Dynamics of a single swimmer

Locomotion by swimming is a crucial aspect to optimize survival strategies of microorganisms such as bacteria, unicellular protozoa, or sperms. Artificial swimmers on the other hand are also expected to play an important role in the nanotechnology of the 21st century, for example, medical treatment will be revolutionized once nanomachines are available that could be used for targeted therapies such as controlled drug delivery. Nano-robots could be employed for environmental cleaning of polluted water, recycling, and processing of waste. Similarly microgears could be used to manipulate and to control flow on sub-micron scales, thus they could be the essential component for the lab-on-a-chip, a nanofluidic devices controlling chemical processing, catalysis, or molecular sieving, similar to the success story of the last five decades of the miniaturization of electronic devices. To gain insight how such swimming agents for future applications in nano-technology could be realized we study a drastically simplified problem. We focus on a single swimmer and use an effective description in terms of stochastic differential equations referred to as Langevin equations in the physics community. Hence the model leaves out the question how the swimming motion originates in the first place, e.g. by flagella beating, shape deformations of the body, chemical gradients close to the surface, light-driven propulsion by external illuminating lasers, or other mechanisms. Furthermore the induced flow field of the surrounding fluid is also ignored which implies that all hydrodynamics is discarded. Yet the model is expected to hold generically on long time and length scales. Albeit the formulation of the model is straightforward and it has been used in numerous publications, a solution beyond the simplest measurable quantity, i.e. the mean-square displacement, has not been achieved. This project fills this gap and aims at an analytical solution for the intermediate scattering function, which is directly measurable, e.g. in a light-scattering experiment. Since it can also be viewed as the characteristic function of the displacements, all moments of the random displacement variable are encoded. Of particular interest is, besides the already known mean-square displacement, the next higher nontrivial moment, the fourth moment of the displacement, or equivalently the non-gaussian parameter. The calculation exploits a mathematical analogy between the motion of a quantum pendulum and the dynamics of the swimmer. Although the analogy is purely mathematical and formal, and not perfect, the developed tools from quantum mechanics can be applied and adjusted to the current problem. The approach to map the problem to a solved problem permits to derive an exact analytical solution which is a rare case in the field of soft matter/biological physics.

Dynamics of a single swimmer Locomotion by swimming is a crucial aspect to optimize survival strategies of microorganisms such as bacteria, unicellular protozoa, or sperms. Artificial swimmers on the other hand are also expected to play an important role in the nanotechnology of the 21st century, for example, medical treatment will be revolutionized once nanomachines are available that could be used for targeted therapies such as controlled drug delivery. Nano-robots could be employed for environmental cleansing of polluted water, recycling, and processing of waste. Similarly, microgears could be used to manipulate and to control flow at sub-micron scales, thus they could be the essential component for the lab-on-a-chip, a nanofluidic device controlling chemical processing, catalysis, or molecular sieving, similar to the success story of the last five decades of the miniaturization of electronic devices. Within the project the paradigmatic model of the active Brownian particle has been investigated and analytic solutions for the dynamics elaborated. In particular, the complete spatio-temporal dynamics has been characterized beyond simple indicators such as the mean-square displacement. The central quantity of interest is the intermediate scattering function which is directly experimentally accessible in light-scattering experiment on suspensions of active particles. Besides the verification by computer simulations, the theoretical predictions have been verified by ingenious experiments by our collaboration at the University of Edingburgh. The theoretical and experimental progress emphasizes impressively the relevance of the active Brownian particle model as paradigm for non-equilibrium phenomena in statistical physics.