Noisy zigzag transition, fluctuations, and thermal bifurcation threshold

Abstract : 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. Many systems exhibit a topological transition as soon as a parameter β, which controls this transition, reaches a threshold β ZZ. An example of such transitions is the " zigzag bifurcation, " which involves a quasi-one-dimensional (1D) chain of interacting particles confined in a narrow channel that forbids any particle crossing [1]. At T = 0 K and for an infinite system, the particles remain aligned until the transverse component of the interparticle forces exceeds the transverse confinement, which may be expressed as β > β ZZ if β is the transverse stiffness [see Fig. 1(a)]. The bifurcation occurs when these two contributions are equal, β = β ZZ. Beyond that, for β < β ZZ , the aligned configuration is energetically unfavorable and mechanically unstable. The linear chain then turns into a 2D zigzag configuration, odd particles up and even down or the symmetric configuration [see Fig. 1(b)]. Experimentally, such transitions and zigzag configurations have been observed in various systems like ions with Coulombic interaction confined in a Paul trap [3,4] or a magnetic Penning trap [5–7], charged particles in a plasma dust interacting with Yukawa interaction [8–11], or colloidal dispersions with screened electrostatic interactions in an annular confinement [12–15]. At the macroscopic scale, we have also observed these zigzag bifurcations with a system of macroscopic metallic beads in electrostatic interaction confined in a finite-sized channel [16] (see Fig. 2). With negligible thermal noise, the zigzag transition is a supercritical pitchfork bifurcation, which can thus be described in the framework of the Landau theory of second order phase transition. In this formalism, an order parameter which characterizes the topological deformation suddenly changes when a control parameter reaches the bifurcation threshold. Among the possible control parameters, we choose in this Introduction the transverse stiffness β, assuming a fixed density and a fixed longitudinal transverse potential [17]. In this configuration, the transverse distance |y| of the particles to the channel axis increases as the confinement decreases (see Fig. 5). It varies from 0 for β > β ZZ to a finite value y(β) = ±(β ZZ − β) 1/2 as the difference β ZZ − β increases. This bifurcation is purely mechanical. The zigzag threshold β ZZ is called the deterministic threshold and noted β ZZ (0) ≡ β ZZ (T = 0). When the confined particles interact with a thermal bath, the topological properties are no longer sufficient to describe the states of the system. Although the equilibrium configurations are independent of the temperature, the thermal fluctuations directly modify the bifurcation scheme. Far from the threshold, when the system is strongly stable, these fluctuations do not modify the stability of the system, but they have a large influence near the bifurcation since the system is then very sensitive to any small perturbation. In particular, just beyond the deterministic bifurcation threshold [β < β ZZ (0)], the thermal fluctuations allow flips between the two symmetric zigzag orderings; therefore, these two equivalent energetic states are randomly occupied. It results in two important consequences. First of all, these flips induce a temperature-dependent shift of the bifurcation threshold. The Brownian system stays in a single zigzag configuration for β < β ZZ (T) < β ZZ (0) only, where the thermal " lower bound " β ZZ (T) characterizes the transverse confinement below which the flips between the energetically equivalent and symmetric zigzag configurations do not happen anymore. The second consequence is a broadening of the variation of y with β around the threshold β ZZ (0), the sharp deterministic bifurcation being replaced by a smooth transition regime for noisy system. Two interpretations are proposed to analyze these noisy bifurcations. The first one still considers the actual bifurcation as a deterministic transition at β = β ZZ (0). In this framework the deterministic transition is only blurred by the thermal noise. On the other hand, several studies focusing on the actual particle displacements preferred to introduce a third " bifurcation region " or " mesostate " defined as the range of parameter values β ZZ (T) < β < β ZZ (0) for which the thermal flips control the long time particle dynamics [18–20]. Whatever the description considered, precise determina-tions of the thresholds are required. The main goal of this article is to discuss the threshold determination in the case of noisy bifurcation. At T = 0 K, β ZZ (0) corresponds to the value of β for which the singularity is observed in the curve y(β). Let us indicate that for finite systems another method has been proposed: β ZZ (0) is characterized by the vanishing of a transverse vibrational frequency of the system [21]. In contrast, for T > 0 K, β ZZ (T) is not so accurate since the y(β) curve is broadened by the thermal fluctuations which smooth out the
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Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2013, 87, pp.62135 - 62135. 〈10.1103/PhysRevE.87.062135〉
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Jean-Baptiste Delfau, Christophe Coste, Michel Saint Jean. Noisy zigzag transition, fluctuations, and thermal bifurcation threshold. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2013, 87, pp.62135 - 62135. 〈10.1103/PhysRevE.87.062135〉. 〈hal-01404829〉

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