The Mathematics and Physics of Diderot. I. On Pendulums and Air Resistance

In this article Denis Diderot's Fifth Memoir of 1748 on the problem of a pendulum damped by air resistance is discussed. Diderot wrote the Memoir in order to clarify an assumption Newton made without further justification in the first pages of the Principia in connection with an experiment to verify the Third Law of Motion using colliding pendulums. To explain the differences between experimental and theoretical values of momentum in the collision experiments he conducted Newton assumed that the bob was retarded by an air resistance $F_R$ proportional to the velocity $v$. By giving Newton's arguments a mathematical scaffolding and recasting his geometrical reasoning in the language of differential calculus, Diderot provides a step-by-step solution guide to the problem and proposes experiments to settle the question about the appropriate form of $F_R$, which for Diderot quadratic in $v$, that is $F_R \sim v^2$. The solution of Diderot is presented in full detail and his results are compared to those obtained from a Lindstedt-Poincare approximation for an oscillator with quadratic damping. It is shown that, up to a prefactor, both coincide. Some results that one can derive from his approach are presented and discussed for the first time. Experimental evidence to support Diderot's or Newton's claims is discussed together with the limitations of their solutions. Some misprints in the original memoir are pointed out.Comment: 31 pages, 8 figures. Submitted to European Physical Journal

Boltzmann and the art of flying

One of the less known facets of Ludwig Boltzmann was that of an advocate of Aviation, one of the most challenging technological problems of his times. Boltzmann followed closely the studies of pioneers like Otto Lilienthal in Berlin, and during a lecture on a prestigious conference he vehemently defended further investments in the area. In this article I discuss his involvement with Aviation, his role in its development and his correspondence with two flight pioneers, Otto Lilienthal e Wilhelm Kress.Comment: 15 pages, no figure

Explosive Ising

We study a two-dimensional kinetic Ising model with Swendsen-Wang dynamics, replacing the usual percolation on top of Ising clusters by explosive percolation. The model exhibits a reversible first-order phase transition with hysteresis. Surprisingly, at the transition flanks the global bond density seems to be equal to the percolation thresholds.Comment: 7 pages, 5 figure

Space Representation of Stochastic Processes with Delay

We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.Comment: 6 pages, 8 figure

Long-range epidemic spreading with immunization

We study the phase transition between survival and extinction in an epidemic process with long-range interactions and immunization. This model can be viewed as the well-known general epidemic process (GEP) in which nearest-neighbor interactions are replaced by Levy flights over distances r which are distributed as P(r) ~ r^(-d-sigma). By extensive numerical simulations we confirm previous field-theoretical results obtained by Janssen et al. [Eur. Phys. J. B7, 137 (1999)].Comment: LaTeX, 14 pages, 4 eps figure