On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related Field

Fluctuations in granular gases

A driven granular material, e.g. a vibrated box full of sand, is a stationary system which may be very far from equilibrium. The standard equilibrium statistical mechanics is therefore inadequate to describe fluctuations in such a system. Here we present numerical and analytical results concerning energy and injected power fluctuations. In the first part we explain how the study of the probability density function (pdf) of the fluctuations of total energy is related to the characterization of velocity correlations. Two different regimes are addressed: the gas driven at the boundaries and the homogeneously driven gas. In a granular gas, due to non-Gaussianity of the velocity pdf or lack of homogeneity in hydrodynamics profiles, even in the absence of velocity correlations, the fluctuations of total energy are non-trivial and may lead to erroneous conclusions about the role of correlations. In the second part of the chapter we take into consideration the fluctuations of injected power in driven granular gas models. Recently, real and numerical experiments have been interpreted as evidence that the fluctuations of power injection seem to satisfy the Gallavotti-Cohen Fluctuation Relation. We will discuss an alternative interpretation of such results which invalidates the Gallavotti-Cohen symmetry. Moreover, starting from the Liouville equation and using techniques from large deviation theory, the general validity of a Fluctuation Relation for power injection in driven granular gases is questioned. Finally a functional is defined using the Lebowitz-Spohn approach for Markov processes applied to the linear inelastic Boltzmann equation relevant to describe the motion of a tracer particle. Such a functional results to be different from injected power and to satisfy a Fluctuation Relation.Comment: 40 pages, 18 figure

Why are we here? An investigation of academic, employability and social facets of business undergraduates' motivation using Thurstone Scaling

In the UK employability is a key university performance measure. This reflects both the tightening graduate employment market and the demands on the sector for greater accountability. The literature on employability considers the implications for institutions and the student motivation literature examines students’ intrinsic and extrinsic goal orientations. This exploratory study complements both areas of work by considering employability, currently deemed an all-pervasive extrinsic goal, as far as students’ motivation is concerned relative to the more conventional drivers of decisions to enter higher education; achieving academic success and social fulfilment. It aims to establish both the significance of employability as a motivating factor and ascertain the degree of association with the academic and social factors as well as profile variables. The research design applies Thurstone attitude scaling. Several hundred business undergraduates were asked to encapsulate why they were on their course. The responses were collated and scored by a set of judges against scales of academic, employability and social motivation. The judges’ scores were used to determine the most appropriate statements to use in the research instrument, which was then used to survey the attitudes of 75 students. The results suggest that employability is a significant aspect of students’ motivation and is associated with the academic and social aspects of motivation. This significance of employability suggests effective learning support strategies are likely to be those that are based on experiential and skill-driven learning alongside more tightly drawn cognitive approaches. The balance of motivational aspects can also inform institutions’ student recruitment

The error of the splitting scheme for solving evolutionary equations

AbstractThe accuracy of splitting method is investigated in an abstract Cauchy problem and is shown to be first order in time for general evolutionary equations except for a special case. A general formula for the leading term is obtained. It is also shown as an immediate consequence of the formula that the accuracy is improved from first order to second order by a simple modification. Such a modification was first proposed by Strang [1] for PDEs. Thus, the Strang result is generalized in the present paper to the case of arbitrary evolutionary equations. In particular, it is valid for practically important cases of integro-differential nonlinear kinetic equations, and therefore, there is no need to make additional error estimations in each particular case

Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibriaThe Patterson-Sullivan embedding and minimal volume entropy for outer space

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→ ∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L¹-norm, as well as various Sobolev norms

ON A BOUNDARY-VALUE PROBLEM FOR THE PLANE BROADWELL MODEL - EXACT-SOLUTIONS AND NUMERICAL-SIMULATION

The initial boundary value problem for the Broadwell model equations in a half infinite channel with an infinitely small hole is considered. It is proved that this boundary value problem has no unique solution for sufficiently large concentration of the gas. There are at least two different solutions, we have constructed them in explicit form. The existence of stationary solutions for the corresponding initial boundary value problem is then numerically investigated. The results indicate a unique asymptotic behavior of the model, very close to one of the two predicted stationary solutions