## Section: Application Domains

### Computational Statistical Mechanics

The orders of magnitude used in the microscopic description of matter are far from the orders of magnitude of the macroscopic quantities we are used to: The number of particles under consideration in a macroscopic sample of material is of the order of the Avogadro number ${\mathcal{N}}_{A}\sim {10}^{23}$, the typical distances are expressed in Å (${10}^{-10}$ m), the energies are of the order of ${k}_{\mathrm{B}}T\simeq 4\times {10}^{-21}$ J at room temperature, and the typical times are of the order of ${10}^{-15}$ s when the proton mass is the reference mass.

To give some insight into such a large number of particles contained in
a macroscopic sample, it is helpful to
compute the number of moles of water on earth. Recall that one mole of water
corresponds to 18 mL, so that a standard glass of water contains roughly
10 moles, and a typical bathtub contains ${10}^{5}$ mol. On the other hand, there
are approximately $1.3\times {10}^{18}$ m${}^{3}$ of water in the oceans,
*i.e.* $7.2\times {10}^{22}$ mol, a number comparable to the
Avogadro number.
This means that inferring the macroscopic behavior of physical systems
described at the microscopic level by the dynamics of
several millions of particles only
is like inferring the ocean's dynamics from hydrodynamics
in a bathtub...

For practical numerical computations of matter at the microscopic level, following the dynamics of every atom would require simulating ${\mathcal{N}}_{A}$ atoms and performing $\mathrm{O}\left({10}^{15}\right)$ time integration steps, which is of course impossible! These numbers should be compared with the current orders of magnitude of the problems that can be tackled with classical molecular simulation, where several millions of atoms only can be followed over time scales of the order of 0.1 $\mu $s.

Describing the macroscopic behavior of matter knowing its microscopic description therefore seems out of reach. Statistical physics allows us to bridge the gap between microscopic and macroscopic descriptions of matter, at least on a conceptual level. The question is whether the estimated quantities for a system of $N$ particles correctly approximate the macroscopic property, formally obtained in the thermodynamic limit $N\to +\infty $ (the density being kept fixed). In some cases, in particular for simple homogeneous systems, the macroscopic behavior is well approximated from small-scale simulations. However, the convergence of the estimated quantities as a function of the number of particles involved in the simulation should be checked in all cases.

Despite its intrinsic limitations on spatial and timescales, molecular simulation has been used and developed over the past 50 years, and its number of users keeps increasing. As we understand it, it has two major aims nowadays.

First, it can be
used as a *numerical microscope*, which allows us to perform
“computer” experiments.
This was the initial motivation for simulations at the microscopic level:
physical theories were tested on computers.
This use of molecular simulation is particularly clear in its historic
development, which was triggered and sustained by the physics of simple
liquids. Indeed, there was no good analytical theory for these systems,
and the observation of computer trajectories was very helpful to guide the
physicists'
intuition about what was happening in the system, for instance the mechanisms
leading to molecular diffusion. In particular,
the pioneering works on Monte-Carlo methods by Metropolis et al, and the first
molecular dynamics
simulation of Alder and Wainwright were performed because of such motivations.
Today, understanding the behavior of matter at the
microscopic level can still be difficult from an experimental viewpoint
(because of the high resolution required, both in time and in
space), or because we simply do not know what to look for!
Numerical simulations are then a valuable tool to test some
ideas or obtain some data to process and analyze in order
to help assessing experimental setups. This is
particularly true for current nanoscale systems.

Another major aim of molecular simulation, maybe even more important than the
previous one,
is to compute macroscopic
quantities or thermodynamic properties,
typically through averages of some functionals of the system.
In this case, molecular simulation is a
way to obtain *quantitative* information on a system,
instead of resorting to approximate theories, constructed for simplified models,
and giving only qualitative answers.
Sometimes, these properties are accessible through experiments,
but in some cases only numerical computations are possible
since experiments may be unfeasible or too costly
(for instance, when high pressure or large temperature regimes are considered,
or when studying materials not yet synthesized).
More generally, molecular simulation is a tool to explore the links between
the microscopic and macroscopic properties of a material, allowing
one to address modelling questions such as “Which microscopic ingredients are
necessary
(and which are not) to observe a given macroscopic behavior?”