By using chiral perturbation theory and the Uehling-Uhlenbeck equation we compute the viscosity of a pion gas, in the low temperature and low density regime, in terms of the temperature and the pion fugacity. The viscosity turns out to be proportional to the square root of the temperature over the pion mass. Next to leading corrections are proportional to the temperature over the pion mass to 3/2.
We consider a model of a quenched disordered geometry in which a random metric is defined on 2 , which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as χ R , with a fluctuation exponent χ ≃ 1 3, while the lateral spread of the minimizing geodesic between two points at a distance L grows as ξ L , with wandering exponent value ξ ≃ 2 3. Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang universality class of surface kinetic roughening, with ξ and χ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.
The discovery of novel entanglement patterns in quantum many-body systems is a prominent research direction in contemporary physics. Here we provide the example of a spin chain with random and inhomogeneous couplings that in the ground state exhibits a very unusual area law violation. In the clean limit, i.e., without disorder, the model is the rainbow chain and has volume law entanglement. We show that, in the presence of disorder, the entanglement entropy exhibits a power-law growth with the subsystem size, with an exponent 1/2. By employing the Strong Disorder Renormalization Group (SDRG) framework, we show that this exponent is related to the survival probability of certain random walks. The ground state of the model exhibits extended regions of short-range singlets (that we term "bubble" regions) as well as rare long range singlet ("rainbow" regions). Crucially, while the probability of extended rainbow regions decays exponentially with their size, that of the bubble regions is power law. We provide strong numerical evidence for the correctness of SDRG results by exploiting the free-fermion solution of the model. Finally, we investigate the role of interactions by considering the random inhomogeneous XXZ spin chain. Within the SDRG framework and in the strong inhomogeneous limit, we show that the above area-law violation takes place only at the free-fermion point of phase diagram. This point divides two extended regions, which exhibit volume-law and area-law entanglement, respectively. arXiv:1807.04179v1 [cond-mat.str-el] A FIG. 1: Setup used in this work. (top) Definition of the random inhomogeneous XX chain.The chain couplings are denoted as Jn = e −|n|h Kn, with n half integer numbers, h a real inhomogeneity parameter, and Kn independent random variables distributed with (4). In this work we focus on the entanglement entropy of a subregion A of length (shaded area in the Figure). Subsystem A starts from the chain center. local, translational invariant models that exhibit area-law violations has been constructed by Movassagh and Shor in Ref. [44]. Their ground-state entanglement entropy is ∝ 1/2 , thus exhibiting a polynomial violation of the area law. Importantly, the exponent of the entanglement growth originates from universal properties of the random walk. This is due to the fact that the ground state of the model is written in terms of a special class of combinatorial objects, called Motzkin paths [51]. A similar result can be obtained [45] using the Fredkin gates [52]. An interesting generalization of Ref. 44 obtained by deforming a colored version of the Motzkin paths has been presented in Ref. 46. The ground-state phase diagram of the model exhibits two phases with area-law and volume-law entanglement, respectively. These are separated by a "special" point, where the ground state displays square-root entanglement scaling.In this paper, we show that unusual area-law violations can be obtained in a one-dimensional inhomogeneous local system in the presence of disorder. Specifically, here we investiga...
The use of Massive Open Online Courses (MOOCs) is increasing worldwide and brings a revolution in education. The application of MOOCs has technological but also pedagogical implications. MOOCs are usually driven by short video lessons, automatic correction exercises, and the technological platforms can implement gamification or learning analytics techniques. However, much more analysis is required about the success or failure of these initiatives in order to know if this new MOOCs paradigm is appropriate for different learning situations. This work aims at analyzing and reporting whether the introduction of MOOCs technology was good or not in a case study with the Khan Academy platform at our university with students in a remedial Physics course in engineering education. Results show that students improved their grades significantly when using MOOCs technology, student satisfaction was high regarding the experience and for most of the different provided features, and there were good levels of interaction with the platform (e.g., number of completed videos or proficient exercises), and also the activity distribution for the different topics and types of activities was appropriate. ß 2016 Wiley Periodicals, Inc. Comput Appl Eng Educ 25:15-25, 2017; View this article online at wileyonlinelibrary.com/ journal/cae;
A model for kinetic roughening of one-dimensional interfaces is presented within an intrinsic geometry framework that is free from the standard small-slope and no-overhang approximations. The model is meant to probe the consequences of the latter on the Kardar-Parisi-Zhang (KPZ) description of non-conserved, irreversible growth. Thus, growth always occurs along the local normal direction to the interface, with a rate that is subject to fluctuations and depends on the local curvature. Adaptive numerical techniques have been designed that are specially suited to the study of fractal morphologies and can support interfaces with large slopes and overhangs. Interface selfintersections are detected, and the ensuing cavities removed. After appropriate generalization of observables such as the global and local surface roughness functions, the interface scaling is seen in our simulations to be of the Family-Vicsek type for arbitrary curvature dependence of the growth rate, KPZ scaling appearing for large sytems sizes and sufficiently large noise amplitudes.
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