What is the Difference Between Canonical and Grand Canonical Ensemble

The key difference between canonical and grand canonical ensemble is that canonical ensemble describes a system in thermal equilibrium with a heat reservoir at a given temperature, whereas grand canonical ensemble describes a system in contact with both a heat reservoir and a particle reservoir.

Canonical ensemble can be described as the statistical ensemble representing a mechanical system’s possible states in thermal equilibrium with a heat bath at a fixed temperature. Grand canonical ensemble can be described as the possible state of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir.

1. Overview and Key Difference
2. What is Canonical Ensemble
3. What is Grand Canonical Ensemble
4. Canonical vs Grand Canonical Ensemble in Tabular Form
5. Summary – Canonical vs Grand Canonical Ensemble

What is Canonical Ensemble?

Canonical ensemble is the statistical ensemble representing a mechanical system’s possible states in thermal equilibrium with a heat bath at a fixed temperature. The system is able to exchange energy with the heat bath, which can make the states of the system differ in total energy.

When considering the principal thermodynamic variable of the canonical ensemble, it is the absolute temperature expressed by “T,” which can determine the probability distribution of states. This parameter also depends on mechanical variables, including the number of particles in the system that is represented by “N” and the volume of the system given by “V.” These parameters can influence the nature of the system by means of internal states. We can call the ensemble with these three parameters as NVT ensemble.

Moreover, there is another parameter known as the free energy, represented by “F,” which is a constant for the ensemble. However, F and other probabilities may vary upon the selection of different N,V, and T values. There are two important roles of F, and it provides a normalization factor for the probability distribution, and many important ensemble averages can be directly calculated from the function.

What is Grand Canonical Ensemble?

Grand canonical ensemble is the possible state of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. In this state, the system can be described as an open state in the sense that the system can exchange energy and particles with a reservoir, which leads to various possible states of the system that can differ in both the total energy and the total number of particles. Moreover, the volume, shape, and other external coordinates are kept constant in all possible states of the system.

Furthermore, we can give the thermodynamic variables regarding the grand canonical ensembles as chemical potential given by µ and absolute temperature. In addition, this ensemble is dependent on mechanical variables such as volume that can influence the nature of the internal state of the system. We can call the grand canonical ensemble as µVT ensemble.

What is the Difference Between Canonical and Grand Canonical Ensemble?

The key difference between canonical and grand canonical ensemble is that a canonical ensemble describes a system in thermal equilibrium with a heat reservoir at a given temperature, whereas a grand canonical ensemble describes a system in contact with both a heat reservoir and a particle reservoir.

The below infographic presents the differences between canonical and grand canonical ensemble in tabular form for side by side comparison.

Summary – Canonical vs Grand Canonical Ensemble

Canonical and grand canonical ensembles are important terms in thermodynamics. The key difference between canonical and grand canonical ensemble is that a canonical ensemble describes a system in thermal equilibrium with a heat reservoir at a given temperature, whereas a grand canonical ensemble describes a system in contact with both a heat reservoir and a particle reservoir.