On dissipation in crackling noise systems

We consider the amount of energy dissipated during individual avalanches at the depinning transition of disordered and athermal elastic systems. Analytical progress is possible in the case of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model for Barkhausen noise, due to an exact mapping between the energy released in an avalanche and the area below a Brownian path until its first zero-crossing. Scaling arguments and examination of an extended mean-field model with internal structure show that dissipation relates to a critical exponent recently found in a study of the rounding of the depinning transition in presence of activated dynamics. A new numerical method to compute the dynamic exponent at depinning in terms of blocked and marginally stable configurations is proposed, and a kind of `dissipative anomaly'- with potentially important consequences for nonequilibrium statistical mechanics- is discussed. We conclude that for depinning systems the size of an avalanche does not constitute by itself a univocal measure of the energy dissipated.Comment: 7 pages, 3 figures, final accepted versio

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.

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.

Performance assessment of Finite Volume methods in transient simulations of hydraulic processes

En esta tesis se presenta el desarrollo de un modelo numérico para simulaciones hidráulicas/hidrológicas. Se considera el acoplamiento de flujos superficiales y subterráneos, poniendo una atención especial a la interacción entre ambos. El flujo superficial está formulado mediante las ecuaciones de aguas poco profundas en 2D, así como el modelo Cero-Inercia en 2D e incluye componentes hidrológicas como lluvia o infiltración. Ambos modelos son discretizados usando un esquema híbrido implícito-explícito en volúmenes finitos. Así mismo, se realiza una comparación exhaustiva entre ambas discretizaciones temporales en términos de precisión y eficiencia en varias aplicaciones tanto sintéticas como reales. La eficiencia del esquema implícito es evaluada en cada caso test con el propósito de su consideración como posible estrategia de aceleración del cálculo para ciertas situaciones. Cuando se consideran simulaciones en las que interviene la transformación de lluvia en escorrentía, tiene especial relevancia la correcta estimación de las pérdidas de agua por infiltración. Con el propósito de mejorar el actual modelo de infiltración deGreen-Ampt, en este trabajo se propone la aplicación de una nueva técnica basada en el cálculo fraccionario, que da lugar a mejoras considerables en los resultados numéricos. Por otro lado, se presentan dos modelos de flujo en el subsuelo: 1) Un modelo 2D de flujo subterráneo en medios porosos basado en la ley de Darcy y la aproximación de Dupuit. El acoplamiento entre la superficie y el subsuelo tiene lugar en la frontera que conecta el nivel freático con la superficie del suelo, lo que da lugar a los procesos de infiltración y exfiltración. 2) Un modelo de drenaje basado en las ecuaciones 1D de aguas poco profundas capaz de simular flujos transitorios en una tubería que pueden llegar a presurizarse localmente. La presurización de la tubería se estima mediante el método de la rendija de Preissmann. En este caso, el acoplamiento con la superficie tiene lugar en localizaciones puntuales como sumideros o alcantarillas, donde se calcula el flujo de intercambio entre modelos. Ambos modelos subsuperficiales son validados de forma independiente mediante varios casos test con solución analítica o datos experimentales. En general, los casos test sintéticos, con solución analítica o datos experimentales presentados en esta tesis ponen de manifiesto la gran aplicabilidad de cada sub-modelo de forma particular y de ambos modelos acoplados. In this thesis, the development of a hydraulic/hydrological numerical simulation model is presented. It considers the coupling of surface-subsurface flows, paying special attention to the interactions among submodels. The surface flow is formulated by means of 2D Shallow Water flow equations and 2D Zero-Inertia flow model and includes hydrological components as rainfall or infiltration. Both models are discretized using a hybrid implicit-explicit finite volume scheme and a full comparison is carried out in terms of accuracy and efficiency in several synthetic and real world applications. The efficiency of the implicit scheme is evaluated in every test case in order to emphasize it as a feasible acceleration technique for certain situations. When rainfall-runoff simulations are considered, a correct estimation of the infiltration water losses is of crucial relevance. With the aim of improving the Green-Ampt infiltration model, a novel technique based on the fractional calculus is combined with the surface flow models, leading to promising improvements in the numerical results. On the other hand, two subsurface submodels are presented in this work: 1) A 2D vertical-averaged groundwater flow model based on Darcy's law and Dupuit approximation. The coupling between surface and groundwater flows takes place in the border connecting the phreatic level with the soil surface, leading to infiltration/exfiltration processes. 2) A drainage model based on 1D Shallow Water equations capable of simulating transient flows in a pipe which can be locally pressurized. The pipe pressurization is estimated by means of the Preissmann slot method. In this case, the coupling with the surface occurs at local points such as manholes, where the exchange surface-sewer flux is calculated. Both subsurface submodels are tested and validated independently with several analytical and experimental cases. Overall, the synthetic, analytical and experimental test cases presented in this thesis point out the good applicability of each submodel and of both coupled models.<br /

Joint probability distributions and fluctuation theorems

We derive various exact results for Markovian systems that spontaneously relax to a non-equilibrium steady-state by using joint probability distributions symmetries of different entropy production decompositions. The analytical approach is applied to diverse problems such as the description of the fluctuations induced by experimental errors, for unveiling symmetries of correlation functions appearing in fluctuation-dissipation relations recently generalised to non-equilibrium steady-states, and also for mapping averages between different trajectory-based dynamical ensembles. Many known fluctuation theorems arise as special instances of our approach, for particular two-fold decompositions of the total entropy production. As a complement, we also briefly review and synthesise the variety of fluctuation theorems applying to stochastic dynamics of both, continuous systems described by a Langevin dynamics and discrete systems obeying a Markov dynamics, emphasising how these results emerge from distinct symmetries of the dynamical entropy of the trajectory followed by the system For Langevin dynamics, we embed the "dual dynamics" with a physical meaning, and for Markov systems we show how the fluctuation theorems translate into symmetries of modified evolution operators.Comment: 39 pages, 1 figure. Minor revision, as suggested by referees. A couple of references and equations added. Acknowledgements slightly modifie