Merge pull request #55 from MobleyLab/mobley

Also a minor PR/minor to-do addressing

Author: Jacob Monroe

paper/basic_training.bib

@@ -890,3 +890,17 @@ @article{Shirts:2007:JPhysChemB
   year = {2007},
   pages = {13052-13063},
 }
+
+@article{Chen:2014:Mol.Simul.,
+  title = {Recent Development and Application of Constant {{pH}} Molecular Dynamics},
+  volume = {40},
+  issn = {0892-7022},
+  doi = {10.1080/08927022.2014.907492},
+  number = {10-11},
+  journal = {Molecular Simulation},
+  author = {Chen, Wei and Morrow, Brian H. and Shi, Chuanyin and Shen, Jana K.},
+  month = aug,
+  year = {2014},
+  keywords = {pH,molecular dynamics,algorithms,electrostatics,pKa calculation},
+  pages = {830-838},
+}

paper/basic_training.tex

@@ -128,8 +128,8 @@ \section{Introduction}
 Thus, for the rest of this document we will restrict ourselves to classical MD.
 
 One other important note is that, within classical molecular simulations, bond breaking and forming is generally not allowed (with notable exceptions such as reactive force fields),  meaning that the overall topology or chemistry of a system will remain constant as a function of time.
-That is, the particles comprising the system move around, but the chemical identity of each molecule in the system is a constant over the course of the simulation (with only partial exceptions, such as the case of constant pH simulations).
-\todo[inline, color={yellow!20}]{DLM: Refs needed here}
+That is, the particles comprising the system move around, but the chemical identity of each molecule in the system is a constant over the course of the simulation (with only partial exceptions, such as the case of constant pH simulations~\cite{Chen:2014:Mol.Simul.}).
+This also means that the notion of pH in molecular simulations primarily refers to the selection of \emph{fixed} protonation states for the components of the system.
 
 Here, we first discuss the scope of this document, then go over some of the fundamental concepts or science topics which provide the underpinnings of molecular simulations, giving references for further reading.
 Then, we introduce a variety of basic simulation concepts and terminology, with additional links to further reading.
@@ -176,7 +176,7 @@ \subsubsection{Key concepts}
 
 Classical molecular models typically consist of point particles carrying mass and electric charge, as well as potentially additional interactions such as van der Waals interactions and bonded interactions of various types.
 Sometimes it is much more efficient to freeze the internal degrees of freedoms and treat the molecule as a rigid body where the particles do not change their relative orientation as the whole body moves; this is commonly done, for example, for rigid models of the water molecule.
-Due to the high frequency of the O-H vibrations, accurately treating water classically would require a very small timestep, so for computational efficiency water is often instead treated as a rigid body.
+Due to the high frequency of the O-H vibrations, accurately treating water classically would require solving the equations of motion with a very small timestep, so for computational efficiency water is often instead treated as a rigid body.
 Keeping specified objects rigid in a simulation involves applying holonomic constraints, where the rigidity is defined by imposing a minimal set of fixed bond lengths and angles through iterative procedures during the numerical integration of the equation of motion (see Section~\ref{sec:integrators} for more on constraints and integrators).
 It is important to understand the concept of point particles, rigid bodies and constraints.
 
@@ -923,8 +923,10 @@ \subsubsection{Desireable integrator properties}
 The latter effect, due to truncation errors, will become obvious if two simulations with different timesteps are compared.
 Shorter timesteps, and hence more steps to achieve a simulation of the same length, will result in \textit{more} drift, since errors get larger with the number of calculations performed by the computer.
 This is exactly opposite to the behavior that is expected for poor energy conservation associated with discretization error, where a shorter timestep will reduce energy drift.
-\todo[inline, color={yellow!20}]{DLM: Probably need to address energy drift, as well as how energy changes should scale with timestep.}
-\todo[inline, color={green!20}]{JIM: I addressed some of this above. I'm not sure exactly about the scalings, though. Can you say for sure how energy drift scales with timestep or number of timesteps?  Isn't this highly algorithm-specific (and maybe even system-specific)?}
+
+Overall, then, integrators do exhibit energy fluctuations that are timestep-dependent.
+All Verlet-equivalent integrators exhibit energy fluctuations which decrease with the square of the timestep~\cite{allen_computer_2017}, which is often an important check when assessing the correctness of an implementation.
+Thus, both energy drift and energy fluctuations are important criteria to understand when assessing integrators, and can be useful measures of simulation quality in the NVE ensemble.
 
 Additionally, it is also desirable that an integrator be computationally efficient.
 Integrator cost mostly appears in the length of the timestep that may be taken while still avoiding discretization error.