# softmatter:Deterministic vs. Stochastic Dynamics

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Like stochastic dynamics, Newtonian dynamics is simply a mathematical sampling tool. However, because Molecular Dynamics (MD) simulations produce phase space trajectories that are sequential in time and hence closely tied to physical reality, it may seem that MD is somehow more exact or realistic than stochastic methods such as Monte Carlo (MC) or Brownian Dynamics. In reality, this is incorrect since the inherent numerical approximations of MD result in a problem that is mathematically poorly conditioned, particularly over long times. Thus, neither MD nor MC are “deterministic” in any true sense of the word, and therefore can be considered equivalent mathematical tools for carrying out molecular simulations, except that they produce microstates in a different order.

With this in mind, it should be noted that there are many situations in which one scheme is generally preferred over the other for practical reasons, such as ease of implementation or the degree of computational cost. For example, the NVE ensemble is seldom implemented via MC, since schemes in which trial moves do not alter the system energy are difficult to generate for most system and therefore are prone to computational inefficiency. Along these lines, sampling thermodynamical ensembles other than NVE for MD can be relatively difficult to implement, so MC methods are often preferred.

Often there are a many competing factors that affect computational efficiency, and thus it is difficult to predict which scheme will perform better for a given application. However, as a general rule, MC simulation is more efficient at low density, where configurational changes are more likely to be accepted (i.e., particles are unlikely to overlap for a given trial move). For dense systems, where only small perturbations to the system result in valid microstates, MD is the preferred method. If all else is equal, MD is the generally preferred, since it yields additional information regarding particle velocities.