On the derivation of a nonlinear generalized Langevin equation

We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the non linear GLE.

On the shape of invading population in anisotropic environments

We analyze the properties of population spreading in environments with spatial anisotropy within the frames of a lattice model of asymmetric (biased) random walkers. The expressions for the universal shape characteristics of the instantaneous configuration of population, such as asphericity A and prolateness S are found analytically and proved to be dependent only on the asymmetric transition probabilities in different directions. The model under consideration is shown to capture, in particular, the peculiarities of invasion in presence of an array of oriented tubes (fibers) in the environment.

Experimental Compliance Breakdown of Industrial Robots

An engineering tool is described, enabling a static compliance breakdown of a machine (with fixed or varying geometry) from a modal analysis performed at a quasi-static frequency. The most interesting feature of the method is that the compliances of the constitutive machine components can be determined independent from the instantaneous configuration of the machine. Real-life test results on a fixed-configuration beamlike structure and on a variable-configuration industrial robot show the accuracy and the power of the method but also reveal some stringent requirements for the motion sensors needed. Some new results on the need for correction of the influence of gravity on the accelerometer signals at low frequencies are also included.