Focal Varieties of Curves of Genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.Comment: This paper consists of 9 page

Work fluctuation theorems for harmonic oscillators

The work fluctuations of an oscillator in contact with a thermostat and driven out of equilibrium by an external force are studied experimentally and theoretically within the context of Fluctuation Theorems (FTs). The oscillator dynamics is modeled by a second order Langevin equation. Both the transient and stationary state fluctuation theorems hold and the finite time corrections are very different from those of a first order Langevin equation. The periodic forcing of the oscillator is also studied; it presents new and unexpected short time convergences. Analytical expressions are given in all cases

Generalized fluctuation relation and effective temperatures in a driven fluid

By numerical simulation of a Lennard-Jones like liquid driven by a velocity gradient \gamma we test the fluctuation relation (FR) below the (numerical) glass transition temperature T_g. We show that, in this region, the FR deserves to be generalized introducing a numerical factor X(T,\gamma)<1 that defines an ``effective temperature'' T_{FR}=T/X. On the same system we also measure the effective temperature T_{eff}, as defined from the generalized fluctuation-dissipation relation, and find a qualitative agreement between the two different nonequilibrium temperatures.Comment: Version accepted for publication on Phys.Rev.E; major changes, 1 figure adde

Fluctuations and response in a non-equilibrium micron-sized system

The linear response of non-equilibrium systems with Markovian dynamics satisfies a generalized fluctuation-dissipation relation derived from time symmetry and antisymmetry properties of the fluctuations. The relation involves the sum of two correlation functions of the observable of interest: one with the entropy excess and the second with the excess of dynamical activity with respect to the unperturbed process, without recourse to anything but the dynamics of the system. We illustrate this approach in the experimental determination of the linear response of the potential energy of a Brownian particle in a toroidal optical trap. The overdamped particle motion is effectively confined to a circle, undergoing a periodic potential and driven out of equilibrium by a non-conservative force. Independent direct and indirect measurements of the linear response around a non-equilibrium steady state are performed in this simple experimental system. The same ideas are applicable to the measurement of the response of more general non-equilibrium micron-sized systems immersed in Newtonian fluids either in stationary or non-stationary states and possibly including inertial degrees of freedom.Comment: 12 pages, submitted to J. Stat. Mech., revised versio

Failure time in the fiber-bundle model with thermal noise and disorder

The average time for the onset of macroscopic fractures is analytically and numerically investigated in the fiber-bundle model with quenched disorder and thermal noise under a constant load. We find an implicit exact expression for the failure time in the low-temperature limit that is accurately confirmed by direct simulations. The effect of the disorder is to lower the energy barrier.Comment: 11 pages, 6 figures; accepted for publication in Phys. Rev.

Extended Heat-Fluctuation Theorems for a System with Deterministic and Stochastic Forces

Heat fluctuations over a time \tau in a non-equilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all \tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small \tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite \tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.Comment: 23 pages, 6 figures. Figures are now in color, Eq. (67) was corrected and a footnote was added on the d-dimensional cas

Failure time and microcrack nucleation

The failure time of samples of heterogeneous materials (wood, fiberglass) is studied as a function of the applied stress. It is shown that in these materials the failure time is predicted with a good accuracy by a model of microcrack nucleation proposed by Pomeau. It is also shown that the crack growth process presents critical features when the failure time is approached.Comment: 13 pages, 4 figures, submitted to Europhysics Letter

Probability density functions of work and heat near the stochastic resonance of a colloidal particle

We study experimentally and theoretically the probability density functions of the injected and dissipated energy in a system of a colloidal particle trapped in a double well potential periodically modulated by an external perturbation. The work done by the external force and the dissipated energy are measured close to the stochastic resonance where the injected power is maximum. We show a good agreement between the probability density functions exactly computed from a Langevin dynamics and the measured ones. The probability density function of the work done on the particle satisfies the fluctuation theorem

Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f\in \C[x_0,...,x_n] of degree $d$, defined by \phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n; f(x)\neq 0\} is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of $\phi_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.Comment: 8 page

Thermodynamic time asymmetry in nonequilibrium fluctuations

We here present the complete analysis of experiments on driven Brownian motion and electric noise in a $RC$ circuit, showing that thermodynamic entropy production can be related to the breaking of time-reversal symmetry in the statistical description of these nonequilibrium systems. The symmetry breaking can be expressed in terms of dynamical entropies per unit time, one for the forward process and the other for the time-reversed process. These entropies per unit time characterize dynamical randomness, i.e., temporal disorder, in time series of the nonequilibrium fluctuations. Their difference gives the well-known thermodynamic entropy production, which thus finds its origin in the time asymmetry of dynamical randomness, alias temporal disorder, in systems driven out of equilibrium.Comment: to be published in : Journal of Statistical Mechanics: theory and experimen