# Rare Events and Enhanced Sampling

Molecular dynamics algorithms have time reversibility and ergodic hypotheses. We assume that all states of a molecule have a probability to be explored (or traversed/sampled) after a sufficiently long simulation. These states include ground states, meta-stable states, and some high-energy states (unstable states). When the simulation reaches equilibrium, the distribution of molecular conformations in the system satisfies

**the Boltzmann distribution**under its**ensemble**.Different states have different probabilities to be sampled. In most cases, the molecules are stuck in a local minimum on the energy surface, and it is difficult to jump over the energy barriers. Therefore, under finite-time simulations, the probability of sampling some high-energy state or another meta-state separated by an energy barrier is very low. These are

**rare events**.Here, it can be compared with Monte Carlo (MC) simulations of a high-dimensional function, starting from a random value of the high-dimensional function to explore the global minima of the function. This can take a lot of time and computational resources.

To increase the probability of rare events occurring in MD simulations, the simulation process can be interfered with using various methods:

Enhanced sampling based on temperature:

Raise the temperature, lower the energy barriers, and increase the probability of rare events.

Replica-Exchange Molecular Dynamics (REMD)[5], selective integrated tempering sampling (SITS)[6], etc.

Enhanced sampling based on bias potential:

Collective variables (CV), are functions of system coordinates. The free energy are defined on collective variables. (Please refer to the difference between free energy and potential energy)

Add bias potential to the given CVs during the simulations, which can push the trajectory out of the local minima on the energy surface to explore other states.

Metadynamics[7], VES[8], RiD[9], etc.

Traditional boosted sampling methods based on bias potential suffer from

**the curse of dimensionality.**