Enhanced Sampling Techniques

Content: The course provides an overview of the most powerful techniques for estimating the probability distribution in molecular systems affected by metastability, in which, therefore, the probability is multimodal. 

• Definition of the mean first passage time and of the average transition time.  Estimate of the probability density and of the free energy from an histogram.
• Qualitative properties of a “good” collective variable (CV). Committor analysis as a test for the quality of a CV
• Umbrella sampling (US): derivation of the US equation for the free energy in  the canonical ensemble
• Estimates of the free energy by US. The optimal bias.
• Adaptive umbrella sampling (AUS). Derivation of the update role of the bias. Stationary solution of the update equations. Regularization of the estimate of the free energy.
• Metadynamics as a limiting case of AUS. Formulation of metadynamics as a Markov process. Estimate of the free energy in metadynamics
• The Weighted Histogram Analysis Method
• Thermodynamic integration.  Estimate of the error. Relationship with Jarzynski theorem.
• Replica exchange. Derivation of the acceptance criterion for two replicas. Scaling of the number of replicas with the number of degrees of freedom. 
• Markov State Modeling: the properties of a transition probability matrix between a set of microstates;  solution of the rate equation on the basis of the eigenvectors. Asymptotic solution (convergence to equilibrium) and stability analysis. Estimate of the transition probability matrix from a MD trajectory.