Abstract
At its most basic level, molecular dynamics is about mapping out complicated point sets using trajectories of a system of ordinary differential equations (or, in Chaps. 6–8, a stochastic-differential equation system). The sets are typically defined as the collection of probable states for a certain system. In the case of Hamiltonian dynamics, they are directly associated to a region of the energy landscape. The trajectories are the means by which we efficiently explore the energy surface. In this chapter we address the design of numerical methods to calculate trajectories.
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Notes
- 1.
Subscripts were used previously to indicate the components of vectors and here they are used to indicate the indexing of timesteps. Although in theory this could lead to some confusion, it normally does not in practice, since we index components in descriptions of details of models and we discuss timesteps in context of defining numerical methods for general classes of systems. Moreover, we use boldface for vectors, so a subscript on a boldface vector indicates a timestep index. When we wish to refer to both the timestep and the component index, we may write z n, i to denote the ith component at timestep n.
- 2.
- 3.
In the multidimensional setting, Taylor’s theorem states that given a C k+1 function \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}\) and a point \({\boldsymbol z}_{0}\), we have
$$\displaystyle\begin{array}{rcl} f({\boldsymbol z}) - f({\boldsymbol z}_{0})& =& \nabla f({\boldsymbol z}_{0}) \cdot ({\boldsymbol z} -{\boldsymbol z}_{0}) + f^{(2)}\langle {\boldsymbol z} -{\boldsymbol z}_{ 0},{\boldsymbol z} -{\boldsymbol z}_{0}\rangle + f^{(3)}\langle {\boldsymbol z} -{\boldsymbol z}_{ 0},{\boldsymbol z} -{\boldsymbol z}_{0},{\boldsymbol z} -{\boldsymbol z}_{0}\rangle {}\\ & & \qquad +\ldots f^{(k)}\langle {\boldsymbol z} -{\boldsymbol z}_{ 0},{\boldsymbol z} -{\boldsymbol z}_{0},\ldots,{\boldsymbol z} -{\boldsymbol z}_{0}\rangle + \mathcal{O}(\|{\boldsymbol z} -{\boldsymbol z}_{0}\|^{k+1}) {}\\ \end{array}$$where \(\nabla f = \partial f/\partial {\boldsymbol z}\) is the gradient, i.e. a vector with m components, f (2) is the m × m Hessian matrix of f (the matrix whose ij component is ∂ 2 f∕∂ z i ∂ z j ), and \(f^{(2)}\langle {\boldsymbol u},{\boldsymbol v}\rangle\), \({\boldsymbol u},{\boldsymbol v} \in \mathbb{R}^{m}\), represents the quadratic form \({\boldsymbol u}^{T}f^{(2)}{\boldsymbol v}\). In a similar way we interpret f (3) as a tensor which we can think of as a m × m × m triply-indexed array, the ijk element being ∂ 3 f∕∂ z i ∂ z j ∂ z k and
$$\displaystyle{f^{(3)}\langle {\boldsymbol u},{\boldsymbol v},{\boldsymbol w}\rangle =\sum _{ i=1}^{m}\sum _{ j=1}^{m}\sum _{ k=1}^{m}(\partial ^{3}f/\partial z_{ i}\partial z_{j}\partial z_{k})u_{i}v_{j}w_{k}.}$$ - 4.
This discussion is a great simplification. Any curve which satisfies this equation will represent a “stationary point” (actually, “stationary curve” would be more accurate) of the classical action. Such curves could include smooth local action minimizers, local action maximizers, or “saddle points” of the actional functional in a generalized sense. Deciding whether a given stationary curve is an actual minimizer of the action would require analysis of the second variation (the coefficient of \(\varepsilon ^{2}\) in the expansion above), which introduces additional complexity. For a more comprehensive treatment, see e.g. [210].
- 5.
Rounding error is the error introduced when numbers are forced into the finite word length representation in a typical digital computer. Adding together two “computer numbers,” then rounding, results in another computer number. Rounding errors may accumulate in long computations, but in molecular dynamics they are normally dominated by the much larger “truncation errors” introduced in the process of discretization, that is, due to replacing the differential equation by a difference equation such as the Euler or Verlet method . For an example of the role of rounding error in the context of constrained molecular dynamics, see [237].
- 6.
The term “secular growth” in this context is a reference to the long-term growth of perturbations in celestial mechanics. For example, the precession of the Earth’s polar axis occurs on a long period relative to its orbital motion and much longer period than its rotation, and so may be classed as a secular motion. In the context of molecular simulations, we use this to refer to accumulation of drift that takes the system steadily away from the energy surface.
- 7.
A homeomorphism is a continuous bijection which has a continuous inverse.
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Leimkuhler, B., Matthews, C. (2015). Numerical Integrators. In: Molecular Dynamics. Interdisciplinary Applied Mathematics, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-16375-8_2
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