On numerical stationary distribution of overdamped Langevin equation in harmonic system

Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time.In this work we study the highly accurate numerical algorithm for the overdamped Langevin equation.In particular,our interest is in the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system.Based on the large friction limit of the underdamped Langevin dynamic scheme,three algorithms for overdamped Langevin equation are obtained.We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory for both one-dimensional case and multi-dimensional case.The accuracy of the stationary distribution of each algorithm is illustrated by comparing with the exact Boltzmann distribution.Our results demonstrate that the“BAOA-limit”algorithm generates an accurate distribution of the harmonic system in a canonical ensemble,within a stable range of time interval.The other algorithms do not produce the exact distribution of the harmonic system.

The Equilibrium of Fractional Derivative and Second Derivative: The Mechanics of a Power-Law Visco-Elastic Solid

This paper investigates the equilibrium of fractional derivative and 2nd derivative, which occurs if the original function is damped (damping of a power-law viscoelastic solid with viscosities η of 0 ≤ η ≤ 1), where the fractional derivative corresponds to a force applied to the solid (e.g. an impact force), and the second derivative corresponds to acceleration of the solid’s centre of mass, and therefore to the inertial force. Consequently, the equilibrium satisfies the principle of the force equilibrium. Further-more, the paper provides a new definition of under- and overdamping that is not exclusively disjunctive, i.e. not either under- or over-damped as in a linear Voigt model, but rather exhibits damping phases co-existing consecutively as time progresses, separated not by critical damping, but rather by a transition phase. The three damping phases of a power-law viscoelastic solid—underdamping, transition and overdamping—are characterized by: underdamping—centre of mass oscillation about zero line;transition—centre of mass reciprocation without crossing the zero line;overdamping—power decay. The innovation of this new definition is critical for designing non-linear visco-elastic power-law dampers and fine-tuning the ratio of under- and overdamping, considering that three phases—underdamping, transition, and overdamping—co-exist consecutively if 0 < η < 0.401;two phases—transition and overdamping—co-exist consecutively if 0.401 < η < 0.578;and one phase— overdamping—exists exclusively if 0.578 < η < 1.

Effective transport of passive particles induced by chiral-active particles in microchannel

Transport of passive particles induced by chiral-active particles in microchannel is investigated by using the overdamped Langevin dynamics simulation in a two-dimensional model system. Due to the chirality of active particles and special structure of microchannel, effective ratchet transport of passive particles is achieved. Effective transport of passive particles depends on the width of microchannel（d）, the density（ρ）, and the angular velocity（ω） of chiral-active particles.There exist optimal parameters for d and ω at which the transport efficiency for passive particles takes its maximal value.This investigation can help us understand the necessity of active motion for living systems to maintain a number of vital processes such as materials transport inside cells and the foraging dynamics of mobile organisms.

CONVERGENCE OF THE CYCLIC REDUCTION ALGORITHM FOR A CLASS OF WEAKLY OVERDAMPED QUADRATICS

In this paper, we establish a convergence result of the cyclic reduction （CR） algorithm for a class of weakly overdamped quadratic matrix polynomials without assumption that the partial multiplicities of the nth largest eigenvalue are all equal to 2. Our result can be regarded as a complement of that by Guo, Higham and Tisseur [SIAM J. Matrix Anal. Appl., 30 （2009）, pp. 1593-1613]. The numerical example indicates that the convergence behavior of the CR algorithm is largely dictated by our theory.Mathematics subject classification： 15A24, 15A48.