Exact maximal height distribution of fluctuating interfaces

We present an exact solution for the distribution P(h(m),L) of the maximal height h(m) (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h(m),L)=L(-1/2)f(h(m)L(-1/2)) for all L>0, where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds, but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.     

Ratchet motion and current reversal of coupled Brownian motors in pulsating symmetric potentials

In this study, we investigate the collective directed transport of coupled Brownian particles in spatially symmetric periodic potentials under time-periodic pulsating modulations. We find that the coupling between two particles can induce symmetry breaking and consequently collective directed motion. Moreover, the direction of motion can be reversed under certain conditions. The dependence of directed current on various parameters is systematically studied. reverse motion can be achieved by modulating the coupling free length and the phase shift of the pulsating potential. The dynamical mechanism of these transport properties is understood in terms of the effective-potential theory and the space-time transformation invariance. The directed transport of coupled Brownian motors can be manipulated and optimized by adjusting the coupling strength, pulsating frequency, or noise intensity.     

Brownian excursions on combs

In this paper we use techniques from Ito excursion theory to analyze Brownian motion on generalized combs. Ito excursion theory is a little-known area of probability theory and we therefore present a brief introduction for the uninitiated. A general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined. We then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs.     

Effect of gain compression above and below threshold on the chirp characteristics of 1.55 µm distributed feedback laser

We have observed and quantified the adiabatic and transient chirp in a directly modulated laser. The wavelength excursion for both chirp terms is well characterized by the phase rate equation model that describes the chirp behavior. In this study, the effect of gain compression and linewidth enhancement factor, below and above threshold, on the chirp characteristics is investigated by simulation. We have observed the trade-off between the two chirp terms i.e., transient part of the chirp is reduced by the strong damping introduced by gain compression, while as the adiabatic part increases with gain compression. We have observed that above threshold the α-factor increases with bias current, which is attributed to the enhancement of gain compression coefficient. It is shown that the higher the maximum gain, the lower the effects of gain compression and lower the α-factor. Finally, the effects of gain compression on the transmission characteristics are investigated.     

Energy fluctuations in a biharmonically driven nonlinear system

We study the fluctuations of work done and dissipated heat of a Brownian particle in a symmetric double well system. The system is driven by two periodic input signals that rock the potential simultaneously. Confinement in one preferred well can be achieved by modulating the relative phase between the drives. We show that in the presence of pumping the stochastic resonance signal is enhanced when analysed in terms of the average work done on the system per cycle. This is in contrast with the case when pumping is achieved by applying an external static bias, which degrades resonance. We analyse the nature of work and heat fluctuations and show that the steady state fluctuation theorem holds in this system.     

Spontaneous symmetry breaking of population in a nonadiabatically driven atomic trap: an Ising-class phase transition

We have observed spontaneous symmetry breaking of the population of Brownian particles between two moving potentials in the spatiotemporally symmetric system. Cold atoms preferentially occupy one of the dynamic double-well potentials, produced in the parametrically driven dissipative magneto-optical trap far from equilibrium, above a critical number of atoms. We find that the population asymmetry, which may be interpreted as the biased Brownian motion, can be qualitatively described by the mean-field Ising-class phase transition. This in situ study may be useful for investigation of dynamic phase transition or temporal behavior of critical phenomena.     

Slowdown for Time Inhomogeneous Branching Brownian Motion

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T ^1/3. We conjecture that this is the worse case correction possible.     

Transverse diffusion induced phase transition in asymmetric exclusion process on a surface

We extend one-dimensional asymmetric simple exclusion process (ASEP) to a surface and show that the effect of transverse diffusion is to induce a continuous phase transition from a constant density phase to a maximal current phase as the forward transition probability p is tuned. The signature of the non-equilibrium transition is evident in the finite size effects near the transition. The results are compared with similar couplings operative only at the boundary. It is argued that the nature of the phases can be interpreted in terms of the modifications of boundary layers.     

Stochastic Spacetime and Brownian Motion of Test Particles

The operational meaning of spacetime fluctuations is discussed. Classical spacetime geometry can be viewed as encoding the relations between the motions of test particles in the geometry. By analogy, quantum fluctuations of spacetime geometry can be interpreted in terms of the fluctuations of these motions. Thus, one can give meaning to spacetime fluctuations in terms of observables which describe the Brownian motion of test particles. We will first discuss some electromagnetic analogies, where quantum fluctuations of the electromagnetic field induce Brownian motion of test particles. We next discuss several explicit examples of Brownian motion caused by a fluctuating gravitational field. These examples include lightcone fluctuations, variations in the flight times of photons through the fluctuating geometry, and fluctuations in the expansion parameter given by a Langevin version of the Raychaudhuri equation. The fluctuations in this parameter lead to variations in the luminosity of sources. Other phenomena that can be linked to spacetime fluctuations are spectral line broadening and angular blurring of distant sources.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

Brownian pump with an unbiased external force

We investigate Brownian pump transport in the presence of an unbiased external force. The pumping system is embedded in a finite region bounded by two particle reservoirs. In the adiabatic limit, we obtain the analytical expressions of the current and the concentration ratio. We find that Brownian particles can be pumped through an asymmetric potential from a particle reservoir at low concentration to one at the same or higher concentration in the presence of an unbiased external force.      

Average Cycle Period in Asymmetrical Flashing Ratchet

The directed motion of a Brownian particle in a flashing potential with various transition probabilities and waiting times in one of two states is studied. An expression for the average cycle period is proposed and the steady current J of the particle is calculated via Langevin simulation. The results show that the optimal cycle period τ_m, which takes the maximum of J, is shifted to a small value when the transition probability λ from the potential on to the potential off decreases, the maximal current appears in the case of the average waiting time in the potential on being longer than in the potential off, and the direction of current depends on the ratio of the average times waiting in two states.     

Constructive influence of noise-flatness in correlation ratchets

The influence of noise-flatness on overdamped motion of Brownian particles in a 1D periodic system with a simple sawtooth potential subjected to both unbiased thermal noise and three-level telegraph noise is considered. The exact formula for the stationary probability flux (current) is presented. The phenomenon of multiple current reversals and some topological properties of the hypersurface of zero current in the parameter space of noises are investigated and illustrated by phase diagrams. The conditions for the existence of four current reversals versus the switching rate of nonequilibrium noise are given. An alternative interpretation of the results in terms of cross-correlation between two dichotomous noises is presented.     

Metastability for a Stochastic Dynamics with a Parallel Heat Bath Updating Rule

We consider the problem of metastability for a stochastic dynamics with a parallel updating rule with single spin rates equal to those of the heat bath for the Ising nearest neighbors interaction. We study the exit from the metastable phase, we describe the typical exit path and evaluate the exit time. We prove that the phenomenology of metastability is different from the one observed in the case of the serial implementation of the heat bath dynamics. In particular we prove that an intermediate chessboard phase appears during the excursion from the minus metastable phase toward the plus stable phase.     

Effect of correlated noises on Brownian motor

The Brownian motor operating between two correlated Gaussian white noises was investigated. The expressions of the current and the energy conversion efficiency of the Brownian motor were analytically derived by exploiting adiabatic approximation. The results indicates that: (i) Regulating the correlation strength λ between the two noises and the ratio D₂/D₁ of the two noise intensities can change the rotational direction of the motor; (ii) For the smaller D₂/D₁, an optimal λ can make the positive current and the efficiency be maximal, and for the smaller λ, an optimal D₂/D₁ also let the positive current be maximal. The results were explained from a viewpoint of modified potentials. The study is of important significance in the aspect of controlling the operation of the Brownian motor.     

Double-temperature ratchet model and current reversal of coupled Brownian motors

On the basis of the transport features and experimental phenomena observed in studies of molecular motors, we propose a double-temperature ratchet model of coupled motors to reveal the dynamical mechanism of cooperative transport of motors with two heads, where the interactions and asynchrony between two motor heads are taken into account. We investigate the collective unidirectional transport of coupled system and find that the direction of motion can be reversed under certain conditions. Reverse motion can be achieved by modulating the coupling strength, coupling free length, and asymmetric coefficient of the periodic potential, which is understood in terms of the effective potential theory. The dependence of the directed current on various parameters is studied systematically. Directed transport of coupled Brownian motors can be manipulated and optimized by adjusting the pulsation period or the phase shift of the pulsation temperature.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.