Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models

Macroscopic models of superconductivity

After giving a description of the basic physical phenomena to be modelled, we begin by formulating a sharp-interface free-boundary model for the destruction of superconductivity by an applied magnetic field, under isothermal and anisothermal conditions, which takes the form of a vectorial Stefan model similar to the classical scalar Stefan model of solid/liquid phase transitions and identical in certain two-dimensional situations. This model is found sometimes to have instabilities similar to those of the classical Stefan model. We then describe the Ginzburg-Landau theory of superconductivity, in which the sharp interface is `smoothed out' by the introduction of an order parameter, representing the number density of superconducting electrons. By performing a formal asymptotic analysis of this model as various parameters in it tend to zero we find that the leading order solution does indeed satisfy the vectorial Stefan model. However, at the next order we find the emergence of terms analogous to those of `surface tension' and `kinetic undercooling' in the scalar Stefan model. Moreover, the `surface energy' of a normal/superconducting interface is found to take both positive and negative values, defining Type I and Type II superconductors respectively. We discuss the response of superconductors to external influences by considering the nucleation of superconductivity with decreasing magnetic field and with decreasing temperature respectively, and find there to be a pitchfork bifurcation to a superconducting state which is subcritical for Type I superconductors and supercritical for Type II superconductors. We also examine the effects of boundaries on the nucleation field, and describe in more detail the nature of the superconducting solution in Type II superconductors - the so-called `mixed state'. Finally, we present some open questions concerning both the modelling and analysis of superconductors

How can macroscopic models reveal self-organization in traffic flow?

In this paper we propose a new modeling technique for vehicular traffic flow, designed for capturing at a macroscopic level some effects, due to the microscopic granularity of the flow of cars, which would be lost with a purely continuous approach. The starting point is a multiscale method for pedestrian modeling, recently introduced in Cristiani et al., Multiscale Model. Simul., 2011, in which measure-theoretic tools are used to manage the microscopic and the macroscopic scales under a unique framework. In the resulting coupled model the two scales coexist and share information, in the sense that the same system is simultaneously described from both a discrete (microscopic) and a continuous (macroscopic) perspective. This way it is possible to perform numerical simulations in which the single trajectories and the average density of the moving agents affect each other. Such a method is here revisited in order to deal with multi-population traffic flow on networks. For illustrative purposes, we focus on the simple case of the intersection of two roads. By exploiting one of the main features of the multiscale method, namely its dimension-independence, we treat one-dimensional roads and two-dimensional junctions in a natural way, without referring to classical network theory. Furthermore, thanks to the coupling between the microscopic and the macroscopic scales, we model the continuous flow of cars without losing the right amount of granularity, which characterizes the real physical system and triggers self-organization effects, such as, for example, the oscillatory patterns visible at jammed uncontrolled crossroads.Comment: 7 pages, 7 figure

Macroscopic equations governing noisy spiking neuronal populations

At functional scales, cortical behavior results from the complex interplay of a large number of excitable cells operating in noisy environments. Such systems resist to mathematical analysis, and computational neurosciences have largely relied on heuristic partial (and partially justified) macroscopic models, which successfully reproduced a number of relevant phenomena. The relationship between these macroscopic models and the spiking noisy dynamics of the underlying cells has since then been a great endeavor. Based on recent mean-field reductions for such spiking neurons, we present here {a principled reduction of large biologically plausible neuronal networks to firing-rate models, providing a rigorous} relationship between the macroscopic activity of populations of spiking neurons and popular macroscopic models, under a few assumptions (mainly linearity of the synapses). {The reduced model we derive consists of simple, low-dimensional ordinary differential equations with} parameters and {nonlinearities derived from} the underlying properties of the cells, and in particular the noise level. {These simple reduced models are shown to reproduce accurately the dynamics of large networks in numerical simulations}. Appropriate parameters and functions are made available {online} for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models

Giant electrocaloric effect in thin film Pb Zr_0.95 Ti_0.05 O_3

An applied electric field can reversibly change the temperature of an electrocaloric material under adiabatic conditions, and the effect is strongest near phase transitions. This phenomenon has been largely ignored because only small effects (0.003 K V^-1) have been seen in bulk samples such as Pb0.99Nb0.02(Zr0.75Sn0.20Ti0.05)0.98O3 and there is no consensus on macroscopic models. Here we demonstrate a giant electrocaloric effect (0.48 K V^-1) in 300 nm sol-gel PbZr0.95Ti0.05O3 films near the ferroelectric Curie temperature of 222oC. We also discuss a solid state device concept for electrical refrigeration that has the capacity to outperform Peltier or magnetocaloric coolers. Our results resolve the controversy surrounding macroscopic models of the electrocaloric effect and may inspire ab initio calculations of electrocaloric parameters and thus a targeted search for new materials.Comment: 5 pages, 4 figure

Significance of the microfluidic concepts for the improvement of macroscopic models of transport phenomena

This paper was presented at the 3rd Micro and Nano Flows Conference (MNF2011), which was held at the Makedonia Palace Hotel, Thessaloniki in Greece. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, Aristotle University of Thessaloniki, University of Thessaly, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute.Complexity of transport phenomena - ranging from macroscopic motion of matter, heat transfer, over to the molecular motions determining the overall flow properties of fluids, or generally aggregation states of matter – inhibited development of a single mathematical model describing simultaneously transport processes at all relevant scales. In classical engineering sciences at each scale level we have different equations, different fundamental variables and different methods of solution [4]. The established basis of the classical fluid dynamics - the Navier-Stokes equations [1, 3] - have apparently nothing in common with molecular physics. At the macroscopic scale of motion the molecular structure of matter and the microscopic molecular motions are ignored (even though they determine the local macroscopic behaviour) [1, 3, 4]. To describe multiphase flows, still other methods must be used – increasing further the number of equations, methods of solution etc. The serious disadvantage of this approach is, that equations describing macroscopic models (Navier-Stokes and there from derived equations), introduce multiple theoretical problems: - higher order continuity requirements [3]; - numerous paradoxes in simple macroscopic flows (Bernoulli eq.); - different equations with different fundamental variables and different methods of solution, build a set of disciplines devoted in principle to a single problem – dynamics of disperse systems