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As far as I understand, a system in equilibrium is which already reached its equilibrium state. And, a non-equilibrium system is which hasn't reached its equilibrium state yet.
Are MD (Molecular Dynamics) and MC (Monte Carlo) both able to study both equilibrium and non-equilibrium systems?
Nike's answer included examples of Monte Carlo in the context of electronic structure. However, Monte Carlo techniques are most popularly employed in statistical physics / thermodynamics - often employing classical force fields - and generally across various disciplines wherever one needs to sample high-dimensional integrals.
In addition to the modeling of equilibria, Monte Carlo methods can be also employed to modeling dynamic processes. An example of this is the kinetic Monte Carlo method, which is often used to model e.g. multi-step reactions.
A topical reference for Monte Carlo methods in materials modeling is a chapter in the Handbook of Materials Modeling.
Monte Carlo methods such as FCIQMC can study non-equilibrium systems: in fact FCIQMC is used for calculating the ground state energy of a system that is already in equilibrium.
Monte Carlo methods can also study non-equilibrium systems, for example in this paper which is about using Monte Carlo methods for real-time quantum dynamics.
Molecular Dynamics certainly can be applied to systems that are not in equilibrium, because it literally simulates the real-time dynamics of a system, which can for example be an equilibrium system of molecules that are not moving until one of them is bombarded by an extremely fast molecule that puts them all out of equilibrium. It can also continue simulating that system until it reaches equilibrium again. Therefore molecular dynamics can also simulate equilibrium systems (why not?) even if the MD equations just say that all accelerations are zero. Molecular dynamics can also simulate systems that are in dynamic equilibrium.
In addition to what others have said, it might help you to know that Monte Carlo methods, at least in the case of simulating rarefied gas flows (Direct Simulation Monte Carlo - DSMC), perform much, much better the further you are from equilibrium. Situations with low speeds / low pressure differences / low temperature differences usually require a prohibitively high computational time, unless you are using a heavily modified version of DSMC. The reason is that the statistical noise hides the (very low) gradients of pressure, temperature, etc, so you need to have a sample of many particles or average over a long enough time interval.
I do not have experience with MD, so please take my opinion with a grain of salt, but I do not see a reason for the method not being able to simulate both types of systems in theory, assuming that you confine your investigation in a very small area and time window. But I suppose you will have better performance in some cases than others.
