## ABSTRACT

This chapter describes the standard methods, Metropolis Monte Carlo (MC) and molecular dynamics (MD). Because MC/MD are relaxation type procedures, they are discussed prior to Chapters 11–13, which are devoted to non-equilibrium problems, where relaxation phenomena are central. MC is based on transition probabilities (TPs), p_{ij}
, from state i to j, which are expected to drive a system to equilibrium. However, only the ratio p_{ij}
/p_{ji}
is completely determined, meaning that p_{ij}
is defined up to a multiplying constant, and thus the number of TPs is unlimited. The efficiency depends on the procedure selected and the type of moves, I → j. Two MC procedures, the “symmetric” and “asymmetric” MC, are presented together with an MC procedure for the grand canonical ensemble. The advantage of MC lies in its simplicity; however, the method becomes inefficient for dense systems (fluids, proteins, etc.) due to rejections, and for self-avoiding walks if treated by local moves. With MD, Newton’s equations of motion are solved numerically, driving the system to equilibrium. The Verlet algorithm and some of its derivatives are described in detail for the micro-canonical and canonical ensembles. MD outperforms MC for dense systems since the coordinates are moved along the calculated gradients and not at random. Still, for proteins, MD is ineffective in leading to protein folding due to the rugged potential energy surface of the protein; therefore, MD should be used within the frameworks of other techniques, e.g., replica exchange (see Chapter 21).