There’s no difference in the phase space, that statement is wrong. There’s one phase space, what matters is how it’s sampled. In molecular dynamics tye sampling is done by generating trajectories from independent initial conditions-so that one doesn’t stay on any particular surface-with Markov chain Monte Carlo equivalent configurations are generated.

The problem both approaches must solve is to sample the phase space, that has dimensionality 6N in both cases. Indeed it’s usual to employ a ``hybrid" approach combining both: MD evolution along with an accept/reject step.

Nice question, yet it is twisted for no reason.

there is no-such more than three D. exist in the world.

Unambiguous empirical evidence shows that nature has three dimension only. Again everything in the universe has journey, period,

Be opened minded read this...

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regards

My though is that the true state of a dynamical system can not be fully determined using just the particle coordinates.

Imagine a particle attached to a massless spring oscillating in X-direction. A phase space consisted just the X-coordinates can not fully describe the state of the system because at each X (except two ends) there are two distinct states: one with positive velocity (moving forward) and one with negative velocity (moving backward). However, a phase space consisted of both coordinates and momenta fully describes the state of the system.

Nonetheless, I'm a bit confused how and why does momenta is involved in the static system as usually refers to Monte Carlo simulation.

The statement is wrong.

The phase space is invariant in both simulations. However, for the MC simulation only position sampling is performed, the velocity or momentum must be considered from the thermal distribution associated with the system.

The phase space has always 6N dimensions. Using molecular dynamics simulations you use both the positions and the momenta. Using Monte Carlo "simulations" you just use the probability for a specific atomic configuration to appear based on the energy difference between this configuration and a fixed one. Monte Carlo is not a "simulation" really. It just creates a sample of atomic positions in the configuration space. So each configuration that it reaches should not be considered as a trajectory, as the "true" trajectory provided by molecular dynamics.

Both methods attempt to estimate the thermodynamically relevant free energy ensemble average (over 3N positions). Because the Monte Carlo (MC) method chooses random coordinates at each step, it turns out that generally more than half of the steps are effectively discarded because the Boltzmann weighting factor exp(-\Delta E/kT) is vanishingly small. The Molecular Dynamics (MD) method avoids this problem by choosing the next set of coordinates by integrating Newton's equations of motion. With the application of the ergodic hypothesis (an infinite time average is equivalent to the ensemble average), MD methods avoid discarding samples but are very limited in their estimation of the infinite time sample.

Choosing between the two methods is, more or less, a matter of taste. MD methods may be somewhat easier to develop and debug because you can watch the model evolve in time: any random jumps or glitches in the motions of the atoms point to a defect in the integration procedure.