A model of scalar turbulent advection in compressible flow is analytically investigated. It is shown that, depending on the dimensionality $d$ of space and the degree of compressibility of the smooth advecting velocity field, the cascade of the scalar is direct or inverse. If $d>4$, the cascade is always direct. For small enough degree of compressibility, the cascade is direct again. Otherwise it is inverse, i.e. very large scales are excited. The dynamical hint for the direction of the cascade is the sign of the Lyapunov exponent for particles separation. Positive Lyapunov exponents are associated to direct cascade and Gaussianity at small scales. Negative Lyapunov exponents lead to inverse cascade, Gaussianity at large scales and strong intermittency at small scales., Comment: 4 pages, RevTex 3.0

Random advection of Lagrangian tracer scalar field $\theta (t,x)$ by a one-dimensional, spatially smooth and short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source concentrated at the integral scale $L$. The statistical properties of both scalar differences and the dissipation field are analytically determined, exploiting the dynamical formulation of the model. The Gaussianity known to be present at small scales for incompressible velocity fields emerges here at large scales ($x\gg L$). These scales are shown to be excited by an inverse cascade of $\theta ^{2}$ and the probability distribution function (PDF) of the corresponding scalar differences to approach the Gaussian form, as larger and larger scales are considered. Small scales ($x\ll L$) statistics is shown to be strongly non-Gaussian. Collapse of scaling exponents for scalar structure functions\thinspace takes place: moments of order $p\ge 1$ scale all linearly, independently of the order $p$. Smooth scaling $x^{p}$ is found for $-1

We expose a functional integration method for the averaging of continuous products $\hat{P}_t$ of $N\times N$ random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum of $\hat{P}_t$. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white noise force field., Comment: Latex, 9 pages

We consider the tails of probability density function (PDF) for the velocity that satisfies Burgers equation driven by a Gaussian large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. For the PDFs of velocity and it's derivatives $u^{(k)} = \partial_x^ku$, the general formula is found: $ln P (|u^{(k)}|) \propto -(|u^{(k)}|/(Re)^k)^{3/(k+1)}$., Comment: 4 pages, RevTeX 3.0, short version of chao-dyn/9603015, submitted to Phys. Rev. Lett

We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient $\partial_xu$ and find out that they correspond to the PDF with $\ln[{\cal P}(\partial_xu)]\propto-(-\partial_xu/{\rm Re})^{3/2}$ where ${\rm Re}$ is the Reynolds number. That stretched exponential form is valid for negative $\partial_xu$ with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences $w$ is steeper than Gaussian, $\ln{\cal P}(w)\sim-(w/u_{\rm rms})^3$, as well as single-point velocity PDF $\ln{\cal P}(u)\sim-(|u|/u_{\rm rms})^3$. For high velocity derivatives $u^{(k)}=\partial_x^ku$, the general formula is found: $\ln{\cal P}(|u^{(k)}|)\propto -(|u^{(k)}|/{\rm Re}^k)^{3/(k+1)}$., Comment: 15 pages, RevTeX 3.0

The problem of the Kosterlitz-Thouless (KT) transition in the highly frustrated XY Kagom\'{e} antiferromagnet is solved. The problem is mapped onto that of the KT transition in the XY ferromagnet on the hexagonal lattice. The transition temperature is found. It is shown that the spin correlation function exponentially decays with distance even in the low-temperature phase, in contrast to the order parameter correlation function, which decays algebraically with distance.

We propose the new method for finding the non-Gaussian tails of probability distribution function (PDF) for solutions of a stochastic differential equation, such as convection equation for a passive scalar, random driven Navier-Stokes equation etc. Existence of such tails is generally regarded as a manifestation of intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration -- {\em instanton}. As an example, we examine the correlation functions of the passive scalar $u$ advected by a large-scale velocity field $\delta$-correlated in time. We find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of $u$. We discuss the simplest instantons for the Navier-Stokes equation., Comment: 15 pages, REVTEX 3.0

Advection of a passive scalar $\theta$ in $d=2$ by a large-scale velocity field rapidly changing in time is considered. The Gaussian feature of the passive scalar statistics in the convective interval was discovered in \cite{95CFKLa}. Here we examine deviations from the Gaussianity: we obtain analytically the simultaneous fourth-order correlation function of $\theta$. Explicit expressions for fourth-order objects, like $\langle(\theta_1-\theta_2)^4\rangle$ are derived., Comment: 4 pages, RevTex 3.0, To appear in JETP Lett. {\bf 61}, (1995)

For a delta-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality $d$. It is shown that anomalous scaling appears in the first order of $1/d$ perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for it's derivatives, in particular, for the dissipation field., Comment: 19 pages, RevTex 3.0, Submitted to Phys.Rev. E, revised version

We investigate the conductance of a 1-D disordered conducting loop with two contacts, immersed in a magnetic flux. We show the appearance in this model of the Al'tshuler-Aronov-Spivak behaviour. We also investigate the case of a chain of loops distributed with finite density: in this case we show that the interference effects due to the presence of the loops can lead to the delocalization of the wave function., Comment: 8 pages; LaTeX; IFUM 463/FT; to appear in Phys. Lett. A

The set of moments and the distribution function of the one-electron current in a one-dimensional disordered ring with arbitrary magnetic flux are calculated., Comment: 10 pages; Plain TeX; IFUM 448/FT; to appear in J. Stat. Phys

The transverse spin autocorrelation function at an arbitrary temperature are calculated rigorously for the system of $N$ uniformly exchanged quantum spins. At large $N$ the correlator in the para-phase is found to have a Gaussian bump at small times and a non vanishing tail at large times. A possible application of those exact results as forming a starting dynamical mean-field approximation for long-range quantum magnets and an accompanied physical picture are discussed., Comment: 6 pages, RevTex 3.0 followed by 2 PostScript figures, Preprint IFUM-467/FT

For a given wave function one can define a quantity $\mu_E$ having a meaning of its inverse spatial size. The Laplace transform of the distribution function $P(\mu_E)$ is calculated analytically for a 1D disordered sample with a finite length $L$., Comment: LaTEX, 7 pages, Preprint IFUM-456/FT, Milano, Jan.1994

Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. That corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach gaussianity with increase in the Peclet number $Pe$ (the pumping-to-diffusion scale ratio). The first $n<\ln Pe$ simultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field: the mean stretching rate, which can be calculated analytically in limiting cases. Non-Gaussian tails of the probability distributions at finite $Pe$ are found to be exponential., Comment: 25 pages, RevTex 3.0, followed by 2 PostScript figures, revised version accepted by Phys.Rev.E

The letter presents new field-theoretical approach to 2D passive scalar problem. The Gaussian form of the distribution for the Lyapunov exponent is derived and its parameters are found explicitly., Comment: 11 pages, RevTex 3.0, IFUM-94/455/January-FT

We introduce a gaussian probability density for the space-time distribution of wormholes, thus taking effectively into account wormhole interaction. Using a mean-field approximation for the free energy, we show that giant wormholes are probabilistically suppressed in a homogenous isotropic ``large'' universe., Comment: 10 pages, Latex