Memory effects for a trapped Brownian particle in viscoelastic shear flows

Simulating structured fluids with tensorial viscoelasticity

We consider an immersed elastic body that is actively driven through a structured fluid by a motor or an external force. The behavior of such a system generally cannot be solved analytically, necessitating the use of numerical methods. However, current numerical methods omit important details of the microscopic structure and dynamics of the fluid, which can modulate the magnitudes and directions of viscoelastic restoring forces. To address this issue, we develop a simulation platform for modeling viscoelastic media with tensorial elasticity. We build on the lattice Boltzmann algorithm and incorporate viscoelastic forces, elastic immersed objects, a microscopic orientation field, and coupling between viscoelasticity and the orientation field. We demonstrate our method by characterizing how the viscoelastic restoring force on a driven immersed object depends on various key parameters as well as the tensorial character of the elastic response. We find that the restoring force depends non-monotonically on the rate of diffusion of the stress and the size of the object. We further show how the restoring force depends on the relative orientation of the microscopic structure and the pulling direction. These results imply that accounting for previously neglected physical features, such as stress diffusion and the microscopic orientation field, can improve the realism of viscoelastic simulations. We discuss possible applications and extensions to the method.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model

The behavior of a stochastic resonate-and-fire neuron model based on a reduction of a fractional noise-driven generalized Langevin equation (GLE) with a power-law memory kernel is considered. The effect of temporally correlated random activity of synaptic inputs, which arise from other neurons forming local and distant networks, is modeled as an additive fractional Gaussian noise in the GLE. Using a first-passage-time formulation, in certain system parameter domains exact expressions for the output interspike interval (ISI) density and for the survival probability (the probability that a spike is not generated) are derived and their dependence on input parameters, especially on the memory exponent, is analyzed. In the case of external white noise, it is shown that at intermediate values of the memory exponent the survival probability is significantly enhanced in comparison with the cases of strong and weak memory, which causes a resonancelike suppression of the probability of spike generation as a function of the memory exponent. Moreover, an examination of the dependence of multimodality in the ISI distribution on input parameters shows that there exists a critical memory exponent α_{c}≈0.402, which marks a dynamical transition in the behavior of the system. That phenomenon is illustrated by a phase diagram describing the emergence of three qualitatively different structures of the ISI distribution. Similarities and differences between the behavior of the model at internal and external noises are also discussed.

Memory effects for a stochastic fractional oscillator in a magnetic field

The problem of random motion of harmonically trapped charged particles in a constant external magnetic field is studied. A generalized three-dimensional Langevin equation with a power-law memory kernel is used to model the interaction of Brownian particles with the complex structure of viscoelastic media (e.g., dusty plasmas). The influence of a fluctuating environment is modeled by an additive fractional Gaussian noise. In the long-time limit the exact expressions of the first-order and second-order moments of the fluctuating position for the Brownian particle subjected to an external periodic force in the plane perpendicular to the magnetic field have been calculated. Also, the particle's angular momentum is found. It is shown that an interplay of external periodic forcing, memory, and colored noise can generate a variety of cooperation effects, such as memory-induced sign reversals of the angular momentum, multiresonance versus Larmor frequency, and memory-induced particle confinement in the absence of an external trapping field. Particularly in the case without external trapping, if the memory exponent is lower than a critical value, we find a resonancelike behavior of the anisotropy in the particle position distribution versus the driving frequency, implying that it can be efficiently excited by an oscillating electric field. Similarities and differences between the behaviors of the models with internal and external noises are also discussed.

Two-dimensional dynamics of a trapped active Brownian particle in a shear flow

We model the two-dimensional dynamics of a pointlike artificial microswimmer diffusing in a harmonic trap subject to the shear flow of a highly viscous medium. The particle is driven simultaneously by the linear restoring force of the trap, the drag force exerted by the flow, and the torque due to the shear gradient. For a Couette flow, elliptical orbits in the noiseless regime, and the correlation functions between the particle's displacements parallel and orthogonal to the flow are computed analytically. The effects of thermal fluctuations (translational) and self-propulsion fluctuations (angular) are treated separately. Finally, we discuss how to extend our approach to the diffusion of a microswimmer in a Poiseuille flow. These results provide an accurate reference solution to investigate, both numerically and experimentally, hydrodynamics corrections to the diffusion of active matter in confined geometries.

Response to a periodic stimulus in a perfect integrate-and-fire neuron model driven by colored noise

The output interspike interval statistics of a stochastic perfect integrate-and-fire neuron model driven by an additive exogenous periodic stimulus is considered. The effect of temporally correlated random activity of synaptic inputs is modeled by an additive symmetric dichotomous noise. Using a first-passage-time formulation, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived in the nonsteady regime, and their dependence on input parameters (e.g., the noise correlation time and amplitude as well as the frequency of an input current) is analyzed. It is shown that an interplay of a periodic forcing and colored noise can cause a variety of nonequilibrium cooperation effects, such as sign reversals of the interspike interval correlations versus noise-switching rate as well as versus the frequency of periodic forcing, a power-law-like decay of oscillations of the serial correlation coefficients in the long-lag limit, amplification of the output signal modulation in the instantaneous firing rate of the neural response, etc. The features of spike statistics in the limits of slow and fast noises are also discussed.

Collective behavior of globally coupled Langevin equations with colored noise in the presence of stochastic resonance

The long-time collective behavior of globally coupled Langevin equations in a dichotomous fluctuating potential driven by a periodic source is investigated. By describing the collective behavior using the moments of the mean field and single-particle displacements, we study stochastic resonance and synchronization using the exact steady-state solutions and related stability criteria. Based on the simulation results and the criterion of the stationary regime, the notable differences between the stationary and nonstationary regimes are demonstrated. For the stationary regime, stochastic resonance with synchronization is discussed, and for the nonstationary regime, the volatility clustering phenomenon is observed.

Statistics of a leaky integrate-and-fire model of neurons driven by dichotomous noise

The behavior of a stochastic leaky integrate-and-fire model of neurons is considered. The effect of temporally correlated random neuronal input is modeled as a colored two-level (dichotomous) Markovian noise. Relying on the Riemann method, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived, and their dependence on noise parameters (such as correlation time and amplitude) is analyzed. Particularly, noise-induced sign reversal and a resonancelike amplification of the kurtosis of the interspike interval distribution are established. The features of spike statistics, analytically revealed in our study, are compared with recently obtained results for a perfect integrate-and-fire neuron model.

Cage effect for the velocity correlation functions of a Brownian particle in viscoelastic shear flows

The long-time limit behavior of velocity correlation functions (VCFs) for an underdamped Brownian particle in an oscillatory viscoelastic shear flow is investigated using the generalized Langevin equation with a power-law memory kernel. The influence of a fluctuating environment is modeled by an additive external fractional Gaussian noise. The exact expressions of the correlation functions of the fluctuating components of velocity for the Brownian particle in the shear plane have been calculated. Also, the particle's angular momentum is found. It is shown that in a certain region of the system parameters an interplay of the shear flow, memory effects, and external noise can induce a bounded long-time behavior of the VCFs, even in the shear flow direction, where in the case of internal noise the velocity process is subdiffusive, i.e., unbounded in time. Moreover, we find resonant behavior of the VCFs and the angular momentum versus the shear oscillation frequency, implying that they can be efficiently excited by oscillatory shear. The role of the initial positional distribution of particles is also discussed.

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: subdiffusion of heavy tracers

We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.