Symmetry for the duration of entropy-consuming intervals

We introduce the violation fraction $\upsilon$ as the cumulative fraction of time that a mesoscopic system spends consuming entropy at a single trajectory in phase space. We show that the fluctuations of this quantity are described in terms of a symmetry relation reminiscent of fluctuation theorems, which involve a function, $\Phi$, which can be interpreted as an entropy associated to the fluctuations of the violation fraction. The function $\Phi$, when evaluated for arbitrary stochastic realizations of the violation fraction, is odd upon the symmetry transformations which are relevant for the associated stochastic entropy production. This fact leads to a detailed fluctuation theorem for the probability density function of $\Phi$. We study the steady-state limit of this symmetry in the paradigmatic case of a colloidal particle dragged by optical tweezers through an aqueous solution. Finally, we briefly discuss on possible applications of our results for the estimation of free-energy differences from single molecule experiments.Comment: 11 pages, 4 figures. Last revised. Version accepted for publication in Phys. Rev.

Duration of local violations of the second law of thermodynamics along single trajectories in phase space

We define the {\it violation fraction} $\nu$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $\tau$. In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. The exponent $\alpha$ is positive and generally depends on the protocol for the external driving forces, being $\alpha=1$ for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of $\langle \nu \rangle$ with $N$ and $\tau$ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.Comment: 8 pages, 2 figures, Final version, as accepted for publication in Phys. Rev.

Nonequilibrium structures and dynamic transitions in driven vortex lattices with disorder

We review our studies of elastic lattices driven by an external force $F$ in the presence of random disorder, which correspond to the case of vortices in superconducting thin films driven by external currents. Above a critical force $F_c$ we find two dynamical phase transitions at $F_p$ and $F_t$, with $F_c<F_p<F_t$. At $F_p$ there is a transition from plastic flow to smectic flow where the noise is isotropic and there is a peak in the differential resistance. At $F_t$ there is a sharp transition to a frozen transverse solid where both the transverse noise and the diffussion fall down abruptly and therefore the vortex motion is localized in the transverse direction. From a generalized fluctuation-dissipation relation we calculate an effective transverse temperature in the fluid moving phases. We find that the effective temperature decreases with increasing driving force and becomes equal to the equilibrium melting temperature when the dynamic transverse freezing occurs.Comment: 8 pages, 3 fig

Intrinsic leakage of the Josephson flux qubit and breakdown of the two-level approximation for strong driving

Solid state devices for quantum bit computation (qubits) are not perfect isolated two-level systems, since additional higher energy levels always exist. One example is the Josephson flux qubit, which consists on a mesoscopic SQUID loop with three Josephson junctions operated at or near a magnetic flux of half quantum. We study intrinsic leakage effects, i.e., direct transitions from the allowed qubit states to higher excited states of the system during the application of pulses for quantum computation operations. The system is started in the ground state and rf- magnetic field pulses are applied at the qubit resonant frequency with pulse intensity $f_p$. A perturbative calculation of the average leakage for small $f_p$ is performed for this case, obtaining that the leakage is quadratic in $f_p$, and that it depends mainly on the matrix elements of the supercurrent. Numerical simulations of the time dependent Schr\"odinger equation corresponding to the full Hamiltonian of this device were also performed. From the simulations we obtain the value of $f_p$ above which the two-level approximation breaks down, and we estimate the maximum Rabi frequency that can be achieved. We study the leakage as a function of the ratio $\alpha$ among the Josephson energies of the junctions of the device, obtaining the best value for minimum leakage ($\alpha\approx0.85$). The effects of flux noise on the leakage are also discussed.Comment: Final improved version. Some figures have changed with new results added. To be published in Phys. Rev.

Vortex dynamics in disordered Josephson junction arrays: from plastic flow to flux flow

We study the dynamics of Josephson junction arrays with positional disorder and driven by an external current. We consider weak magnetic fields, corresponding to a frustration $f=n+1/25$ with $n$ integer. We find that above the critical current $i_c$ there is a plastic flow of vortices, where most of the vortices are pinned and only a few vortices flow through channels. This dynamical regime is characterized by strong fluctuations of the total vorticity. The number of the flow channels grow with increasing bias current. At larger currents there is a dynamical regime characterized by the homogeneous motion of all the vortices, i.e. a flux flow regime. We find a dynamical phase transition between the plastic flow and the flux flow regimes when analyzing voltage-voltage correlation functions.Comment: 9 pages. 3 Figures available upon request. Presented in the Workshop on Josephson Junction Arrays, ICTP (August 1995). To appear in Physica B (1996

Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data