Synchronization and fluctuations: Coupling a finite number of stochastic units.

It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.

A continuous-time persistent random walk model for flocking.

A classical random walker is characterized by a random position and velocity. This sort of random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers," including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles.

Three-stage stochastic pump: Another type of Onsager-Casimir symmetry and results far from equilibrium.

The stochastic thermodynamic analysis of a time-periodic single particle pump sequentially exposed to three thermochemical reservoirs is presented. The analysis provides explicit results for flux, thermodynamic force, entropy production, work, and heat. These results apply near equilibrium as well as far from equilibrium. In the linear response regime, a different type of Onsager-Casimir symmetry is uncovered. The Onsager matrix becomes symmetric in the limit of zero dissipation.

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions.

We study the role of the tail and the range of interaction in a spatially structured population of two-state on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long-range interactions) to a disordered one (short-range interactions). Depending on the interaction kernel, the transition may be smooth (second order) or abrupt (first order). We analyze the transient, which may present different routes to the steady state with vastly different time scales.

Stochastic thermodynamics for a periodically driven single-particle pump.

We present the stochastic thermodynamic analysis of a time-periodic single-particle pump, including explicit results for flux, thermodynamic force, entropy production, work, heat, and efficiency. These results are valid far from equilibrium. The deviations from the linear (Onsager) regime are discussed.

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases.

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators.

Onsager coefficients for systems with periodic potentials.

We carry out the thermodynamic analysis of a Markovian stochastic engine, driven by a spatially and temporally periodic modulation in a d-dimensional space. We derive the analytic expressions for the Onsager coefficients characterizing the linear response regime for the isothermal transfer of one type of work (a driver) to another (a load), mediated by a stochastic time-periodic machine. As an illustration, we obtain the explicit results for a Markovian kangaroo process coupling two orthogonal directions and find extremely good agreement with numerical simulations. In addition, we obtain and discuss expressions for the entropy production, power, and efficiency for the kangaroo process.

Kramers' rate for systems with multiplicative noise.

Kramers' rate for the passage of trajectories X(t) over an energy barrier due to thermal or other fluctuations is usually associated with additive noise. We present a generalization of Kramers' rate for systems with multiplicative noise. We show that the expression commonly used in the literature for multiplicative noise is not correct, and we present results of numerical integrations of the Langevin equation for dX(t)/dt evolving in a quartic bistable potential which corroborate our claim.

Dynamical approach to weakly dissipative granular collisions.

Granular systems present surprisingly complicated dynamics. In particular, nonlinear interactions and energy dissipation play important roles in these dynamics. Usually (but admittedly not always), constant coefficients of restitution are introduced phenomenologically to account for energy dissipation when grains collide. The collisions are assumed to be instantaneous and to conserve momentum. Here, we introduce the dissipation through a viscous (velocity-dependent) term in the equations of motion for two colliding grains. Using a first-order approximation, we solve the equations of motion in the low viscosity regime. This approach allows us to calculate the collision time, the final velocity of each grain, and a coefficient of restitution that depends on the relative velocity of the grains. We compare our analytic results with those obtained by numerical integration of the equations of motion and with exact ones obtained by other methods for some geometries.

Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation.

We consider an array of units each of which can be in one of three states. Unidirectional transitions between these states are governed by Markovian rate processes. The interactions between units occur through a dependence of the transition rates of a unit on the states of the units with which it interacts. This coupling is nonlocal, that is, it is neither an all-to-all interaction (referred to as global coupling), nor is it a nearest neighbor interaction (referred to as local coupling). The coupling is chosen so as to disfavor the crowding of interacting units in the same state. As a result, there is no global synchronization. Instead, the resultant spatiotemporal configuration is one of clusters that move at a constant speed and that can be interpreted as traveling waves. We develop a mean field theory to describe the cluster formation and analyze this model analytically. The predictions of the model are compared favorably with the results obtained by direct numerical simulations.

Globally coupled stochastic two-state oscillators: fluctuations due to finite numbers.

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a 1/N expansion and the Langevin equation via standard methods of Itô calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits N → ∞ and t → ∞ (t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

Synchronization of globally coupled two-state stochastic oscillators with a state-dependent refractory period.

We present a model of identical coupled two-state stochastic units, each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition.

Pulse propagation in a chain of o-rings with and without precompression.

We implement a binary collision approximation to study pulse propagation in a chain of o-rings. In particular, we arrive at analytic results from which the pulse velocity is obtained by simple quadrature. The predicted pulse velocity is compared to the velocity obtained from the far more resource-intensive numerical integration of the equations of motion. We study chains without precompression, chains precompressed by a constant force at the chain ends (constant precompression), and chains precompressed by gravity (variable precompression). The application of the binary collision approximation to precompressed chains provides an important generalization of a successful theory that had up to this point only been implemented to chains without precompression, that is, to chains in a sonic vacuum.

Energy transport in a one-dimensional granular gas.

We study heat conduction in one-dimensional granular gases. In particular, we consider two mechanisms of viscous dissipation during intergrain collisions. In one, the dissipative force is proportional to the grain's velocity and dissipates not only energy but also momentum. In the other, the dissipative force is proportional to the relative velocity of the grains and therefore conserves momentum even while dissipating energy. This allows us to explore the role of momentum conservation in the heat conduction properties of this one-dimensional nonlinear system. We find normal thermal conduction whether or not momentum is conserved.

Short-pulse dynamics in strongly nonlinear dissipative granular chains.

We study the energy decay properties of a pulse propagating in a strongly nonlinear granular chain with damping proportional to the relative velocity of the grains. We observe a wave disturbance that at low viscosities consists of two parts exhibiting two entirely different time scales of dissipation. One part is an attenuating solitary wave, dominated by discreteness and nonlinearity effects as in a dissipationless chain, and has the shorter lifetime. The other is a purely dissipative shocklike structure with a much longer lifetime and exists only in the presence of dissipation. The range of viscosities and initial configurations that lead to this complex wave disturbance are explored.

Numerical study of A+A-->0 and A+B-->0 reactions with inertia.

Using numerical methods the authors study the annihilation reactions A+A-->0 and A+B-->0 in one and two dimensions in the presence of inertial contributions to the motion of the particles. The particles move freely following Langevin dynamics at a fixed temperature. The authors focus on the role of friction.

Observation of two-wave structure in strongly nonlinear dissipative granular chains.

In a strongly nonlinear viscous granular chain impacted by a single grain we observe a wave disturbance that consists of two parts exhibiting two time scales of dissipation. Above a critical viscosity there is no separation of the two pulses, and the dissipation and nonlinearity dominate the shocklike attenuating pulse.