# Molecular Simulation/Thermodynamic ensembles

The entire universe is governed by the laws of thermodynamics through the transfer of energy between matter. This process is too complex to consider directly, instead we consider several simplifications where part of the universe (i.e., the system) is considered separately from the rest of its surroundings. The system houses the process or thermodynamic state under consideration, whereas the surroundings contain everything else.

The system can be described using an ensemble, also known as a statistical ensemble, which is an idealization consisting a large number of copies of a system, considered all at once, each of which represents a possible state that the real system might be in. A thermodynamic ensemble is a statistical ensemble that is in statistical equilibrium. It provides a way to derive the properties of a real thermodynamic systems from the laws of classical and quantum mechanics.

We can consider different systems with different degrees of separation from their surroundings, from completely isolated (e.g., microcanonical ensemble) to completely open to its surroundings (e.g., grand canonical ensemble). However, for the purposes of molecular simulations only three main ensembles will be considered. Namely, the microcanonical, canonical, and isothermal-isobaric ensembles

## Microcanonical ensemble

The microcanonical ensemble is a system that is completely isolated from its surroundings. There is no transfer of energy or matter between the system and the surroundings, and the volume of the system remains fixed. In other words, this is a physical system where the total number of particles (N), the total volume (V), and the total energy (E) are constant. The microcanonical ensemble is typically abbreviated NVE.

The total energy of the system is constant, but there is no defined temperature of the microcanonical ensemble. Temperature can only be defined for systems interacting with their surroundings. Isolated systems, such as microcanonical, are only described in terms of their composition, volume, and total energy. In statistical mechanics, a microcanonical ensemble is used to represent the possible states of a system which have a specified total energy.

## Canonical ensemble

In the canonical ensemble, the volume of the system is fixed, and energy can transfer across the boundary between system and surroundings, but matter cannot. We can describe systems as being immersed in a heat bath at a temperature (T), where the heat bath is much larger than the system. No amount of heat put off by the system will significantly increase the temperature of the surroundings. The canonical ensemble applies to systems of any size; while it is necessary to assume that the relative size of the heat bath is very large, the system itself may be small or large. Now because the system and surroundings are in thermal contact, the system will transfer heat (q) to and from the surroundings until they are in thermal equilibrium. Therefore, unlike the microcanonical ensemble, the temperature of the canonical ensemble can be a defined constant (T). This ensemble is typically abbreviated in terms of constant number of particles, total volume, and temperature (NVT).

The canonical ensemble is a statistical mechanics ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at some fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ only in total energy. In the microcanonical ensemble, the total energy is fixed, but in the canonical ensemble the energy is no longer constant. It can take on a range of values depending on the temperature. The canonical ensemble is important when attempting to describe the Helmholtz free energy of a system, which is the maximum amount of work a system can do at a constant volume (V) and temperature (T).

## Isothermal-Isobaric ensemble

In the isothermal-isobaric ensemble, energy can transfer across the boundary, but matter cannot. The volume of the system can change such that the internal pressure of the system matches the pressure exerted on the system by its surroundings. Similar to the canonical ensemble, we can describe the isothermal-isobaric ensemble as a system immersed in a heat bath at a temperature (T), where the heat bath is much larger than the system. No amount of heat put off by the system will significantly increase the temperature of the surroundings.

The isothermal-isobaric ensemble is a statistical mechanical ensemble that maintains a constant total number of particles, and constant temperature (T) and pressure (p), typically abbreviated NpT. This ensemble plays an important role in chemistry since the majority of important chemical reactions are carried out under constant pressure conditions. The isothermal-isobaric ensemble is also important when attempting to describe the Gibbs free energy of a system, which is the maximum amount of work a system can do at constant pressure (p) and temperature (T).