Before revision:  Drawing from the mean-field approach in Statistical Mechanics and Physics, we assume a large number of agents interacting in the order book . By taking the 'continuum' limit we obtain a set of nonlinear differential equations, the core of our dynamical theory of price formation. And we explicitly solve them using Fourier analysis. One could talk as well of a "micro-macro" approach of equilibrium, where the market price is the consequence of each ("microscopic") agent behaving with respect to his preferences and to global ("macroscopic") information. When a large market order (or metaorder) perturbs the market, our model recovers the square-root law of impact , providing new insights on the price formation process. In addition, we give various limiting cases, examples and possible extensions .
After revision:  Drawing from the mean-field assumption that views the order book through its average density. We show how price impact develops in this new model, providing a flexible family of solutions that can potentially be calibrated to real data. While no closed-form solution is provided, we complement our theoretical investigation with extensive numerical results, including a simulation scheme for the entire order book .