Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. We assume that these constraints are selected by properties of forcing and dissipation. We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by a linear [`(w)]-y\overline{\omega}-\psi relationship. Furthermore, we prove that the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived. These equations can serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.     

Statistical mechanics of Fofonoff flows in an oceanic basin

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h = by which represents either a real bottom topography or the β-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function ψ. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or β-effect).     

Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

We review and connect different variational principles that have been proposed to settle the dynamical and thermodynamical stability of two-dimensional incompressible and inviscid flows governed by the 2D Euler equation. These variational principles involve functionals of a very wide class that go beyond the usual Boltzmann functional. We provide relaxation equations that can be used as numerical algorithms to solve these optimization problems. These relaxation equations have the form of nonlinear mean field Fokker-Planck equations associated with generalized “entropic” functionals [P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)].     

Kinetic theory of semiconductor cascade laser based on quantum wells and wires

The paper presents a numerical solution of a system of nonlinear equations for the electron distribution functions in the upper and lower subbands between which lasing transitions occur and the number of nonequilibrium optical phonons in semiconducting cascade lasers based on quantum wells and wires. For the case of quantum wells, we propose an analytical solution of this system of equations, which is a generalization of the previously found solution [V. F. Elesin and Yu. V. Kopaev, Zh. éksp. Teor. Fiz. 108, 2186 (1995) [JETP 81, 1192 (1995)]; V. F. Elesin and Yu. V. Kopaev, Sol. St. Commun. 96, 897 (1995)] in a wider range of injection rates. The threshold injection rate can be significantly reduced owing to reabsorption and accumulation of nonequilibrium optical phonons, nonparabolicity of the subbands and different effective masses of electrons in different subbands. In the case of quantum wires, the threshold injection rate is considerably lower, and its decrease is even larger than in quantum wells. It is remarkable that, owing to the lower electron-electron relaxation rate in the one-dimensional case, the decrease in the threshold injection rate may be two or three orders of magnitude. The relation between the density of states and threshold current has also been studied. Zh. éksp. Teor. Fiz. 111, 681–695 (February 1997)     

Casimir friction in terms of moving harmonic oscillators: equivalence between two different formulations

The Casimir friction problem can be dealt with in a simplified way by considering two harmonic oscillators moving with constant relative velocity. Recently we calculated the energy dissipation ΔE for such a case [Europhys. Lett. 91, 60003 (2010); Eur. Phys. J. D 61, 335 (2011)]. A recent study of Barton [New J. Phys. 12, 113044 (2010)] seemingly leads to a different result for the dissipation. If such a discrepancy really were true, it would imply a delicate difficulty for the basic theory of Casimir friction. In the present note we show that the expressions for ΔE are in fact physically equivalent, at T = 0.     

Dynamics of two coupled chaotic systems driven by external signals

Setting-up a controlled or synchronized state in a space-time chaotic structure targeting an unstable periodic orbit is a key feature of many problems in high dimensional physical, electronics, biological and ecological systems (among others). Formerly, we have shown numerically and experimentally that phase synchronization [M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 4193 (1997)] can be achieved in time dependent hydrodynamic flows [D. Maza, A. Vallone, H.L. Mancini, S. Boccaletti, Phys. Rev. Lett. 85, 5567 (2000)]. In that case the flow was generated in a small container with inhomogeneous heating in order to have a single roll structure produced by a Bénard-Marangoni instability [E.L. Koshmieder, Bénard Cells and Taylor Vortices (Cambridge University Press, 1993)]. Phase synchronization was achieved by a small amplitude signal injected at a subharmonic frequency obtained from the measured Fourier temperature spectrum. In this work, we analyze numerically the effects of driving two previously synchronized chaotic oscillators by an external signal. The numerical system represents a convective experiment in a small container with square symmetry, where boundary layer instabilities are coupled by a common flow. This work is an attempt to control this situation and overcome some difficulties to select useful frequency values for the driving force, analyzing the influence of different harmonic injection signals on the synchronization in a system composed by two identical chaotic Takens-Bogdanov equations (TBA and TBB) bidirectionally coupled.     

Growing mechanisms in the QKPZ equation aaand the DPD models

The roughening of interfaces moving in inhomogeneous media is investigated by numerical integration of the phenomenological stochastic differential equation proposed by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889 (1986)] with quenched noise (QKPZ) [Phys. Rev. Lett. 74, 920 (1995)]. We express the evolution equations for the mean height and the roughness into two contributions: the local and the lateral one in order to compare them with the local and the lateral contributions obtained for the directed percolation depinning models (DPD) introduced independently by Tang and Leschhorn [Phys. Rev A 45, R8309 (1992)] and Buldyrev et al. [Phys. Rev A 45, R8313 (1992)]. These models are classified in the same universality class of the QKPZ although the mechanisms of growth are quite different. In the DPD models the lateral contribution is a coupled effect of the competition between the local growth and the lateral one. In these models the lateral contribution leads to an increasing of the roughness near the criticality while in the QKPZ equation this contribution always flattens the roughness. Received 7 April 2000 and Received in final form 7 March 2001     

Effect of charge exchange on the energy distribution of fast multiply charged ions propagating through matter

A technique for calculating, in the diffusion approximation, energy distributions of multiply-charged ions with an arbitrary number of charge states propagating through matter has been suggested. Examples of numerical solutions of kinetic equations taking into account charge exchange between ions and matter are given. A compact solution for the special case of two charge states has been found. The calculations are compared to experimental data. Zh. éksp. Teor. Fiz. 111, 2226–2236 (June 1997)     

Bianchi type-V string cosmological models in general relativity

Bianchi type-V string cosmological models in general relativity are investigated. To get the exact solution of Einstein’s field equations, we have taken some scale transformations used by Camci et al [Astrophys. Space Sci. 275, 391 (2001)]. It is shown that Einstein’s field equations are solvable for any arbitrary cosmic scale function. Solutions for particular forms of cosmic scale functions are also obtained. Some physical and geometrical aspects of the models are discussed.     

Twin-correlations in atoms

We discuss the possibility of preparing an atomic sample of atoms with minimum fluctuations in the difference between populations of two levels. A first scheme involves absorption of twin beams of light, and it presents a variant of a recent proposal for atomic spin squeezing within an excited state manifold [Kuzmich et al., Phys. Rev. Lett. 79, 4782 (1997)]. A second scheme involves atoms with two stable states, and we suggest that by use of quantum non-demolition detection and feed-back optical pumping, we may ensure a perfect agreement between the number of atoms in these two states. Received: 14 May 1998 / Revised: 10 August 1998 / Accepted: 8 October 1998     

Fourier transform and smoothness of solutions of a class of semilinear elliptic degenerate equations with double characteristics

This paper is a continuation of our earlier note [V. T. T. Hien and N. M. Tri, “Analyticity of Solutions of Semililnear Equations with Double Characteristics,” J. Math. Anal. Appl. 337, 1249–1260 (2008)]. Here we prove the analyticity of solutions of a class of semilinear elliptic degenerate equations with double characteristics by using the Fourier transform.     

Effects of high-j intruder states in the fine structure of superdeformed bands

All known cases of ΔI=4 bifurcation in superdeformed bands are analyzed on the basis of the theory proposed by the author in a previous paper [I. M. Pavlichenkov, Phys. Rev. C 55, 1275 (1997)]. It is shown that the high-j intruder single-particle states play a critical role in the phenomenon. The bands in which fine structure can be observed are predicted. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 12, 759–764 (25 December 1997)     

Asymptotic and effective coarsening exponents in surface growth models

We consider a class of unstable surface growth models, ?_t z = -?_x J\partial_t z = -\partial_x {\cal J} , developing a mound structure of size λ and displaying a perpetual coarsening process, i.e. an endless increase in time of λ. The coarsening exponents n, defined by the growth law of the mound size λ with time, λ∼tⁿ, were previously found by numerical integration of the growth equations [A. Torcini, P. Politi, Eur. Phys. J. B 25, 519 (2002)]. Recent analytical work now allows to interpret such findings as finite time effective exponents. The asymptotic exponents are shown to appear at so large time that cannot be reached by direct integration of the growth equations. The reason for the appearance of effective exponents is clearly identified.     

Strong field effects in the system of two interacting Rydberg atoms

The ionization dynamics of two interacting Rydberg atoms in a strong laser field has been investigated. Each atom has been described in the “two discrete levels + continuum” model. Quasienergy states in this system, which describe field-dressed atoms, have been studied. It has been shown that one of the quasienergy states corresponds to the formation of an atomic state stable against ionization, which leads to the interference stabilization regime first observed in the case of individual atoms in [M. V. Fedorov and A. M. Movsesian, J. Phys. B 21, L155 (1988)]. Methods for creating entangled states in this system have been proposed and the dynamics of entanglement in the process of interaction with the laser field has been analyzed.     

Ionisation of phosphorus, arsenic, antimony, and bismuth by electron impact

Total ionization cross sections of neutral phosphorus, arsenic, antimony, and bismuth atoms by electron impact are reported and compared to the only available experimental results by Freund et al. [Phys. Rev. A 41, 3575 (1990)]. These calculations take into account the possibilities that some target atoms used in the experiments were in metastable states close to the ground state, the excitation-autoionization of nsnp⁴ excited states may be substantial, and the ions produced in experiments may be in excited, low-lying metastable states. The cross sections for direct ionization calculations are based on the BEB model by Kim and Rudd [Phys. Rev. A 50, 3954 (1994)]. Plane-wave Born cross sections scaled by the method developed by Kim [Phys. Rev. A 64, 3954 032713 (2001)] are used to determine the contributions from excitation-autoionization. The combination of the BEB model and the scaled Born cross sections is in agreement with the experimental data by Freund et al. These theoretical data are useful to experimentalists and can be used to complete data tables needed for plasma or astrophysical studies.     

Full separability criterion for tripartite quantum systems

In this paper, an intuitive approach is employed to generalize the full separability criterion of tripartite quantum states of qubits to the higher-dimensional systems [Phys. Rev. A 72, 022333 (2005)]. A distinct characteristic of the present generalization is that less restrictive conditions are needed to characterize the properties of full separability. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the Kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for weakly mixed tripartite quantum states and investigate the full separability of some weakly mixed states.     

On the calculation of steady-state magnetohydrodynamic flows of liquid metals in circular ducts of a rectangular cross section

A numerical code has been designed to calculate two-dimensional steady-state magnetohydrodynamic (MHD) flows of incompressible conducting fluids (liquid metals) in linear and circular thin-wall ducts of a rectangular cross section. The flows are caused by the Lorentz force J × B that appears when an electric current passes through a fluid placed in a vertical uniform magnetic field. The code is the generalization of the well-known iteration Gauss-Seidel method to the case of a set of elliptical equations. The method proposed can be used to calculate steady-state flows over wide ranges of Hartmann (Ha = 1–10³) and Reynolds (Re = 1–10⁶) numbers.     

Randomness and a step-like distribution of pile heights in avalanche models

This paper considers a one-parameter family of sand-piles. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean pile height is used to describe the crossover. The height densities corresponding to the models with relaxation of both types approach one another as the parameter increases. Relaxation is supposed to deal with the local losses of grains by a fixed amount. In that case the densities show a step-like behaviour in contrast to the peaked shape found in the models with the local loss of grains down to a fixed level [S. Lübeck, Phys. Rev. E 62, 6149 (2000)]. A spectral approach based on the long-run properties of the pile height considers the models with deterministic and random relaxation more accurately and distinguishes between the two cases for admissible parameter values.     

Scaling and universality in models of step bunching: the “C<Superscript>+</Superscript>–C<Superscript>-</Superscript>” model

We recently introduced a novel model of step flow crystal growth – the so-called “C⁺–C^-” model [B. Ranguelov et al., C.R. Acad. Bulgare Sci. 60, 389 (2007)]. In this paper we aim to develop a complete picture of the model’s behaviour in the framework of the notion of universality classes. The basic assumption of the model is that the reference (“equilibrium”) densities used to compute the supersaturation might be different on either side of a step, so C_L/C_R ≠ 1 (L/R stands for left/right in a step train descending from left to right), and that this will eventually cause destabilization of the regular step train. Linear stability analysis considering perturbation of the whole step train shows that the vicinal is always unstable when the condition C_L /C_R >1 is fulfilled. Numerical integration of the equations of step motion combined with an original monitoring scheme(s) results in obtaining the exact size- and time- scaling of the step bunches in the limit of long times (including the numerical prefactors). Over a broad range of parameters the surface morphology is characterized by the appearance of the minimal interstep distance at the beginning of the bunches (at the trailing edge of the bunch) and may be described by a single universality class, different from those already generated by continuum theories [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002), J. Krug et al., Phys. Rev. B 71, 045412 (2005)]. In particular, the scaling of the minimal interstep distance l_min in the new universality class is shown to be l_min = (S_n /N)^1/(n+1), where N is the number of steps in the bunch, n is the exponent in the step-step repulsion law U ~ 1/d₀ ⁿ for two steps placed a distance d₀ apart and S_n is a combination of the model parameters. It is also shown that N scales with time with universal exponent 1/2 independent of n. For the regime of slow diffusion it is obtained for the first time that the time scaling depends only on the destabilization parameter C_L/C_R. The bunching outside the parameter region where the above scaling exists cannot be assigned to a specific universality class and thus should be considered non-universal.