Dimensional phase transition in small Yukawa clusters

Ring structural transitions in strongly coupled dusty plasmas

This paper presents a numerical study of ring structural transitions in strongly coupled dusty plasma confined in a ring-shaped (quartic) potential well with a central barrier, whose axis of symmetry is parallel to the gravitational attraction. It is observed that increasing the amplitude of the potential leads to a transition from a ring monolayer structure (rings of different diameters nested within the same plane) to a cylindrical shell structure (rings of similar diameter aligned in parallel planes). In the cylindrical shell state, the ring's alignment in the vertical plane exhibits hexagonal symmetry. The ring transition is reversible, but exhibits hysteresis in the initial and final particle positions. As the critical conditions for the transitions are approached, the transitional structure states exhibit zigzag instabilities or asymmetries on the ring alignment. Furthermore, for a fixed amplitude of the quartic potential that results in a cylinder-shaped shell structure, we show that additional rings in the cylindrical shell structure can be formed by decreasing the curvature of the parabolic potential well, whose axis of symmetry is perpendicular to the gravitational force, increasing the number density, and lowering the screening parameter. Finally, we discuss the application of these findings to dusty plasma experiments with ring electrodes and weak magnetic fields.

Phase transition of three-dimensional finite-sized charged dust clusters in a plasma environment

The dynamics of a harmonically trapped three-dimensional Yukawa ball of charged dust particles immersed in plasma is investigated as function of external magnetic field and Coulomb coupling parameter using molecular dynamics simulation. It is shown that the harmonically trapped dust particles organize themselves into nested spherical shells. The particles start rotating in a coherent order as the magnetic field reaches a critical value corresponding to the coupling parameter of the system of dust particles. The magnetically controlled charged dust cluster of finite size undergoes a first-order phase transition from disordered to ordered phase. At sufficiently high coupling and strong magnetic field, the vibrational mode of this finite-sized charged dust cluster freezes, and the system retains only rotational motion.

Transverse single-file diffusion and enhanced longitudinal diffusion near a subcritical bifurcation

A quasi-one-dimensional system of repelling particles undergoes a configurational phase transition when the transverse confining potential decreases. Below a threshold, it becomes energetically favorable for the system to adopt one of two staggered raw patterns, symmetric with respect to the system axis. This transition is a subcritical pitchfork bifurcation for short range interactions. As a consequence, the homogeneous zigzag pattern is unstable in a finite zigzag amplitude range [h_{C1},h_{C2}]. We exhibit strong qualitative effects of the subcriticality on the thermal motions of the particles. When the zigzag amplitude is close enough to the limits h_{C1} and h_{C2}, a transverse vibrational soft mode occurs which induces a strongly subdiffusive behavior of the transverse fluctuations, similar to single-file diffusion. On the contrary, the longitudinal fluctuations are enhanced, with a diffusion coefficient which is more than doubled. Conversely, a simple measurement of the thermal fluctuations allows a precise determination of the bifurcation thresholds.

Interaction, coalescence, and collapse of localized patterns in a quasi-one-dimensional system of interacting particles

We study the path toward equilibrium of pairs of solitary wave envelopes (bubbles) that modulate a regular zigzag pattern in an annular channel. We evidence that bubble pairs are metastable states, which spontaneously evolve toward a stable single bubble. We exhibit the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive, whereas it is repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: It is attractive for NF systems and repulsive for F systems and decreases exponentially with the bubbles distance. Moreover, for NF systems, the bubbles come closer and eventually merge as a single bubble, in a coalescence process. We also evidence a collapse process, in which one bubble shrinks in favor of the other one, overcoming an energetic barrier in phase space. This process is relevant for both NF systems and F systems. In NF systems, the coalescence prevails at low temperature, whereas thermally activated jumps make the collapse prevail at high temperature. In F systems, the path toward equilibrium involves a collapse process regardless of the temperature.

Thermal motion of a nonlinear localized pattern in a quasi-one-dimensional system

We study the dynamics of localized nonlinear patterns in a quasi-one-dimensional many-particle system near a subcritical pitchfork bifurcation. The normal form at the bifurcation is given and we show that these patterns can be described as solitary-wave envelopes. They are stable in a large temperature range and can diffuse along the chain of interacting particles. During their displacements the particles are continually redistributed on the envelope. This change of particle location induces a small modulation of the potential energy of the system, with an amplitude that depends on the transverse confinement. At high temperature, this modulation is irrelevant and the thermal motion of the localized patterns displays all the characteristics of a free quasiparticle diffusion with a diffusion coefficient that may be deduced from the normal form. At low temperature, significant physical effects are induced by the modulated potential. In particular, the localized pattern may be trapped at very low temperature. We also exhibit a series of confinement values for which the modulation amplitudes vanishes. For these peculiar confinements, the mean-square displacement of the localized patterns also evidences free-diffusion behavior at low temperature.

Avalanches, plasticity, and ordering in colloidal crystals under compression

Using numerical simulations we examine colloids with a long-range Coulomb interaction confined in a two-dimensional trough potential undergoing dynamical compression. As the depth of the confining well is increased, the colloids move via elastic distortions interspersed with intermittent bursts or avalanches of plastic motion. In these avalanches, the colloids rearrange to minimize their colloid-colloid repulsive interaction energy by adopting an average lattice constant that is isotropic despite the anisotropic nature of the compression. The avalanches take the form of shear banding events that decrease or increase the structural order of the system. At larger compression, the avalanches are associated with a reduction of the number of rows of colloids that fit within the confining potential, and between avalanches the colloids can exhibit partially crystalline or anisotropic ordering. The colloid velocity distributions during the avalanches have a non-Gaussian form with power-law tails and exponents that are consistent with those found for the velocity distributions of gliding dislocations. We observe similar behavior when we subsequently decompress the system, and find a partially hysteretic response reflecting the irreversibility of the plastic events.

Hysteretic and intermittent regimes in the subcritical bifurcation of a quasi-one-dimensional system of interacting particles

In this article, we study the effects of white Gaussian additive thermal noise on a subcritical pitchfork bifurcation. We consider a quasi-one-dimensional system of particles that are transversally confined, with short-range (non-Coulombic) interactions and periodic boundary conditions in the longitudinal direction. In such systems, there is a structural transition from a linear order to a staggered row, called the zigzag transition. There is a finite range of transverse confinement stiffnesses for which the stable configuration at zero temperature is a localized zigzag pattern surrounded by aligned particles, which evidences the subcriticality of the bifurcation. We show that these configurations remain stable for a wide temperature range. At zero temperature, the transition between a straight line and such localized zigzag patterns is hysteretic. We have studied the influence of thermal noise on the hysteresis loop. Its description is more difficult than at T=0 K since thermally activated jumps between the two configurations always occur and the system cannot stay forever in a unique metastable state. Two different regimes have to be considered according to the temperature value with respect to a critical temperature T_{c}(τ_{obs}) that depends on the observation time τ_{obs}. An hysteresis loop is still observed at low temperature, with a width that decreases as the temperature increases toward T_{c}(τ_{obs}). In contrast, for T>T_{c}(τ_{obs}) the memory of the initial condition is lost by stochastic jumps between the configurations. The study of the mean residence times in each configuration gives a unique opportunity to precisely determine the barrier height that separates the two configurations, without knowing the complete energy landscape of this many-body system. We also show how to reconstruct the hysteresis loop that would exist at T=0 K from high-temperature simulations.

Subcriticality of the zigzag transition: A nonlinear bifurcation analysis

When repelling particles are confined by a transverse potential in quasi-one-dimensional geometry, the straight line equilibrium configuration becomes unstable at small confinement, in favor of a staggered row that may be inhomogeneous or homogeneous. This conformational phase transition is a pitchfork bifurcation called the zigzag transition. We study the zigzag transition in infinite and periodic finite systems with short-range interactions. We provide numerical evidence that in this case the bifurcation is subcritical since it exhibits phase coexistence and hysteretic behavior. The physical mechanism responsible for the change in the bifurcation character is the nonlinear coupling between the transverse soft mode at the transition and the longitudinal Goldstone mode linked to the translational or rotational invariance of the zigzag pattern. An asymptotic analysis, near the bifurcation threshold and assuming an infinite system, gives an explicit expression for the normal form of the bifurcation. We establish the subcriticality, and we describe with excellent precision the inhomogeneous zigzag patterns observed in the simulations. A direct test of the physical mechanism responsible for the bifurcation character evidences a quantitative agreement.

Linear instability of a zigzag pattern

Interacting particles confined in a quasi-one-dimensional channel are physical systems which display various equilibrium patterns according to the interparticle interaction and the transverse confinement potential. Depending on the confinement, the particles may be distributed along a straight line, in a staggered row (zigzag), or in a configuration in which the linear and zigzag phases coexist (distorted zigzag). In order to clarify the conditions of existence of each configuration, we have studied the linear stability of the zigzag pattern. We find an acoustic transverse mode that destabilizes the zigzag configuration for short-range interaction potentials, and we calculate the interaction range above which this instability disappears. In particular, we recover the unconditional stability of zigzag patterns for Coulomb interactions. We show that the domain of existence for the distorted zigzag patterns is accurately described by our linear stability analysis. We also emphasize the complexity of finite size effects. Last, we provide a criterion for the onset of instability in the thermodynamic limit and propose a biphasic model that explains some characteristics of the distorted zigzag patterns.

Longitudinal and Transverse Single File Diffusion in Quasi-1D Systems

We review our recent results on Single File Diffusion (SFD) of a chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We exhibit new behaviors specifically associated to small systems and to small damping. The fluctuation dynamics is explained by the decomposition of the particles' motion in the normal modes of the chain. For longitudinal fluctuations, we emphasize the relevance of the soft mode linked to the translational invariance of the system to the long time SFD behavior. We show that close to the zigzag threshold, the transverse fluctuations also exhibit the SFD behavior, characterized by a mean square displacement that increases as the square root of time. This cannot be explained by the single file ordering, and the SFD behavior results from the strong correlation of the transverse displacements of neighbouring particles near the bifurcation. Extending our analytical modelization, we demonstrate the existence of this subdiffusive regime near the zigzag transition, in the thermodynamic limit. The zigzag transition is a supercritical pitchfork bifurcation, and we show that the transverse SFD behavior is closely linked to the vanishing of the frequency of the zigzag transverse mode at the bifurcation threshold.

[Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in files with longitudinal movements. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics,28 expanding files,63 and files with force and advanced formulations.29

Interaction force in a vertical dust chain inside a glass box

Small number dust particle clusters can be used as probes for plasma diagnostics. The number of dust particles as well as cluster size and shape can be easily controlled employing a glass box placed within a Gaseous Electronics Conference (GEC) rf reference chamber to provide confinement of the dust. The plasma parameters inside this box and within the larger plasma chamber have not yet been adequately defined. Adjusting the rf power alters the plasma conditions causing structural changes of the cluster. This effect can be used to probe the relationship between the rf power and other plasma parameters. This experiment employs the sloshing and breathing modes of small cluster oscillations to examine the relationship between system rf power and the particle charge and plasma screening length inside the glass box. The experimental results provided indicate that both the screening length and dust charge decrease as rf power inside the box increases. The decrease in dust charge as power increases may indicate that ion trapping plays a significant role in the sheath.

Nonequilibrium finite dust clusters: connecting normal modes and wakefields

The dynamic properties of finite three-dimensional dust clusters in a dusty plasma under the influence of an ion focus are studied by normal modes. The mode analysis has been extended to account for the ion focus using the point-charge model for the horizontal interaction of the dust particles. From that, an analytical model for a few-particle system is derived accounting for three distinct dynamical regimes at different focus strengths, namely, absolutely unstable and fully stable configurations as well as an unstable oscillatory regime. The techniques of normal mode analysis (NMA) and instantaneous normal modes (INM) extended by the ion focus have been applied to dust systems in the experiment and compared to the model. From this, the ion focus strength has been derived. The specific sensitivity of NMA and INM allows one to identify equilibrium configurations in this nonequilibrium environment for these finite clusters.

Noisy zigzag transition, fluctuations, and thermal bifurcation threshold

We study the zigzag transition in a system of particles with screened electrostatic interaction, submitted to a thermal noise. At finite temperature, this configurational phase transition is an example of noisy supercritical pitchfork bifurcation. The measurements of transverse fluctuations allow a complete description of the bifurcation region, which takes place between the deterministic threshold and a thermal threshold beyond which thermal fluctuations do not allow the system to flip between the symmetric zigzag configurations. We show that a divergence of the saturation time for the transverse fluctuations allows a precise and unambiguous definition of this thermal threshold. Its evolution with the temperature is shown to be in good agreement with theoretical predictions from noisy bifurcation theory.

Dimensional transitions in small Yukawa clusters

We provide a detailed analysis of structural transitions leading to rapid changes in the dimensionality of small Yukawa clusters. These transformations are induced by variations in the shape of confinement as well as the screening strength. We show that even in the most primitive systems composed of only a few strongly interacting particles, the order parameter exhibits a power-law behavior in the vicinity of the critical point of the continuous transition. The critical exponent γ = 1/2 is found to be universal in all studied cases, which is consistent with the general theory of continuous phase transitions.

Enhanced fluctuations of interacting particles confined in a box

We study the position fluctuations of interacting particles aligned in a finite cell that avoid any crossing in equilibrium with a thermal bath. The focus is put on the influence of the confining force directed along the cell length. We show that the system may be modeled as a 1D chain of particles with identical masses, linked with linear springs of varying spring constants. The confining force may be accounted for by linear springs linked to the walls. When the confining force range is increased toward the inside of the chain, a paradoxical behavior is exhibited. The outermost particles fluctuations are enhanced, whereas those of the inner particles are reduced. A minimum of fluctuations is observed at a distance of the cell extremities that scales linearly with the confining force range. Those features are in very good agreement with the model. Moreover, the simulations exhibit an asymmetry in their fluctuations which is an anharmonic effect. It is characterized by the measurement of the skewness, which is found to be strictly positive for the outer particles when the confining force is short ranged.

Instability of ion kinetic waves in a weakly ionized plasma

The fundamental higher-order Landau plasma modes are known to be generally heavily damped. We show that these modes for the ion component in a weakly ionized plasma can be substantially modified by ion-neutral collisions and a dc electric field driving ion flow so that some of them can become unstable. This instability is expected to naturally occur in presheaths of gas discharges at sufficiently small pressures and thus affect sheaths and discharge structures.

Structural transitions in laterally compressed two-dimensional Coulomb clusters

We model structural transitions of small-size Wigner crystals in laterally compressed two-dimensional traps. Ground and metastable configurations are calculated and their transformations are linked to conspicuous changes in the heat capacity of the system. We show that various types of structural transitions are reflected by characteristic features in the behavior of the heat capacity. For deeper understanding, results produced by the Monte Carlo numerical calculations are compared to predictions of simple one-dimensional models.