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The existence and search for thermodynamic phase transitions is of unfadinginterest. In this paper, we present numerical evidence of dynamical phasetransitions occurring in boundary driven systems with a constrained integratedcurrent. It is shown that certain models exhibit a discontinuous transitionbetween two different density profiles and a continuous transition between atime-independent and a time-dependent profile. We also checked that the KMPmodel does not exhibit phase transition in a range much larger than previouslyexplored.

Numerical Study of Continuous and Discontinuous Dynamical Phase Transitions for Boundary Driven Systems