umbrella sampling learning

Q1: what is umbrella sampling?
The umbrella sampling applies bias energy to an interested system on some CVs. It is similar to heating a system. The motion of particles starts accelerating. The bare events, i.e. the some un-accessible near ground states, could be explored. The benefits of umbrella sampling is that US can obtain the relative free energy profiles.

Q2: what is free energy in simulation? Why free energy is used in MD simulations?

The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system minus the amount of energy that cannot be used to perform work. This unusable energy is given by the entropy of a system multiplied by the temperature of the system.

Internal energy has two major components, kinetic energy and potential energy. The kinetic energy is due to the motion of the system's particles (translations, rotations, vibrations), and the potential energy is associated with the static rest mass energy of the constituents of matter, static electric energy of atoms within molecules or crystals, and the static energy of chemical bonds. 
Enthalpy:  It includes the system's internal energy or thermodynamic potential (a state function), as well as its volume and pressure (the energy required to "make room for it" by displacing its environment, which is an extensive quantity).

Entropy: Entropy is a measure of the number of specific ways in which a system may be arranged, often taken to be a measure of disorder, or a measure of progressing towards thermodynamic equilibrium. The entropy of an isolated system never decreases, because isolated systems spontaneously evolve towards thermodynamic equilibrium, which is the state of maximum entropy.  Entropy is a logarithmic measure of the number of states with significant probability of being occupied. If all microstates have equal probability, the entropy is k_b*ln(the number of micro states)

In MD simulations, there are several common free energy types.
Common Free Energy Type
● Solvation Free Energy / Transfer Free Energy
● Binding Free energy
● Confomational Free Energy 

Ergodic hypothesis
time average = ensemble average

free energy difference:
ΔG = -RT ln (P1/P0)

Accurate calculations of absolute free energy is nearly impossible due to insufficient sampling in a
finite length and time scale simulation.  Need different methods to estimate free energy.
● Common method
- Thermodynamical integration
- Free energy perturbation
- Umbrella sampling
- Potential of mean force

Running US simulation
● Choose the reaction coordinate
● Choose the number of windows and the biasing potential
● Run the simulations
● Compute time series for the value of the reaction coordinate (histograms)
● Apply the WHAM equations