Collective Variable Theory Research Articles

Gas-liquid phase equilibrium in ionic fluids: Coulomb versus non-Coulomb interactions

Using the collective variables theory, we study the effect of competition between Coulomb and dispersion forces on the gas-liquid phase behaviour of a model ionic fluid, i.e. a charge-asymmetric primitive model with additional short-range attractive interactions. Both the critical parameters and the coexistence envelope are calculated in a one-loop approximation as a function of the parameter $\alpha$ measuring the relative strength of the Coulomb to short-range interactions. We found the very narrow region of $\alpha$ bounded from the both sides by tricritical points which separates the models with "nonionic" and "Coulombic" phase behaviour. This is at variance with the result of available computer simulations where no tricritical point is found for the finely-discretized lattice version of the model.

Gas-liquid phase coexistence and crossover behavior of binary ionic fluids with screened Coulomb interactions.

We study the effects of an interaction range on the gas-liquid phase diagram and the crossover behavior of a simple model of ionic fluids: an equimolar binary mixture of equisized hard spheres interacting through screened Coulomb potentials which are repulsive between particles of the same species and attractive between particles of different species. Using the collective variables theory, we find explicit expressions for the relevant coefficients of the effective φ{4} Ginzburg-Landau Hamiltonian in a one-loop approximation. Within the framework of this approximation, we calculate the critical parameters and gas-liquid phase diagrams for varying inverse screening length z. Both the critical temperature scaled by the Yukawa potential contact value and the critical density rapidly decrease with an increase of the interaction range (a decrease of z) and then for z<0.05 they slowly approach the values found for a restricted primitive model (RPM). We find that gas-liquid coexistence region reduces with an increase of z and completely vanishes at z≃2.78. Our results clearly show that an increase in the interaction range leads to a decrease of the crossover temperature. For z≃0.01, the crossover temperature is the same as for the RPM.

Nonlinear Schrödinger solitons oscillate under a constant external force

odinger equation with an external timeindependent force of the form f (x) = r exp(−iKx). Here the solitons travel with an oscillating velocity and all other characteristics of the solitons (amplitude, width, momentum, and phase) also oscillate. This behavior was predicted by a collective variable theory and confirmed by simulations. However, the reason for these oscillations remains unclear. Moreover, the spectrum of the oscillations exhibits a second strong peak, in addition to the intrinsic soliton peak. We show that the additional frequency belongs to a certain extended linear mode (which we refer to as a phonon for short) close to the lower band edge of the phonon continuum. Initially the soliton is at rest. When it starts to move it is deformed, begins to oscillate, and excites the above phonon mode such that the total momentum in a certain moving frame is conserved. In this frame the phonon does not move. However, not only does the soliton move in the homogeneous, time-periodic field of the phonon, but it also oscillates.

Spatial inhomogeneities in ionic liquids, charged proteins, and charge stabilized colloids from collective variables theory

Effects of size and charge asymmetry between oppositely charged ions or particles on spatial inhomogeneities are studied for a large range of charge and size ratios. We perform a stability analysis of the primitive model of ionic systems with respect to periodic ordering using the collective variables-based theory. We extend previous studies [Ciach et al., Phys. Rev. E 75, 051505 (2007)] in several ways. First, we employ a nonlocal approximation for the reference hard-sphere fluid which leads to the Percus-Yevick pair direct correlation functions for the uniform case. Second, we use the Weeks-Chandler-Anderson regularization scheme for the Coulomb potential inside the hard core. We determine the relevant order parameter connected with the periodic ordering and analyze the character of the dominant fluctuations along the λ lines. We show that the above-mentioned modifications produce large quantitative and partly qualitative changes in the phase diagrams obtained previously. We discuss possible scenarios of the periodic ordering for the whole range of size and charge ratios of the two ionic species, covering electrolytes, ionic liquids, charged globular proteins or nanoparticles in aqueous solutions, and charge-stabilized colloids.

Green’s function of quasiparticles in Calogero model and quantum hydrodynamics

We find that correlation functions at one dimension are crucially affected by the curvature of the bare single particle spectrum. Parabolic curvature leads to two closely related phenomena: the Green's function exhibits oscillation (as a function of the coordinate), while the polarization operator acquires support in part of the frequency-momentum plane. We calculated the Green's function using the WKB approximation for collective variables theory \cite{JevickiSakita}. Within this approach, the single particle Green's function is related to a quantum soliton \cite{Polychronakos}. The finite support of the polarization operator is due to periodic density waves.

Optimization of soliton ratchets in inhomogeneous sine-Gordon systems

Unidirectional motion of solitons can take place, although the applied force has zero average in time, when the spatial symmetry is broken by introducing a potential V(x) , which consists of periodically repeated cells with each cell containing an asymmetric array of strongly localized inhomogeneities at positions xi. A collective coordinate approach shows that the positions, heights, and widths of the inhomogeneities (in that order) are the crucial parameters so as to obtain an optimal effective potential Uopt that yields a maximal average soliton velocity. Uopt essentially exhibits two features: double peaks consisting of a positive and a negative peak, and long flat regions between the double peaks. Such a potential can be obtained by choosing inhomogeneities with opposite signs (e.g., microresistors and microshorts in the case of long Josephson junctions) that are positioned close to each other, while the distance between each peak pair is rather large. These results of the collective variable theory are confirmed by full simulations for the inhomogeneous sine-Gordon system.

Behavior of different ansätze in the generalized projection operator method

Abstract Using the recently reported generalized projection operator method for the nonlinear Schrodinger equation, we derive the generalized pulse parameters equations for ansatze like hyperbolic secant and raised cosine functions. In general, each choice of the phase factor θ in the projection operator gives a different set of ordinary differential equations. For θ = 0 or θ = π /2, the corresponding projection operator scheme is equivalent to the Lagrangian variation method or the bare approximation of the collective variable theory. We prove that because of the inherent symmetric property between the pulse parameters of a Gaussian ansatz results the same set of pulse parameters equations for any value of the generalized projection operator parameter θ . Finally we prove that after the substitution of the ansatze function, the Lagrange function simplifies to the same functional form irrespective of the ansatze used because of a special property shared by all the anatze chosen in this work.

Ratchet behavior in nonlinear Klein-Gordon systems with pointlike inhomogeneities

We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lattice of pointlike inhomogeneities. We explain the underlying rectification mechanism within a collective coordinate framework, which shows that such a system behaves as a rocking ratchet for point particles. Careful attention is given to the kink width dynamics and its role in the transport. We also analyze the robustness of our kink rocking ratchet in the presence of noise. We show that the noise activates unidirectional motion in a parameter range where such motion is not observed in the noiseless case. This is subsequently corroborated by the collective variable theory. An explanation for this phenomenon is given.

Comment on “Soliton ratchets induced by excitation of internal modes”

Very recently Willis [Phys. Rev. E 69, 056612 (2004)] have used a collective variable theory to explain the appearance of a nonzero energy current in an ac-driven, damped sine-Gordon equation. In this Comment, we prove rigorously that the time-averaged energy current in an ac-driven nonlinear Klein-Gordon system is strictly zero.

Vortex motion in a finite-size easy-plane ferromagnet and application to a nanodot

We study the motion of a non-planar vortex in a circular easy-plane ferromagnet, which imitates a magnetic nanodot. Analysis was done using numerical simulations and a new collective variable theory which includes the coupling of Goldstone-like mode with the vortex center. Without magnetic field the vortex follows a spiral orbit which we calculate. When a rotating in-plane magnetic field is included, the vortex tends to a stable limit cycle which exists in a significant range of field amplitude B and frequency $\omega$ for a given system size L. For a fixed $\omega$, the radius R of the orbital motion is proportional to L while the orbital frequency $\Omega$ varies as 1/L and is significantly smaller than $\omega$. Since the limit cycle is caused by the interplay between the magnetization and the vortex motion, the internal mode is essential in the collective variable theory which then gives the correct estimate and dependency for the orbit radius $R\sim B L/\omega$. Using this simple theory we indicate how an ac magnetic field can be used to control vortices observed in real magnetic nanodots.

Nonintegrable Schrödinger discrete breathers

In an extensive numerical investigation of nonintegrable translational motion of discrete breathers in nonlinear Schrödinger lattices, we have used a regularized Newton algorithm to continue these solutions from the limit of the integrable Ablowitz–Ladik lattice. These solutions are shown to be a superposition of a localized moving core and an excited extended state (background) to which the localized moving pulse is spatially asymptotic. The background is a linear combination of small amplitude nonlinear resonant plane waves and it plays an essential role in the energy balance governing the translational motion of the localized core. Perturbative collective variable theory predictions are critically analyzed in the light of the numerical results.

Importance of the Internal Shape Mode in Magnetic Vortex Dynamics

We investigate the motion of a nonplanar vortex in a circular easy-plane magnet with a rotating in-plane magnetic field. Our numerical simulations of the Landau-Lifshitz equations show that the vortex tends to a circular limit trajectory, with an orbit frequency which is lower than the driving field frequency. To describe this we develop a new collective variable theory by introducing additional variables which account for the internal degrees of freedom of the vortex core, strongly coupled to the translational motion. We derive the evolution equations for these collective variables and find limit-cycle solutions whose characteristics are in qualitative agreement with the simulations of the many-spin system.

Generalized projection operator method to derive the pulse parameters equations for the nonlinear Schrödinger equation

We present a novel projection operator method for deriving the ordinary differential equations (ODEs) which describe the pulse parameters dynamics of an ansatz function for the nonlinear Schrödinger equation. In general, each choice of the phase factor θ in the projection operator gives a different set of ODEs. For θ=0 or π/2, we prove that the corresponding projection operator scheme is equivalent to the Lagrangian method or the bare approximation of the collective variable theory. Which set of ODEs best approximates the pulse parameter dynamics depends on the ansatz used.

Scattering of vortex pairs in 2D easy-plane ferromagnets

Vortex-antivortex pairs in 2D easy-plane ferromagnets have characteristics of solitons in two dimensions. We investigate numerically and analytically the dynamics of such vortex pairs. In particular we simulate numerically the head-on collision of two pairs with different velocities for a wide range of the total linear momentum of the system. If the momentum difference of the two pairs is small, the vortices exchange partners, scatter at an angle depending on this difference, and form two new identical pairs. If it is large, the pairs pass through each other without losing their identity. We also study head-tail collisions. Two identical pairs moving in the same direction are bound into a moving quadrupole in which the two vortices as well as the two antivortices rotate around each other. We study the scattering processes also analytically in the frame of a collective variable theory, where the equations of motion for a system of four vortices constitute an integrable system. The features of the different collision scenarios are fully reproduced by the theory. We finally compare some aspects of the present soliton scattering with the corresponding situation in one dimension.

Collective variable theory for optical solitons in fibers

We present a projection-operator method to express the generalized nonlinear Schrödinger equation for pulse propagation in optical fibers, in terms of the pulse parameters, called collective variables, such as the pulse width, amplitude, chirp, and frequency. The collective variable (CV) equations of motion are derived by imposing a set of constraints on the CVs to minimize the soliton dressing during its propagation. The lowest-order approximation of this CV approach is shown to be equivalent to the variational Lagrangian method. Finally, we demonstrate the application of this CV theory for pulse propagation in dispersion-managed optical fiber links.

A collective-variable theory for nonlinear coherent excitations in classical Hamiltonian systems

We derive a collective-variable theory for nonlinear coherent excitations that is based on Dirac’s formalism of constrained Hamiltonian systems. No approximations are involved in this theory in the sense that we are able to formulate equations of motion for a new set of variables, including the collective variables themselves, which are equivalent to the original set of Hamilton’s equation of motion. We show that the dynamics of collective excitations has to be classified according to the rank of the so-called gyromatrix . Two extreme cases are particularly important: in the first case (zero gyromatrix) collective excitations behave very much like Newtonian particles of classical mechanics, while in the second case (regular gyromatrix) collective excitations are similar to charged, massless particles in an external magnetic field.

Note on collective variable theory of nonlinear Schrödinger solitons

Inconsistencies arise in a recently developed collective variable theory of nonlinear Schrodinger solitons, as a result of a particular formulation of the energy-conservation principle in terms of the time derivative of the phase of the original field. We show that the inconsistencies are resolved either by correctly reformulating the energy-conservation principle or by directly averaging the nonlinear Schrodinger equation.

Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

We study the effects of finite temperature on the dynamics of non-planar vortices in the classical, two-dimensional anisotropic Heisenberg model with XY- or easy-plane symmetry. To this end, we analyze a generalized Landau-Lifshitz equation including additive white noise and Gilbert damping. Using a collective variable theory with no adjustable parameters we derive an equation of motion for the vortices with stochastic forces which are shown to represent white noise with an effective diffusion constant linearly dependent on temperature. We solve these stochastic equations of motion by means of a Green's function formalism and obtain the mean vortex trajectory and its variance. We find a non-standard time dependence for the variance of the components perpendicular to the driving force. We compare the analytical results with Langevin dynamics simulations and find a good agreement up to temperatures of the order of 25% of the Kosterlitz-Thouless transition temperature. Finally, we discuss the reasons why our approach is not appropriate for higher temperatures as well as the discreteness effects observed in the numerical simulations.

Magnon modes and magnon-vortex scattering in two-dimensional easy-plane ferromagnets

We calculate the magnon modes in the presence of a vortex on a circular system, combining analytical calculations in the continuum limit with a numerical diagonalization of the discrete system. The magnon modes are expressed by the $S$ matrix for magnon-vortex scattering, as a function of the parameters and the size of the system and for different boundary conditions. Certain quasilocal translational modes are identified with the frequencies which appear in the trajectory $\stackrel{\ensuremath{\rightarrow}}{X}(t)$ of the vortex center in recent molecular dynamics simulations of the full many-spin model. Using these quasilocal modes we calculate the two parameters of a third-order equation of motion for $\stackrel{\ensuremath{\rightarrow}}{X}(t).$ This equation was recently derived by a collective variable theory and describes very well the trajectories observed in the simulations. Both parameters, the vortex mass and the factor in front of $\stackrel{\ensuremath{\rightarrow}}{X}⃛,$ depend strongly on the boundary conditions.

Removal of singularities from collective-variable theory by incorporating relativistic invariance.

The equations of motion which govern the evolution of collective variables introduced into a nonlinear Klein-Gordon system are obtained by projecting the nonlinear field equation onto vectors in Hilbert space. For systems in the zero-kink sector, which admit breather and kink-antikink dynamics, the magnitude of some vectors onto which the field equation is projected become zero at particular instants in time. At these instants in time, the projections are not defined, and consequently the collective-variable equations of motion themselves are not defined: the transformation from the original field variables to the collective variables becomes singular. Although the singularity problem has already appeared in various contexts in the literature, it has not been resolved in the context of a collective-variable theory which contains collective variables in addition to that of the center of mass. For such cases, we show the singularities are removed by correctly taking into account the relativistic invariance of the underlying theory and give an example Ansatz which admits a well-defined collective-variable theory which would otherwise have been singular. We comment on the increased complexity of the present Ansatz relative to the utility of collective-variable theory.