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Average displacement of a diffusing particle from its starting point is zero at all times. Hence, a population of such particles may only be transported in a particular direction by applying a drift-inducing external potential. However, a potential that exerts a force whose space-average is zero is not expected to result in transport (i.e. a flux whose space-average is non-zero). One such potential is the 'ratchet' potential, named so because its asymmetric sawtooth shape resembles the mechanical rectifying device called a ratchet. Surprisingly, if such a potential is turned on and off periodically or even stochastically, it results in a transport of the particles in the direction of its gentler slope. This 'thermal ratchets model' explains several transport phenomena observed in nature, including the transport of vital materials by molecular walkers along the cytoskeleton inside the living cell. Several papers in the 1990s studied this model and the resultant flux via analytical, computational and experimental approaches. In this thesis we employ a new approach of study, via a numerical solution of the drift-diffusion equation, which offers several advantages and the promise of new insights over the older methods. With this method we have been able to probe the previously unexplored regime of high alternating rate that does not allow adiabatic relaxation. One of the more surprising results is that even in such a case the system settles into periodic dynamics asymptotically.

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... By Eq. (2), the steady state condition c t = 0 enforces uniform flux over space. It may be shown that the flux in fact vanishes everywhere [11], unless there exists an additional external force, in which case there is a uniform flux in its direction. However, switching the potential at finite intervals prevents the attainment of the steady state, but achieves transport even in the absence of any further external force. ...

The thermal ratchets model toggles a spatially periodic asymmetric potential to rectify random walks and achieve transport of diffusing particles. We numerically solve the governing equation for the full dynamics of an infinite 1D ratchet model in response to periodic switching. Transient aperiodic behavior is observed that converges asymptotically to the period of the switching. We study measures of the transport rate, the transient lifetime, and an equivalent of `amplitude', then investigate their dependence on various
properties of the system, along with other features of the transient and asymptotic dynamics.

While many papers in the last few years have dealt with various equations euphemistically called ratchets, the original Feyman two-temperature setup has been left largely unchallenged. We present here a look at the details of how this famous engine actually generates motion from a temperature difference.

Numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons. The first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of Bashforth and Adams for linear multistep methods and Runge for Runge–Kutta methods. Other reasons, which of course apply to numerical analysis in general, are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics. This survey paper follows many of the main strands in the developments of these methods, both for general problems, stiff systems, and for many of the special problem types that have been gaining in significance as the century draws to an end. © 2000 Elsevier Science B.V. All rights reserved.

Aimed at senior undergraduates and graduate students in science and biomedical engineering, this text explores the architecture of a cell's envelope and internal scaffolding, and the properties of its soft components. The book first discusses the properties of individual flexible polymers, networks and membranes, and then considers simple composite assemblages such as bacteria and synthetic cells. The analysis is performed within a consistent theoretical framework, although readers can navigate from the introductory material to results and biological applications without working through the intervening mathematics. This, together with a glossary of terms and appendices providing quick introductions to chemical nomenclature, cell structure, statistical mechanics and elasticity theory, make the text suitable for readers from a variety of subject backgrounds. Further applications and extensions are handled through problem sets at the end of each chapter and supplementary material available on the Internet. Written for students in the burgeoning, new inter-disciplinary field of biological physics Aimed at a multidisciplinary audience; background material from physics and biology is included in four appendices Chapters permit the reader to skip the mathematical derivations without losing the most important results and applications

Designed to teach fundamental ideas as opposed to physics by formula.
The primary goal is to expose basic properties of the atom, focusing on
the description of experiments and data, both historical and current,
used to establish physics principles. Contains 250 carefully worked
single concept problems which demonstrate the thinking behind the answer
and yield numerically significant results. Prerequisites include some
exposure to classical mechanics and electromagnetism.