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When studied at finite temperature, Yang-Mills theories in $3+1$ dimensionsdisplay the presence of confinement/deconfinement phase transitions, which areknown to be of first order -- the $SU(2)$ gauge theory being the exception.Theoretical as well as phenomenological considerations indicate that it isessential to establish a precise characterisation of these physical systems inproximity of such phase transitions. We present and test a new method to studythe critical region of parameter space in non-Abelian quantum field theories onthe lattice, based upon the Logarithmic Linear Relaxation (LLR) algorithm. Weapply this method to the $SU(3)$ Yang Mills lattice gauge theory, and performextensive calculations with one fixed choice of lattice size. We identify thecritical temperature, and measure interesting physical quantities near thetransition. Among them, we determine the free energy of the model in thecritical region, exposing for the first time its multi-valued nature with anumerical calculation from first principles, providing this novel evidence insupport of a first order phase transition. This study sets the stage for futurehigh precision measurements, by demonstrating the potential of the method.

First-order phase transitions in Yang-Mills theories and the density of state method