Contents -- Preface -- 1. Introduction -- 1.1 Computational statistical physics: some landmarks -- 1.1.1 Some orders of magnitude -- 1.1.2 Aims of molecular simulation -- 1.1.2.1 An example: the equation of state of Argon -- 1.2 Microscopic description of physical systems -- 1.2.1 Interactions -- 1.2.1.1 Boundary conditions -- 1.2.1.2 Potential functions -- 1.2.2 Dynamics of isolated systems -- 1.2.2.1 The Hamiltonian dynamics -- 1.2.2.2 Equivalent reformulations -- 1.2.2.3 Properties of the Hamiltonian dynamics -- 1.2.2.4 Numerical integration -- 1.2.3 Thermodynamic ensembles -- 1.2.3.1 The microcanonical ensemble -- 1.2.3.2 The canonical ensemble -- 1.2.3.3 Other thermodynamic ensembles -- 1.3 Free energy and its numerical computation -- 1.3.1 Absolute free energy -- 1.3.1.1 Definition -- 1.3.1.2 Relationship with macroscopic thermodynamics -- 1.3.2 Relative free energies -- 1.3.2.1 Alchemical transitions -- 1.3.2.2 Transitions indexed by a reaction coordinate -- 1.3.2.3 A typical alchemical transition: Widom insertion -- 1.3.2.4 A typical transition indexed by a reaction coordinate: Dimer in a solvent -- 1.3.3 Free energy and metastability -- 1.3.3.1 A simple example of metastable dynamics -- 1.3.3.2 Entropic and energetic barriers -- 1.3.3.3 Free energy biased sampling -- 1.3.4 Computational techniques -- 1.3.4.1 Thermodynamic integration -- 1.3.4.2 Methods based on straightforward sampling -- 1.3.4.3 Nonequilibrium dynamics -- 1.3.4.4 Adaptive dynamics -- 1.4 Summary of the mathematical tools and structure of the book -- 2. Sampling methods -- 2.1 Markov chain methods -- 2.1.1 Some background material on the theory of Markov chains -- 2.1.2 The Metropolis-Hastings algorithm -- 2.1.2.1 Presentation of the method -- 2.1.2.2 Mathematical properties -- 2.1.2.3 Some examples of proposition transition kernels -- 2.1.3 Hybrid Monte-Carlo.

2.1.4 Generalized Metropolis-Hastings variants -- 2.1.4.1 Presentation of the method -- 2.1.4.2 Mathematical properties -- 2.1.4.3 Relationship with the Hybrid Monte-Carlo algorithm -- 2.2 Continuous stochastic dynamics -- 2.2.1 Mathematical background on Markovian continu- ous processes -- 2.2.1.1 Infinitesimal generator of diffusion processes -- 2.2.1.2 Properties of time-homogeneous diffusion processes -- 2.2.2 Overdamped Langevin process -- 2.2.2.1 Detailed balance and ergodicity -- 2.2.2.2 Time discretization and numerical implementation -- 2.2.3 Langevin process -- 2.2.3.1 Detailed balance and ergodicity -- 2.2.3.2 Time discretization and numerical implementation -- 2.2.4 Overdamped limit of the Langevin dynamics -- 2.2.4.1 Limit of the stochastic processes -- 2.2.4.2 Overdamped limit of the numerical schemes -- 2.3 Convergence of sampling methods -- 2.3.1 Sampling errors -- 2.3.1.1 Bias -- 2.3.1.2 Statistical errors -- 2.3.1.3 Practical computation of error bars -- 2.3.2 Rate of convergence for stochastic processes -- 2.3.2.1 Longtime convergence -- 2.3.2.2 A possible quantification of metastability -- 2.3.2.3 Obtaining logarithmic Sobolev inequalities -- 2.4 Methods for alchemical free energy differences -- 2.4.1 Free energy perturbation -- 2.4.1.1 General idea of the method -- 2.4.1.2 Expansions using the distribution of energy differences -- 2.4.1.3 Staging -- 2.4.1.4 Umbrella sampling -- 2.4.2 Bridge sampling -- 2.4.2.1 Presentation of the method -- 2.4.2.2 Derivation of the optimal function -- 2.4.2.3 Numerical strategy -- 2.4.2.4 Numerical illustration -- 2.5 Histogram methods -- 2.5.1 Principle of histogram methods -- 2.5.1.1 Free energy as an approximated canonical average -- 2.5.1.2 Combining partial samples -- 2.5.2 Extended bridge sampling -- 2.5.2.1 Presentation of the method -- 2.5.2.2 Recovering canonical averages.

2.5.2.3 Application to the model problem -- 3. Thermodynamic integration and sampling with constraints -- 3.1 Introduction: The alchemical setting -- 3.1.1 General strategy -- 3.1.2 Numerical application -- 3.2 The reaction coordinate case: configurational space sampling -- 3.2.1 Reaction coordinate and free energy -- 3.2.1.1 Marginal and conditional probability measures -- 3.2.1.2 The free energy -- 3.2.1.3 The case of a non-standard scalar product -- 3.2.2 The mean force -- 3.2.3 Sampling measures on submanifolds of Rn -- 3.2.3.1 Geometrical notation -- 3.2.3.2 Projected dynamics -- 3.2.3.3 Ergodicity of the projected dynamics -- 3.2.3.4 Softly and rigidly constrained dynamics -- 3.2.4 Sampling measures on submanifolds of Rn: dis- cretization -- 3.2.4.1 Rewriting of the projected dynamics (3.52) using Lagrange multipliers -- 3.2.4.2 Discretization of the projected dynamics (3.52) -- 3.2.4.3 Consistency of the predictor-corrector schemes. -- 3.2.4.4 Ergodicity of the numerical schemes and time discretiza- tion error -- 3.2.5 Computing the mean force -- 3.2.5.1 Methods based on the sampling of the conditional mea- sures v (.

3.3.3.1 Definition of the constrained dynamics -- 3.3.3.2 Properties of the constrained dynamics -- 3.3.4 Constrained Langevin processes -- 3.3.4.1 Definition of the dynamics -- 3.3.5 Numerical implementation -- 3.3.5.1 Numerical schemes for the Hamiltonian part -- 3.3.5.2 Fluctuation-dissipation part -- 3.3.5.3 Numerical schemes obtained by a splitting strategy -- 3.3.5.4 Metropolization -- 3.3.5.5 Exact sampling of constrained overdamped processes -- 3.3.6 Thermodynamic integration with constrained Langevin processes -- 3.3.6.1 Free energy -- 3.3.6.2 The case of molecular constraints -- 3.3.6.3 The mean force -- 3.3.6.4 Free energy from Lagrange multipliers -- 3.3.6.5 Numerical illustration -- 4. Nonequilibrium methods -- 4.1 The Jarzynski equality in the alchemical case -- 4.1.1 Markovian nonequilibrium simulations -- 4.1.2 Importance weights of nonequilibrium simulations -- 4.1.3 Practical implementation -- 4.1.4 Degeneracy of weights -- 4.1.4.1 Work distributions -- 4.1.4.2 An analytical example -- 4.1.4.3 Reducing the width of work distributions -- 4.1.5 Error analysis -- 4.1.5.1 One-sided averages -- 4.1.5.2 Double-sided averages -- 4.1.5.3 Influence of the parameters -- 4.1.5.4 Numerical results for Widom insertion -- 4.2 Generalized Jarzynski-Crooks fluctuation identity -- 4.2.1 Derivation of the identity -- 4.2.2 Relationship with standard equalities in the physics and chemistry literature -- 4.2.3 Numerical strategies -- 4.3 Nonequilibrium stochastic methods in the reaction coordi- nate case -- 4.3.1 Overdamped nonequilibrium dynamics -- 4.3.1.1 Definition of the switched dynamics -- 4.3.1.2 Jarzynski-Crooks identity -- 4.3.1.3 Numerical implementation -- 4.3.2 Hamiltonian and Langevin nonequilibrium dynamics -- 4.3.2.1 Generalized free energy and notation -- 4.3.2.2 Dynamics and generators -- 4.3.2.3 Definition of the work.

4.3.2.4 Jarzynski-Crooks identity -- 4.3.2.5 Numerical schemes -- 4.3.3 Numerical results -- 4.4 Path sampling strategies -- 4.4.1 The path ensemble -- 4.4.1.1 Equilibrium paths -- 4.4.1.2 Switching paths -- 4.4.2 Sampling switching paths -- 4.4.2.1 General sampling strategy -- 4.4.2.2 Shooting moves -- 4.4.2.3 Brownian tube moves for stochastic dynamics -- 4.4.2.4 Weighted path ensembles -- 4.4.2.5 Importance sampling -- 4.4.2.6 Efficiency of the path sampling approach -- 4.4.2.7 Application to Widom insertion -- 5. Adaptive methods -- 5.1 Adaptive algorithms: A general framework -- 5.1.1 Updating formulas -- 5.1.1.1 Observed free energy and mean force -- 5.1.1.2 Updating the bias with the observed quantities -- 5.1.1.3 Consistency of adaptive methods. -- 5.1.1.4 Generalized adaptive importance sampling strategies -- 5.1.1.5 Adaptive biasing force or adaptive biasing potential? -- 5.1.2 Extended dynamics -- 5.1.3 Discretization methods -- 5.1.3.1 Approximation of probability measures -- 5.1.3.2 Approximations based on kernel density estimation: regu- larization and mathematical results -- 5.1.3.3 Discretization of functions defined on the reaction coordi- nate space -- 5.1.3.4 Approximation of the law based on trajectorial averages -- 5.1.4 Classical examples of adaptive methods -- 5.1.4.1 Metadynamics -- 5.1.4.2 Wang-Landau -- 5.1.4.3 The Adaptive Biasing Force method -- 5.1.4.4 An ABF method in extended space -- 5.1.4.5 The self-healing umbrella sampling method -- 5.1.5 Numerical illustration -- 5.2 Convergence of the adaptive biasing force method -- 5.2.1 Presentation of the studied ABF dynamics -- 5.2.1.1 Notation and definitions -- 5.2.1.2 The ABF dynamics -- 5.2.1.3 Reformulation as a nonlinear partial differential equation -- 5.2.2 Precise statements of the convergence results -- 5.2.2.1 Decomposition of the entropy.

5.2.2.2 Convergence of the adaptive dynamics (5:60)-(5:61).