Linear instability of a zigzag pattern

Wave spectroscopy in a driven granular material

Driven granular media constitute model systems in out-of-equilibrium statistical physics. By assimilating the motions of granular particles to those of atoms, by analogy, one can obtain macroscopic equivalent of phase transitions. Here, we study fluid-like and crystal-like two-dimensional states in a driven granular material. In our experimental device, a tunable magnetic field induces and controls remote interactions between the granular particles. We use high-speed video recordings to analyse the velocity fluctuations of individual particles in stationary regime. Using statistical averaging, we find that the particles self-organize into collective excitations characterized by dispersion relations in the frequency-wavenumber space. These findings thus reveal that mechanical waves analogous to condensed matter phonons propagate in driven granular media. When the magnetic coupling is weak, the waves are longitudinal, as expected for a fluid-like phase. When the coupling is stronger, both longitudinal and transverse waves propagate, which is typically seen in solid-like phases. We model the dispersion relations using the spatial distribution of particles and their interaction potential. Finally, we infer the elastic parameters of the granular assembly from equivalent sound velocities, thus realizing the spectroscopy of a granular material.

Buckling of two-dimensional plasma crystals with nonreciprocal interactions

Laboratory realizations of two-dimensional (2D) plasma crystals typically involve monodisperse microparticles confined into horizontal monolayers in radio-frequency (rf) plasma sheaths. This gives rise to the so-called plasma wakes beneath the microparticles. The presence of wakes renders the interactions in such systems nonreciprocal, a fact that can lead to a quite different behavior from the one expected for their reciprocal counterparts. Here we examine the buckling of a hexagonal 2D plasma crystal, occurring as the confinement strength is decreased, taking explicitly into account the nonreciprocity of the system via a well-established point-wake model. We observe that for a finite wake charge, the monolayer hexagonal crystal undergoes a transition first to a bilayer hexagonal structure, unrealizable in harmonically confined reciprocal Yukawa systems, and subsequently to a bilayer square structure. Our theoretical results are confirmed by molecular dynamics simulations for experimentally relevant parameters, indicating the potential of their observation in state-of-the-art experiments with 2D complex plasmas.

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.

Enhancement of Brownian motion for a chain of particles in a periodic potential

The transport of particles in very confined channels in which single file diffusion occurs has been largely studied in systems where the transverse confining potential is smooth. However, in actual physical systems, this potential may exhibit both static corrugations and time fluctuations. Some recent results suggest the important role played by this nonsmoothness of the confining potential. In particular, quite surprisingly, an enhancement of the Brownian motion of the particles has been evidenced in these kinds of systems. We show that this enhancement results from the commensurate effects induced by the underlying potential on the vibrational spectra of the chain of particles, and from the effective temperature associated with its time fluctuations. We will restrict our derivation to the case of low temperatures for which the mean squared displacement of the particles remains smaller than the potential period.

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.