Brownian motion: From Einstein to Mandelbrot. On analysing diabetics cells

The erythrocytes in flow are continually changing their shape in a synchronized erratic, chaotic or random fashion. Then, there is a natural crossover between the mathematical field of Brownian motion and biology. Eugene Wigner, one of the important physicists of the twenty's century claimed: “that the enormous usefulness of mathematics in natural sciences is something bordering on the mysterious and that there is no rational explanation for it”. In the first part of this talk I will present, avoiding mathematic background and hard definitions, the random motion described by scotch biologist Robert Brown in 1927 and the mathematical description given in his first paper of Brownian motion by Albert Einstein in 1905, then the generalization based on fractionary Brownian motion and fractal self- affinities proposed by Benoit Mandelbrot and Harold Hurst in 1953.In the second part of the talk. I am going to present the application of ordinary and fractionary Brownian motion to characterize two different populations of human being red blood cells: healthy individuals and diabetic patients. The cells deformability is studied using photometric readings (time series) of light intensity variations, along the major axis of the elliptical diffraction pattern. The series were obtained by ektacytrometry, using a home-made device called Rheometer which was developed and constructed in our Laboratory and patented by CONICET. The photometrically recorded series are used to obtain rheological measurements of RBCs subjected to well defined shear stress during which the cells behaviour is like the one in capillaries. The recorded time series obtained have 5100 data points for creep shear stress process, and 5100 data points for relaxation process. By the very beginning the photometric series are studied by shuffle surrogates in order to validate the results. Then mathematical quantifier such as ordinary Brownian motion and fractionary Brownian motion, Correlation Coefficient, and Hurst are the mathematical quantifiers applied. The nonlinear mathematical results obtained can not only characterize but also quantify the different red blood cells populations, by applying ordinary and fractionary Brownian motion  which cannot be analyzed on the linear mathematical field and not at all in the light of contemporary notions of  nonlinear statistic mechanics and Brownian motion.