Molecular Dynamics¶

General workflow of molecular dynamics (MD) is shown below.

flowchart TD
    id1[Initial structure]
    id2[Force Field]
    id25[Setting initial velocity]
    id3[Compute forces and energy]
    id4[Solve equation of motion]
    id45[Update velocity and position]
    id46[Update thermostat or barostat]
    id5{Reached<br>maximum steps?}
    id6[Finished: trajectory]
    id1 --> id25
    id2 --> id3
    id25 --> id3
    id3 --> id4
    id4 --> id45
    id45-->id46
    id46 -->id5 
    id5 -->|yes|id6
    id5 -->|no|id3

Statistical Mechanics¶

Statistical mechanics bridges macroscopic properties (temperature, pressure, and heat capacity) with microscopic parameters (position, velocity, and ordering). There is a nice online course for this topic.

Partition Function¶

Partition function is the key of the statistical mechanics, which is defined as \(Z=\sum_i e^{-\beta E_i}\) where \(\beta=\frac{1}{k_BT}\) and \(i\) is a microstate. It is basically a sum over all microstates of a system. You can get all thermodynamical properties of a system once you know its partition function exactly.

In order to compute partition function, you need to sample microstates of your system. There are two ways: Molecular Dynamics (MD) and Monte Carlo (MC). MD and MC should both yield to the same results if microstates are fully sampled, i.e. ergodicity requirement is met. In MC, you're sampling the system randomly whereas in MD you're sampling the system using Newtonian mechanics.

Why MD?¶

Ergodic Hypothesis Compliance: Systems that comply with the ergodic hypothesis allow the use of molecular dynamics (MD) simulations to determine macroscopic thermodynamic properties.
Time and Ensemble Averages: In ergodic systems, time averages from a single MD simulation correspond to microcanonical ensemble averages.
Terminology and Concept: MD is often referred to as "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics."
Purpose and Insight: MD simulations animate natural forces, predicting future behaviors and providing insights into molecular motion at the atomic scale.

Constraints¶

An ensemble is a collection of systems sharing one or more macroscopic characteristics but each being in a unique microstate. Depends on the property you're interested in, you can run MD at different constraints.

Ensemble	Name	Constraints	Comments
NVE	microcanonical ensemble	number of particles, lattice, and energy	kinetic energy and potential energy are conserved
NVT	canonical ensemble	number of particles, lattice, and temperature	temperature is controlled by a thermostat
NPT	isothermal-isobaric ensemble	number of particles, pressure, and temperature	temperature is controlled by a thermostat, pressure is controlled by a barostat
\(\rm{\mu}\)VT	grand canonical ensemble	chemical potential, lattice, and temperature	useful for exploring single phase (phase diagram) because you only get a single phase when you have a constant \(\rm{\mu}\)

Thermostat and barostat are the algorithms to control temperature and pressure of the system.

Force Fields¶

Force field is a model to compute interatomic force and total energy. Different force fields are summarized below:

	Classical force fields	Ab initio methods	Machine learning potentials
Model complexity	Simple analytical model	Quantum mechanical model	Machine learning model
Model building	Hard, need to understand deeply for each material	Very easy, no need to build force field	Medium, need to collect data and train the model
Speed and cost	Very fast and very cheap to compute (\(10^6\) to \(10^7\) atoms in nano-seconds)	Slow and expensive: \(10^2\) to \(10^3\) atoms in \(10^2\) pico-seconds	Fast and cheap: \(10^3\) to \(10^4\) atoms in nano-seconds
Transferability	Poor, need to build potential for each system	Good, can be used for most systems, even for unknown system	Depends on your dataset

Machine learning potential (MLP) stays between classical force fields and ab initio methods.