Diffusion enhancement and autoparametric resonance.

The possibility of an autoparametric resonance in an isolated many-particle system induces a specific behavior of the particles in the presence of thermal noise. In particular, the variance associated with a resonant mode, and consequently that of the associated particles, is strongly increased compared to what it would have in the absence of parametric resonance. In this paper we consider a dimer submitted to a periodic potential for which there are only two modes, the center of mass motion and the internal vibration mode. This is the simplest system which is dynamically rich enough to exhibit an autoparametric excitation of the internal vibrations by the center of mass motion. The consequences of this autoparametric excitation on the particles diffusion will be discussed according to the stiffness of the interaction and to the initial energy of the dimer, the relevant parameters which characterize this dynamics.

Parametric resonance in a conservative system of coupled nonlinear oscillators.

We study a dimer in a periodic potential well, which is a conservative but nonintegrable system. This seemingly simple system exhibits a surprisingly rich dynamics. Using a systematic asymptotic analysis, we demonstrate that the translation mode of the dimer (center of mass motion) may induce a parametric resonance of the oscillatory mode. No external forcing occurs, thus this system belongs to the class of autoparametric systems. When the dimer energy is such that both particles are trapped in neighboring potential wells, we derive the relevant amplitude equations for the eigenmodes (center of mass motion and relative motion) and show that they are integrable. In the opposite limit, when the dimer slides along the external potential so that the center of mass motion is basically a translation, we also exhibit autoparametric amplification of the relative motion. In both cases our calculations provide reliable estimates of the relevant parameters for the autoparametric resonance to appear. Moreover, the comparison between the numerical integration of the actual system and the asymptotic analysis evidences an excellent quantitative agreement.

Motions of a dimer on a periodic potential.

A dimer on a periodic potential is a simple system that exhibits a surprisingly rich dynamics. This system is conservative, but it is nonlinear and nonintegrable. In a previous work, we evidenced the autoparametric excitation of the relative motion by the center of mass in two limiting cases (very small or very large initial energy, compared to the external potential depth). We extend these results for arbitrary initial energy. The relevant control parameters are the dimer initial energy and the stiffness of the link between the two particles. In this parameter plane, we build a behavior map which classifies the available dynamical regimes of the dimer. The parameters plane can be separated into domains in which the dimer particles are either trapped in adjacent potential wells, slide along the potential, or exhibit more complex motions in which the particles jumps to farthest well or in which the center-of-mass motion is neither monotonous nor periodic. We discuss the thresholds between these domains.

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.

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.

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.

Single-file diffusion of particles in a box: transient behaviors.

We consider a finite number of particles with soft-core interactions, subjected to thermal fluctuations and confined in a box with excluded mutual passage. Using numerical simulations, we focus on the influence of the longitudinal confinement on the transient behavior of the longitudinal mean squared displacement. We exhibit several power laws for its time evolution according to the confinement range and to the rank of the particle in the file. We model the fluctuations of the particles as those of a chain of springs and point masses in a thermal bath. Our main conclusion is that actual system dynamics can be described in terms of the normal oscillation modes of this chain. Moreover, we obtain complete expressions for the physical observables, in excellent agreement with our simulations. The correct power laws for the time dependency of the mean squared displacement in the various regimes are recovered, and analytical expressions of the prefactors according to the relevant parameters are given.

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.

Single-file diffusion of particles with long-range interactions: damping and finite-size effects.

We study the single file diffusion of a cyclic chain of particles that cannot cross each other, in a thermal bath, with long-ranged interactions and arbitrary damping. We present simulations that exhibit new behaviors specifically associated with systems of small numbers of particles and with small damping. In order to understand those results, we present an original analysis based on the decomposition of the particles' motion in the normal modes of the chain. Our model explains all dynamic regimes observed in our simulations and provides convincing estimates of the crossover times between those regimes.

Real-time single-cell response to stiffness.

Living cells adapt to the stiffness of their environment. However, cell response to stiffness is mainly thought to be initiated by the deformation of adhesion complexes under applied force. In order to determine whether cell response was triggered by stiffness or force, we have developed a unique method allowing us to tune, in real time, the effective stiffness experienced by a single living cell in a uniaxial traction geometry. In these conditions, the rate of traction force buildup dF/dt was adapted to stiffness in less than 0.1 s. This integrated fast response was unambiguously triggered by stiffness, and not by force. It suggests that early cell response could be mechanical in nature. In fact, local force-dependent signaling through adhesion complexes could be triggered and coordinated by the instantaneous cell-scale adaptation of dF/dt to stiffness. Remarkably, the effective stiffness method presented here can be implemented on any mechanical setup. Thus, beyond single-cell mechanosensing, this method should be useful to determine the role of rigidity in many fundamental phenomena such as morphogenesis and development.

Comment on "Elastic constants from microscopic strain fluctuations".

Sengupta [Phys. Rev. E 61, 1072 (2000)] presented an elegant and simple finite-size scaling method for the calculation of elastic constants and their corresponding correlation lengths, which is suitable for many finite discrete systems considered through simulations or experiments. We take into account a mathematical finite-size effect that was neglected by the authors in order to propose a more accurate method. Consequences on the authors' results are also discussed.

Single file diffusion in macroscopic Wigner rings.

The single file diffusion in a circular channel of millimetric charged balls is studied. The evolution in time of the mean square displacement is shown to be subdiffusive, but slower than the powerlike t1/2 behavior observed in circular colloidal systems or predicted in one-dimensional infinite systems.