The earthquake-like model with a continuous distribution of static thresholds is used to describe the properties of solid friction. The evolution of the model is reduced to a master equation which can be solved analytically. This approach naturally describes stick-slip and smooth sliding regimes of tribological systems within a framework which separates the calculation of the friction force from the studies of the properties of the contacts.PACS numbers: 81.40. Pq; 46.55.+d; 61.72.Hh In spite of its crucial practical importance, friction is still not fully understood [1]. It raises questions at many scales, from the atomic scale studied nowadays by atomic force microscopy to the macroscopic scale of a solid block sliding on an other. A simple mesoscopic model has been introduced to bridge the gap in scales and describe the main experimental observations, such as stick slip or smooth sliding, in terms of the properties of local contacts. This widely used Burridge-Knopoff spring-block model [2], initially introduced to study earthquakes (EQ model), has been developed by Olami, Feder and Christensen [3]. It describes the contacts in terms of elastic springs and junctions that break at a critical force. Computer simulations [4,5] showed that the EQ model may reproduce the experimentally the observed stick-slip and smooth-sliding regimes, including the role of velocity and temperature, if the model is at least two-dimensional and various assumptions on the properties of the contacts are made.The drawback of such a simulation approach is that heavy calculations with different parameter sets or contact properties are required to determine the main features of the model, and it is hard to draw conclusions of general validity. The calculations may be tedious because a large number of contacts and investigations on very long evolution times are necessary to get meaningful statistics and to make sure that the calculation has reached asymptotic properties which are not influenced by the initial conditions. Moreover almost all the studies based on the EQ model assume for simplicity, and to reduce the parameter space to explore, that all contacts have identical properties. It turns out that, as we show below, this limit is singular and may lead to qualitatively incorrect conclusions.Here we introduce a master equation (ME) approach which is much more efficient than simulations and can be solved analytically in cases which are particularly rel- * obraun@iop.kiev.ua; http://www.iop.kiev.ua/~obraun † Michel.Peyrard@ens-lyon.fr evant. It provides a deeper understanding of friction analyzed at the mesoscale in terms of the statistical properties of the contacts. This splits the study of friction in two independent parts: (i) the calculation of the friction force given by the master equation provided the statistical properties of the contacts are known, (ii) the study of the contacts and their statistics, which needs inputs from the microscopic scale. Many aspects such as the interaction between the contacts and their aging can ...