Strong clustering of noninteracting, sliding passive scalars driven by fluctuating surfaces

Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale ω between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value ω=ω_{c} when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ω_{c} in terms of the coupling parameters of the system. We show that ω_{c} is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ω_{c} from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ω^{*} of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix A and compressibility matrix K. Our exact calculations show that Onsager-type current symmetry relation AK=KA^{T} can be satisfied if and only if ω^{*}=ω_{c}. Our coarse-grained model calculations can be easily generalized to other coupled systems.

Lattice models of random advection and diffusion and their statistics

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighboring random sites. The model belongs to a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases some models studied in the literature, such as those of Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation of the advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and a mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

Dynamic multiscaling in stochastically forced Burgers turbulence

We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of interval collapse time, which we define as the time taken for a spatial interval, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponents of the moments of various orders of these interval collapse times, we show that (a) there is not one but an infinity of characteristic time scales and (b) the probability distribution function of the interval collapse times is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to higher dimensions, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.

Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall

We present an exact solution for the height distribution of the KPZ equation at any time t in a half space with flat initial condition. This is equivalent to obtaining the free-energy distribution of a polymer of length t pinned at a wall at a single point. In the large t limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben-Arous-Péché transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the polymer at large time. We also unveil curious identities in distribution between partition functions in half-space and certain partition functions in full space for Brownian-type initial condition.

Kinetic roughening in active interfaces

The essential features of many interfaces driven out of equilibrium are described by the same equation—the Kardar-Parisi-Zhang (KPZ) equation. How do living interfaces, such as the cell membrane, fit into this picture? In an endeavour to answer such a question, we proposed in [F. Cagnetta, M. R. Evans, D. Marenduzzo, PRL 120, 258001 (2018)] an idealised model for the membrane of a moving cell. Here we discuss how the addition of simple ingredients inspired by the dynamics of the membrane of moving cells affects common kinetic roughening theories such as the KPZ and Edwards-Wilkinson equations.

Interface growth driven by a single active particle

We study pattern formation, fluctuations, and scaling induced by a growth-promoting active walker on an otherwise static interface. Active particles on an interface define a simple model for energy-consuming proteins embedded in the plasma membrane, responsible for membrane deformation and cell movement. In our model, the active particle overturns local valleys of the interface into hills, simulating growth, while itself sliding and seeking new valleys. In one dimension, this "overturn-slide-search" dynamics of the active particle causes it to move superdiffusively in the transverse direction while pulling the immobile interface upward. Using Monte Carlo simulations, we find an emerging tentlike mean profile developing with time, despite large fluctuations. The roughness of the interface follows scaling with the growth, dynamic, and roughness exponents, derived using simple arguments as β=2/3, z=3/2, and α=1/2, respectively, implying a breakdown of the usual scaling law β=α/z, due to very local growth of the interface. The transverse displacement of the puller on the interface scales as ∼t^{2/3} and the probability distribution of its displacement is bimodal, with an unusual linear cusp at the origin. Both the mean interface pattern and probability distribution display scaling. A puller on a static two-dimensional interface also displays aspects of scaling in the mean profile and probability distribution. We also show that a pusher on a fluctuating interface moves subdiffusively leading to a separation of timescale in pusher motion and interface response.

Statistical mechanics of a single active slider on a fluctuating interface

We study the statistical mechanics of a single active slider on a fluctuating interface, by means of numerical simulations and theoretical arguments. The slider, which moves by definition towards the interface minima, is active as it also stimulates growth of the interface. Even though such a particle has no counterpart in thermodynamic systems, active sliders may provide a simple model for ATP-dependent membrane proteins that activate cytoskeletal growth. We find a wide range of dynamical regimes according to the ratio between the timescales associated with the slider motion and the interface relaxation. If the interface dynamics is slow, the slider behaves like a random walker in a random environment, which, furthermore, is able to escape environmental troughs by making them grow. This results in different dynamic exponents to the interface and the particle: the former behaves as an Edward-Wilkinson surface with dynamic exponent 2, whereas the latter has dynamic exponent 3/2. When the interface is fast, we get sustained ballistic motion with the particle surfing a membrane wave created by itself. However, if the interface relaxes immediately (i.e., it is infinitely fast), particle motion becomes symmetric and goes back to diffusive. Due to such a rich phenomenology, we propose the active slider as a toy model of fundamental interest in the field of active membranes and, generally, whenever the system constituent can alter the environment by spending energy.

Response to a small external force and fluctuations of a passive particle in a one-dimensional diffusive environment

We investigate the long-time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant D. We consider two cases: (a) The particle is pulled forward by a small external constant force and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large D limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the crossover to the true asymptotic behavior.

Time evolution of intermittency in the passive slider problem

How does a steady state with strong intermittency develop in time from an initial state which is statistically random? For passive sliders driven by various fluctuating surfaces, we show that the approach involves an indefinitely growing length scale which governs scaling properties. A simple model of sticky sliders suggests scaling forms for the time-dependent flatness and hyperflatness, both measures of intermittency and these are confirmed numerically for passive sliders driven by a Kardar-Parisi-Zhang surface. Aging properties are studied via a two-time flatness. We predict and verify numerically that the time-dependent flatness is, remarkably, a nonmonotonic function of time with different scaling forms at short and long times. The scaling description remains valid when clustering is more diffuse as for passive sliders evolving through Edwards-Wilkinson driving or under antiadvection, although exponents and scaling functions differ substantially.

Phase Segregation of Passive Advective Particles in an Active Medium

Localized contractile configurations or asters spontaneously appear and disappear as emergent structures in the collective stochastic dynamics of active polar actomyosin filaments. Passive particles which (un)bind to the active filaments get advected into the asters, forming transient clusters. We study the phase segregation of such passive advective scalars in a medium of dynamic asters, as a function of the aster density and the ratio of the rates of aster remodeling to particle diffusion. The dynamics of coarsening shows a violation of Porod behavior; the growing domains have diffuse interfaces and low interfacial tension. The phase-segregated steady state shows strong macroscopic fluctuations characterized by multiscaling and intermittency, signifying rapid reorganization of macroscopic structures. We expect these unique nonequilibrium features to manifest in the actin-dependent molecular clustering at the cell surface.

Replica Symmetry Breaking in Trajectories of a Driven Brownian Particle

We study a Brownian particle passively driven by a field obeying the noisy Burgers' equation. We demonstrate that the system exhibits replica symmetry breaking in the path ensemble with the initial position of the particle being fixed. The key step of the proof is that the path ensemble with a modified boundary condition can be exactly mapped onto the canonical ensemble of directed polymers.

Condensation and intermittency in an open-boundary aggregation-fragmentation model

We study real space condensation in aggregation-fragmentation models where the total mass is not conserved, as in phenomena such as cloud formation and intracellular trafficking. We study the scaling properties of the system with influx and outflux of mass at the boundaries using numerical simulations, supplemented by analytical results in the absence of fragmentation. The system is found to undergo a phase transition to an unusual condensate phase, characterized by strong intermittency and giant fluctuations of the total mass. A related phase transition also occurs for biased movement of large masses, but with some crucial differences which we highlight.

Bidirectional transport on a dynamic lattice

Bidirectional variants of stochastic many particle models for transport by molecular motors show a strong tendency to form macroscopic clusters on static lattices. Inspired by the fact that the microscopic tracks for molecular motors are dynamical, we study the influence of different types of lattice dynamics on stochastic bidirectional transport. We observe a transition toward efficient transport (corresponding to the dissolution of large clusters) controlled by the lattice dynamics.