Make sure that you distinguish between internal energy (U=PE+KE) and Helmholtz or Gibbs free energy. Then think of what thermodynamic ensemble you are simulating. For example in MD, if you simulate an NVE ensemble, this corresponds to an isolated system. We know from the second law of thermodynamics that an equilibrium then forms when the entropy (S) maximizes. In an NVT ensemble an equilibrium forms when the Helmholtz free energy (U-TS) minimizes and in an NPT ensemble when the Gibbs free energy (U-TS+PV) minimizes. Thus, the driving forces toward equilibrium are different in different ensembles, and is not just a generic 'energy minimum', but in the end it all comes down to exchange between (configurational and kinetic) degrees of freedom due to particle interactions. In conclusion, you don't use a criterion to ensure energy minimization when evolving the equations of motion. The laws of physics do this for you.

Molecular Dynamics isn't about minimizing the energy-it's about conserving energy.

Brownian dynamics isn't about minimizing the energy, either-it's about ensuring equilibrium with the external bath.

Neither are of direct relevance for phase separation, nor for driving the system to the minima of the energy.

Make sure that you distinguish between internal energy (U=PE+KE) and Helmholtz or Gibbs free energy. Then think of what thermodynamic ensemble you are simulating. For example in MD, if you simulate an NVE ensemble, this corresponds to an isolated system. We know from the second law of thermodynamics that an equilibrium then forms when the entropy (S) maximizes. In an NVT ensemble an equilibrium forms when the Helmholtz free energy (U-TS) minimizes and in an NPT ensemble when the Gibbs free energy (U-TS+PV) minimizes. Thus, the driving forces toward equilibrium are different in different ensembles, and is not just a generic 'energy minimum', but in the end it all comes down to exchange between (configurational and kinetic) degrees of freedom due to particle interactions. In conclusion, you don't use a criterion to ensure energy minimization when evolving the equations of motion. The laws of physics do this for you.

Brownian Dynamics BD is less computational intensive than Molecular Dynamics MD. Check Molecular Biology.