46,677 research outputs found

    Continuous-time statistics and generalized relaxation equations

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    Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.Comment: 12 pages, submitted to EPJ

    Continuous-time statistics and generalized relaxation equations

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    Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic

    Quartic Graphs with Minimum Spectral Gap

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    Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with nn vertices is (1+o(1))3n22π2(1+o(1)) \frac{3n^2}{2\pi^2}. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected kk-regular graph on nn vertices is at least (1+o(1))2kπ23n2(1+o(1))\frac{2k\pi^2}{3n^2}, and the bound is attained for at least one value of kk. We determine the structure of connected quartic graphs on nn vertices with minimum spectral gap which enable us to show that the minimum spectral gap of connected quartic graph on nn vertices is (1+o(1))4π2n2(1+o(1))\frac{4\pi^2}{n^2}. From this result, the Aldous--Fill conjecture follows for k=4k=4.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1907.0373

    A Nanoflare Distribution Generated by Repeated Relaxations Triggered by Kink Instability

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    Context: It is thought likely that vast numbers of nanoflares are responsible for the corona having a temperature of millions of degrees. Current observational technologies lack the resolving power to confirm the nanoflare hypothesis. An alternative approach is to construct a magnetohydrodynamic coronal loop model that has the ability to predict nanoflare energy distributions. Aims: This paper presents the initial results generated by such a model. It predicts heating events with a range of sizes, depending on where the instability threshold for linear kink modes is encountered. The aims are to calculate the distribution of event energies and to investigate whether kink instability can be predicted from a single parameter. Methods: The loop is represented as a straight line-tied cylinder. The twisting caused by random photospheric motions is captured by two parameters, representing the ratio of current density to field strength for specific regions of the loop. Dissipation of the loop's magnetic energy begins during the nonlinear stage of the instability, which develops as a consequence of current sheet reconnection. After flaring, the loop evolves to the state of lowest energy where, in accordance with relaxation theory, the ratio of current to field is constant throughout the loop and helicity is conserved. Results: The results suggest that instability cannot be predicted by any simple twist-derived property reaching a critical value. The model is applied such that the loop undergoes repeated episodes of instability followed by energy-releasing relaxation. Hence, an energy distribution of the nanoflares produced is collated. Conclusions: The final energy distribution features two nanoflare populations that follow different power laws. The power law index for the higher energy population is more than sufficient for coronal heating.Comment: 13 pages, 18 figure
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