Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics

A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in the separation coordinate between two possible reactants. This boundary condition can be interpreted as an idealization of a reactive interaction potential model, in which a potential barrier must be surmounted before reactions can occur. In this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation, where molecules react instantly upon reaching the peak of the barrier. The limiting boundary condition is derived by the method of matched asymptotic expansions, and shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased. Limiting boundary conditions for the same interaction potential in both the overdamped Fokker-Planck equation (Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are investigated. It is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the Kramers equation solution converges to the solution of the Fokker-Planck equation.Comment: 23 pages, 2 figure

Free associations mirroring self- and world-related concepts: Implications for personal construct theory, psycholinguistics and philosophical psychology

People construe reality by using words as basic units of meaningful categorization. The present theory-driven study applied the method of a free association task to explore how people express the concepts of the world and the self in words. The respondents were asked to recall any five words relating with the word world. Afterwards they were asked to recall any five words relating with the word self. The method of free association provided the respondents with absolute freedom to choose any words they wanted. Such free recall task is suggested as being a relatively direct approach to the respondents’ self- and world-related conceptual categories, without enormous rational processing. The results provide us, first, with associative ranges for constructs of the world and the self, where some associative dimensions are defined by semantic polarities in the meanings of peripheral categories (e.g., Nature vs. Culture). Second, our analysis showed that some groups of verbal categories that were associated with the words world and self are central, while others are peripheral with respect to the central position. Third, the analysis of category networks revealed that some categories play the role of a transmitter, mediating the pathway between other categories in the network

Mathematical Modelling of Turning Delays in Swarm Robotics

We investigate the effect of turning delays on the behaviour of groups of differential wheeled robots and show that the group-level behaviour can be described by a transport equation with a suitably incorporated delay. The results of our mathematical analysis are supported by numerical simulations and experiments with e-puck robots. The experimental quantity we compare to our revised model is the mean time for robots to find the target area in an unknown environment. The transport equation with delay better predicts the mean time to find the target than the standard transport equation without delay.Comment: Submitted to the IMA Journal of Applied Mathematic

Refining self-propelled particle models for collective behaviour

Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density

Expansion of the human mitochondrial proteome by intra- and inter-compartmental protein duplication

The human mitochondrial proteome is shown to have expanded due to duplication of protein encoding genes and re-localization of these duplicated proteins