Rosa María Velasco

Entropy Production: Its Role in Non-Equilibrium Thermodynamics

It is unquestionable that the concept of entropy has played an essential role both in the physical and biological sciences. However, the entropy production, crucial to the second law, has also other features not clearly conceived. We all know that the main difficulty is concerned with its quantification in non-equilibrium processes and consequently its value for some specific cases is limited. In this work we will review the ideas behind the entropy production concept and we will give some insights about its relevance.

Work-fluctuation theorems for a particle in an electromagnetic field

The theoretical study about the transient and stationary fluctuation theorems is extended to include the effects of electromagnetic fields on a charged Brownian particle. In particular, we consider a harmonic trapped Brownian particle under the action of a constant magnetic field pointing perpendicular to a plane and a time-dependent electric field acting on this plane. The electric field is seen to be responsible for the motion of the center in the harmonic trap, giving as a result a time-dependent dragging. Our study is focused on the solution of the Smoluchowski equation associated with the over-damped Langevin equation and also considers two particular cases for the motion of the harmonic trap minimum. The first one is produced by a linear time-dependent electric field and, in the second case an oscillating electric field produces a circular motion. In this last case we have found resonant behavior in the mean work when the electric field is tuned with Larmors frequency. Some comparisons are made with other works in the absence of the magnetic field.

Kerner's free-synchronized phase transition in a macroscopic traffic flow model with two classes of drivers

The development of synchronized flow has been a challenge in traffic flow studies, and so far the approaches based on macroscopic models have yet to provide a good level of understanding. In this work we present a continuum model which considers two vehicle classes, based on kinetic traffic flow theory for aggressive drivers. The numerical solutions show that the transition F ? S occurs under specific conditions, qualitatively in agreement with Kerners three-phase theory.

An improved second-order continuum traffic model

We construct a second-order continuum traffic model by using an iterative procedure in order to derive a constitutive relation for the traffic pressure which is similar to the Navier–Stokes equation for ordinary fluids. Our second-order traffic model represents an improvement on the traffic model suggested by Kerner and Konhauser since the iterative procedure introduces, in the constitutive relation for the traffic pressure, a density-dependent viscosity coefficient. By using a finite-difference scheme based on the Steger–Warming flux splitting, we investigate the solution of our improved second-order traffic model for specific problems like shock fronts in traffic and freeway-lane drop.

Inconsistency in the Moment's method for solving the Boltzmann equation

Abstract Moment’s methods devised to solve the Boltzmann kinetic equation for a simple, inert, dilute gas exhibit inconsistencies. These are brought up in a general way for the ordinary Boltzmann equation (BE), the Bhatnagar-Gross-Krook (BGK) model, and the Mott-Smith ansatz. Although also present in the Chapman-Enskog method the nature of its perturbation expansion in terms of Knudsen’s number takes care of the problem in a natural way. We show that indeed another step based on this idea is required to arrive at a closure condition in the moment’s solution. Also, a general proof is offered showing why these inconsistencies appear when appealing to moment expansions.

The Entropy Production Distribution in Non-Markovian Thermal Baths

In this work we study the distribution function for the total entropy production of a Brownian particle embedded in a non-Markovian thermal bath. The problem is studied in the overdamped approximation of the generalized Langevin equation, which accounts for a friction memory kernel characteristic of a Gaussian colored noise. The problem is studied in two physical situations: (i) when the particle in the harmonic trap is subjected to an arbitrary time-dependent driving force; and (ii) when the minimum of the harmonic trap is arbitrarily dragged out of equilibrium by an external force. By assuming a natural non Markovian canonical distribution for the initial conditions, the distribution function for the total entropy production becomes a non Gaussian one. Its characterization is then given through the first three cumulants.

TRAVELING WAVES, CATASTROPHES AND BIFURCATIONS IN A GENERIC SECOND ORDER TRAFFIC FLOW MODEL

We consider the macroscopic, second order model of Kerner–Konhauser for traffic flow given by a system of PDE. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We prove that under generic conditions on the so-called fundamental diagram, the surface of critical points has a fold or cusp catastrophe and each fold point gives rise to a Takens–Bogdanov bifurcation. In particular, limit cycles arising from a Hopf bifurcation give place to traveling wave solutions of the PDE.

Clusters in the helbing's improved model

The formation of clusters in Helbings improved model is studied by an iterative method. It is shown that after certain density we will always obtain a density profile which has the structure of a soliton. Its characteristics such as the amplitude and width are determined by the parameters in the model.

A Multi-class Vehicular Flow Model for Aggressive Drivers

The kinetic theory approaches to vehicular traffic modelling have given very good results in the understanding of the dynamical phenomena involved [3, 8]. In this work, we deal with the kinetic approach modelling of a traffic situation where there are many classes of aggressive drivers [5]. Their aggressiveness is characterised through their relaxation times. The reduced Paveri-Fontana equation is taken as a starting point to set the model. It contains the usual drift terms and the interactions between drivers of the same class, as well as the corresponding one between different classes. The reference traffic state used in the kinetic treatment is determined by a dimensionless parameter. The balance equations for the density and average speed for each class are obtained through the usual methods in the kinetic theory. In this model, we consider that each class of drivers preserve the corresponding aggressiveness, in such a way that there will be no adaptation effects [6]. It means that the number of drivers in a class is conserved. As preliminary results, we have obtained a closure relation to derive the Euler-like equations for two drivers classes. Some characteristics of the model are explored with the usual methods.

The Stability Analysis of a Macroscopic Traffic Flow Model with Two-Classes of Drivers

One of the most important objectives in the development of traffic theories is the improvement of traffic conditions. To achieve this goal, it is important a good understanding of multistyle and/or multilane traffic. In this work, we summarize the traffic model presented in Mendez and Velasco (FTC J Phys A Math Theor 46(46):462001, 2013) and additionally include the stability analysis of the same. The presented traffic model considers different driving styles, different vehicle types or both, for a two-classes of vehicles in which a model for the average desired speed is introduced (the aggressive drivers model) (Mendez and Velasco, Transp Res Part B 42:782–797, 2008; Velasco and Marques, Phys Rev E 72:046102, 2005). The kinetic model was solved for the steady and homogeneous state and also we obtained the local distribution function from an information entropy maximization procedure. The macroscopic traffic model is constructed by the usual methods in kinetic theory and a method akin with the Maxwellian iterative procedure is accomplished in order to close the macroscopic model for the mixture, where only the densities are present as relevant quantities. The linear stability analysis is carried out in order to have an insight of the unstable traffic regions of the model, which is very helpful in the numerical solution.