Discretization of fractional differential equations by a piecewise constant approximation

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation

Motor properties from persistence: a linear molecular walker lacking spatial and temporal asymmetry

The stepping direction of linear molecular motors is usually defined by a spatial asymmetry of the motor, its track, or both. Here we present a model for a molecular walker that undergoes biased directional motion along a symmetric track in the presence of a temporally symmetric chemical cycle. Instead of using asymmetry, directionality is achieved by persistence. At small load force the walker can take on average thousands of steps in a given direction until it stochastically reverses direction. We discuss a specific experimental implementation of a synthetic motor based on this design and find, using Langevin and Monte Carlo simulations, that a realistic walker can work against load forces on the order of picoNewtons with an efficiency of ~18%, comparable to that of kinesin. In principle, the walker can be turned into a permanent motor by externally monitoring the walker's momentary direction of motion, and using feedback to adjust the direction of a load force. We calculate the thermodynamic cost of using feedback to enhance motor performance in terms of the Shannon entropy, and find that it reduces the efficiency of a realistic motor only marginally. We discuss the implications for natural protein motor performance in the context of the strong performance of this design based only on a thermal ratchet

Dynamical continuous time random walk

© 2015, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg. We consider a continuous time random walk model in which each jump is considered to be dynamical process. Dissipative launch velocity and hopping time in each jump is the key factor in this model. Within the model, normal diffusion and anomalous diffusion is realized theoretically and numerically in the force free potential. Besides, external potential can be introduced naturally, so the random walker’s behavior in the linear potential and quartic potential is discussed, especially the walker with Lévy velocity in the quartic potential, bimodal behavior of the spatial distribution is observed, it is shown that due to the inertial effect induced by damping term, there exists transition from unimodality to bimodality for the walker’s spatial stationary distribution