Large compact clusters and fast dynamics in coupled nonequilibrium systems

Abstract

We demonstrate particle clustering on macroscopic scales in a coupled nonequilibrium system where two species of particles are advected by a fluctuating landscape and modify the landscape in the process. The phase diagram generated by varying the particle-landscape coupling, valid for all particle density and in both one and two dimensions, shows novel nonequilibrium phases. While particle species are completely phase separated, the landscape develops macroscopically ordered regions coexisting with a disordered region, resulting in coarsening and steady state dynamics on time scales which grow algebraically with size, not seen earlier in systems with pure domains.PACS numbers: 64.75.Gh, 68.43.Jk Particle clustering is important in many natural physical and biological phenomena, for instance, the formation of sediments [1] and protein clustering on a biological membrane [2]. Evidently, it is important to understand processes that cause clustering in different physical contexts, and how these processes influence the properties of the cluster and the time taken to form it. Often, large-scale clustering is driven by interactions with an external medium which itself has correlations in space and time [3][4][5]. An important physical effect in such systems is the back-influence of the particles on the medium. This interaction can aid clustering, or destroy it. If a cluster does form, it may be compact and robust, or a dynamic object that undergoes constant reorganization. The formation time may grow exponentially with the size, or as a power law. Given this wealth of possibilities, it is important to look for an understanding, within simple models, of the circumstances under which different sorts of macroscopically clustered states occur.In this letter, we derive the phase diagram of a simple model system as we vary the interaction between the environment and particles. In the process, we unmask a novel non-equilibrium phase of particles with compact clustering and rich and rapid dynamics coexisting with a macroscopically organized landscape. The model has partial overlap with the lattice gas model of Lahiri and Ramaswamy (LR) for sedimenting colloidal crystals [6,7], but the new phases manifest themselves outside the LR regime. Our results hold in both one and two dimensions.The model consists of two sets of particles moving stochastically in a fluctuating potential energy landscape. Particles try to minimize their energy by (a) moving along the local potential gradient of the landscape and (b) modifying the landscape around them in such a way as to lower the energy further. The model is generic but we discuss it in the language of particles confined to move on a fluctuating surface in the presence of gravity, where the particles can locally distort the surface shape to further lower the energy (see Fig. 1). One of the particle species is considered lighter and the other is heavier; we use the name LH (Light-heavy) model to describe the system. Process (b) affects the landscape dynamics quite differently in parts...

Ordered phases in coupled nonequilibrium systems: Dynamic properties

“…1 denotes the position of V , and the mean-squared fluctuation of this position is denoted as σ 2 0 (t). Our earlier results in [4] show that σ 2 0 (t) grows diffusively with time with a diffusion constant ∼ 1/N for short times t N 2 , then saturates at a value ∼ N for large times N 2 t exp(bN ). This means that V diffuses around this mean position C, but stays confined within a region of size ∼ √ N , and this region is explored by V over an algebraic time-scale ∼ N 2 .…”

“…We consider an untilted surface with equal number of upslope and downslope bonds so that i τ i+1/2 = 0. Our model shows three distinct well-ordered phases as the parameters b and b are varied [4], corresponding to whether L particles tend to impart an upward push, no push, or a downward push to the fluctuating surface. In [5] we have presented a detailed characterization of the static properties of these ordered phases.…”

“…As a result of this current, the landscape, along with the particle clusters shows a downward motion with an average velocity 1/N . In [4] this phase was named as "infinitesimal fall with phase separation" since for very large N , the velocity of the downward fall becomes infinitesimally small. The coexistence of an ordered valley along with the symmetrically fluctuating part of the landscape, gives rise to rich steady state dynamics with time-scales growing algebraically with N for the landscape, and exponentially with N for the particles.…”

“…The entire system carries a finite current of upslope and downslope bonds in the steady state resulting in a net downward motion with finite velocity. The fact that unlike IPS phase, the velocity of the downward fall remains finite in this case even for large systems, explains the name 'fast fall with phase separation' used in [4].…”

“…In an earlier work [4] we had measured the dynamical exponent which describes the coarsening during approach to the steady state, and had obtained z = 2 for both IPS and FPS phases. In the IPS phase, the same value of z describes steady state dynamics as well.…”