Non-ergodicity of Nose-Hoover dynamics

The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat "mass" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity

A Metropolis-Adjusted Nosé-Hoover Thermostat

We present a Monte Carlo technique for sampling from the canonical distribution in molecular dynamics. The method is built upon the Nosé-Hoover constant temperature formulation and the generalized hybrid Monte Carlo method. In contrast to standard hybrid Monte Carlo methods only the thermostat degree of freedom is stochastically resampled during a Monte Carlo step

Highly degenerate diffusions for sampling molecular systems

This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique

Variational and linearly implicit integrators, with applications

We show that symplectic and linearly implicit integrators proposed by Zhang & Skeel (1997, Cheap implicit symplectic integrators. Appl. Numer. Math., 25, 297–302) are variational linearizations of Newmark methods. When used in conjunction with penalty methods (i.e., methods that replace constraints by stiff potentials), these integrators permit coarse time-stepping of holonomically constrained mechanical systems and bypass the resolution of nonlinear systems. Although penalty methods are widely employed, an explicit link to Lagrange multiplier approaches appears to be lacking; such a link is now provided (in the context of two-scale flow convergence (Tao, M., Owhadi, H. & Marsden, J. E. (2010) Nonintrusive and structure-preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul., 8, 1269–1324). The variational formulation also allows efficient simulations of mechanical systems on Lie groups

Development of a machine learning potential for nucleotides in water

Recent experimental research showed that nucleotides, under favorable conditions of temperature and concentration, can self-assemble into liquid crystals. The mechanism involves the stacking of nucleotides into columnar aggregates. It has been proposed that this ordered structure can favor the polymerization of long nucleotide chains, which is a fundamental step toward the so called “RNA world”. In this thesis, starting from ab initio molecular dynamics simulations, at the density func- tional theory level, an all-atom potential for nucleotides in water, based on an implicit neural network representation, has been developed. Its stability and accuracy have been tested and its predictions on simple model systems have been compared with data generated both ab initio and using currently available empirical force field for nucleic acids

Thermodynamics and structure of methionine enkephalin using the statistical temperature molecular dynamics algorithm

Kim, Straub, and Keyes introduced the statistical temperature molecular dynamics (STMD) algorithm to overcome broken ergodicity by sampling a non­-Boltzmann flat energy histogram as noted in Kim, Straub, and Keyes, Phys. Rev. Lett. 97: 050601 (2007). Canonical averages are calculated via reweighting to the desired temperature. While STMD is promising, its application has been almost entirely to simple or model systems. In this dissertation the implementation of STMD into the biosimulation package CHARMM is used to simulate the methionine enkephalin pentamer peptide with a methione terminal cap in a droplet of CHARMM TIP3P water molecules. Chain thermodynamics is analyzed from the novel perspective of the statistical temperature as a function of potential energy, $TS(U), automatically generated by STMD. Both the minimum in the slope of$TS(U), and the peak in the heat capacity as a function of temperature, calculated via reweighting, indicate a collapse transition at Tθ ≈ 253K. Distributions of dihedral angles are obtained as a function of temperature. Rotamer regions found in the literature are reproduced, along with unique regions not found previously, including with advanced algorithms, indicating the power of STMD enhanced sampling

Generalised Langevin equation: asymptotic properties and numerical analysis

In this thesis we concentrate on instances of the GLE which can be represented in a Markovian form in an extended phase space. We extend previous results on the geometric ergodicity of this class of GLEs using Lyapunov techniques, which allows us to conclude ergodicity for a large class of GLEs relevant to molecular dynamics applications. The main body of this thesis concerns the numerical discretisation of the GLE in the extended phase space representation. We generalise numerical discretisation schemes which have been previously proposed for the underdamped Langevin equation and which are based on a decomposition of the vector field into a Hamiltonian part and a linear SDE. Certain desirable properties regarding the accuracy of configurational averages of these schemes are inherited in the GLE context. We also rigorously prove geometric ergodicity on bounded domains by showing that a uniform minorisation condition and a uniform Lyapunov condition are satisfied for sufficiently small timestep size. We show that the discretisation schemes which we propose behave consistently in the white noise and overdamped limits, hence we provide a family of universal integrators for Langevin dynamics. Finally, we consider multiple-time stepping schemes making use of a decomposition of the fluctuation-dissipation term into a reversible and non-reversible part. These schemes are designed to efficiently integrate instances of the GLE whose Markovian representation involves a high number of auxiliary variables or a configuration dependent fluctuation-dissipation term. We also consider an application of dynamics based on the GLE in the context of large scale Bayesian inference as an extension of previously proposed adaptive thermostat methods. In these methods the gradient of the log posterior density is only evaluated on a subset (minibatch) of the whole dataset, which is randomly selected at each timestep. Incorporating a memory kernel in the adaptive thermostat formulation ensures that time-correlated gradient noise is dissipated in accordance with the fluctuation-dissipation theorem. This allows us to relax the requirement of using i.i.d. minibatches, and explore a variety of minibatch sampling approaches

Molecular Dynamics Simulation

Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...