In order to identify the basic conditions for thermal rectification we investigate a simple model with non-uniform, graded mass distribution. The existence of thermal rectification is theoretically predicted and numerically confirmed, suggesting that thermal rectification is a typical occurrence in graded systems, which are likely to be natural candidates for the actual fabrication of thermal diodes. In view of practical implications, the dependence of rectification on the asymmetry and system's size is studied. The study of the underlying dynamical mechanisms which determine the macroscopic properties of heat conduction has opened the fascinating possibility to control the heat current. In particular a model of a thermal rectifier has been proposed [1] and since then, the phenomenon of thermal rectification has been intensively investigated [2][3][4][5][6][7][8][9] in order to analyze and improve the rectification effect, including experimental realizations [10].However, as correctly pointed out in Ref.[3], most recurrent proposals of a thermal diode, based on the sequential coupling of two or three segments with different anharmonic potentials, are difficult to be experimentally implemented and the rectification power typically decays to zero with increasing the system size. In addition, most investigations so far have been based on numerical simulations and a much better theoretical understanding is highly desirable both for fundamental reasons as well as for obtaining useful hints for the actual realization of devices with satisfactory rectification power.Along these lines, graded materials are attracting more and more interest: Papers with numerical [5][6][7], analytical [8,9] and even experimental [10] studies have appeared recently in the literature.The present paper addresses the fundamental dynamical mechanisms that lead to rectification. Our strategy is to consider a simple model that contains the minimal ingredients we theoretically judge to be necessary to rectification, and compare the numerical results with the theoretical predictions. As the features of our model are also shared by more realistic models such as anharmonic chains of oscillators, we conjecture that the obtained results may have practical implications as well. Our study allows us to understand the basic ingredients behind rectification, to describe non-trivial and important properties of the heat flow, and to show that rectification in graded materials could be a ubiquitous phenomenon.We consider a chain [11] of elastically colliding particles of two kinds referred to, in the following, as "bars" L R
FIG. 1: (Color online)The schematic plot of our model. Dotted lines divide elementary unit cells. In each cell there is a bar which is subjected to elastic collisions with both cell boundaries and with neighboring bullets. The first (last) cell is coupled to a heat bath at temperature τL (τR).and "bullets", respectively. (See Fig. 1.) Each bar is confined inside a cell of unit length; that is, besides elastic collisions with its neighbor...
