Ion-ion correlations: an improved one-component plasma correction

Based on a Debye-Hueckel approach to the one-component plasma we propose a new free energy for incorporating ionic correlations into Poisson-Boltzmann like theories. Its derivation employs the exclusion of the charged background in the vicinity of the central ion, thereby yielding a thermodynamically stable free energy density, applicable within a local density approximation. This is an improvement over the existing Debye-Hueckel plus hole theory, which in this situation suffers from a "structuring catastrophe". For the simple example of a strongly charged stiff rod surrounded by its counterions we demonstrate that the Poisson-Boltzmann free energy functional augmented by our new correction accounts for the correlations present in this system when compared to molecular dynamics simulations.Comment: 5 pages, 2 figures, revtex styl

A Generalization of the Stillinger-Lovett Sum Rules for the Two-Dimensional Jellium

In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge $q$ immersed in a neutralizing background, the fixing of one of the $q$-charges induces a screening cloud of the charge density whose zeroth and second moments are determined just by the Stillinger-Lovett sum rules. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge $Z q$ immersed in the bulk interior of the 2D jellium with the coupling constant $\Gamma=\beta q^2$ ($\beta$ is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge $Z>-2/\Gamma$. The derivation is based on a mapping technique of the 2D jellium at the coupling $\Gamma$ = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of $\Gamma$ corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling $\Gamma$ the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye-H\"uckel limit $\Gamma\to 0$ and at the free-fermion point $\Gamma=2$. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.Comment: 16 page

Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations

We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibson, who observed that the vorticity vector {\boldmath\omega} aligns with the intermediate eigenvector of the strain matrix $S$, we study this problem in the context of both the three-dimensional Euler and Navier-Stokes equations using the variables \alpha = \hat{{\boldmath\xi}}\cdot S\hat{{\boldmath\xi}} and {\boldmath\chi} = \hat{{\boldmath\xi}}\times S\hat{{\boldmath\xi}} where \hat{{\boldmath\xi}} = {\boldmath\omega}/\omega. This introduces the dynamic angle $\phi (x,t) = \arctan(\frac{\chi}{\alpha})$, which lies between {\boldmath\omega} and S{\boldmath\omega}. For the Euler equations a closed set of differential equations for $\alpha$ and {\boldmath\chi} is derived in terms of the Hessian matrix of the pressure $P = \{p_{,ij}\}$. For the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn out to be the Lagrangian fixed point solutions of the equivalent (\alpha,{\boldmath\chi}) equations with a corresponding angle $\phi = 0$. Under certain assumptions for more general flows it is shown that there is an attracting fixed point of the (\alpha,\bchi) equations which corresponds to positive vortex stretching and for which the cosine of the corresponding angle is close to unity. This indicates that near alignment is an attracting state of the system and is consistent with the formation of Burgers-like structures.Comment: To appear in Nonlinearity Nov. 199

Fluctuations de charge et de masse dans un sel fondu binaire a la limite hydrodynamique

We show, by a hydrodynamic calculation, that binary molten salts behave essentially like simple neutral liquids. A recent molecular dynamics simulation exhibited an optic mode, due to charge density oscillations. The hydrodynamic study leads only to the damping of this mode, due to interdiffusion, and to a sound mode, which had not been observed in the computer simulation.Nous montrons, par une étude hydrodynamique, que les sels fondus binaires se comportent essentiellement comme un fluide simple non chargé. Une simulation de dynamique moléculaire récente avait mis en évidence un mode propre optique, correspondant à des oscillations de charge. Seul, l'amortissement de ces oscillations, dû à l'interdiffusion, est observé à la limite hydrodynamique. Par contre, le son qui n'apparaissait pas dans l'étude de dynamique moléculaire est mis en évidence ici

Local interaction between vorticity and shear in a perfect incompressible fluid

We show from a simple model related to the Euler equations that the flow of an incompressible and inviscid fluid diverges in a finite time. For this we look at the local interaction between vorticity and shear by neglecting the gradients of these two quantities in their equations of motion. A non linear system of 8 first order differential equations is obtained whose asymptotic behaviour can be easily obtained. The two largest eigenvalues of the shear tensor diverge to + ∞ and the smallest one to — ∞. The vorticity vector also diverges and lies along the eigenvector of the shear tensor which corresponds to the (positive) intermediate eigenvalue, thus giving a positive sign to the energy spreading function of the von Kármán-Howarth equation. At the same time the rotation of the shear principal axis stops.Nous montrons par un modèle simple tiré des équations d'Euler que l'écoulement d'un fluide parfait incompressible devient divergent au bout d'un temps fini. Pour cela nous considérons l'interaction locale entre la vorticité et le cisaillement en négligeant les gradients de ces deux grandeurs dans leurs équations du mouvement. Il en résulte un système de 8 équations différentielles non linéaires du premier ordre dont le comportement asymptotique peut être facilement extrait. Les 3 valeurs propres du tenseur de cisaillement divergent vers + ∞ pour les deux plus grandes et — ∞ pour la plus petite. Le vecteur vorticité diverge aussi en se couchant sur le vecteur propre du tenseur de cisaillement qui correspond à la valeur propre intermédiaire (positive), donnant ainsi un signe positif à la fonction de transfert de l'énergie de l'équation de von Kármán-Howarth. En même temps la rotation du référentiel des axes propres du cisaillement s'arrête

Improved Debye Hückel theory for one-and multicomponent plasmas

When applied to classical plasmas which are neutralized by a uniform background, the well known Debye Hückel theory is only valid in the weak coupling limit. The main reason for this limitation is that the mean field approximation, upon which the Debye Hückel theory is based, neglects correlations between ions of the screening cloud. Here we show how it is possible to take into account the correlation contributions to the internal energy by properly modifying the formula which determines the internal energy as a function of the mean field. The internal energy we obtain correctly exhibits the two asymptotic behaviours in the weak and strong coupling limits. Moreover, at strong coupling, we get the well known linear law for the internal energy of multicomponent plasmas as a function of the concentrations, and its relationship to the perfect screening condition is proved. Finally, we show a simple model for phase separation of binary mixtures neutralized by electrons under high pressure.La théorie bien connue de Debye Hückel appliquée aux plasmas classiques neutralisés par un fond continu n'est valide que lorsque le couplage est suffisamment faible. La raison essentielle de ceci est que l'approximation de champ moyen sur laquelle est fondée la théorie de Debye Hückel néglige les corrélations entre les ions du nuage d'écrantage. Nous montrons ici comment il est possible de tenir compte de la contribution de ces corrélations à l'énergie interne en modifiant de façon adéquate la formule donnant l'énergie en fonction du champ moyen. L'énergie ainsi obtenue a bien les deux bons comportements limites en couplages faible et fort. De plus, en couplage fort, la linéarité dans les concentrations de l'énergie interne des plasmas à plusieurs composants est montrée et son lien avec la condition d'écrantage parfait est établi. Enfin nous présentons un modèle simple de démixtion sous forte pression des mélanges binaires en présence d'électrons