Gibbs rotational ensemble. The Gibbs rotational ensemble represents the possible states of a mechanical system in thermal and rotational equilibrium at temperature  and angular velocity . The Jaynes procedure can be used to obtain this ensemble. An ensemble is the set of microstates corresponding to a given macrostate.

The Gibbs rotational ensemble assigns a probability  to a given microstate characterized by energy  and angular momentum  for a given temperature  and rotational velocity .

where  is the partition function

Derivation
The Gibbs rotational ensemble can be derived using the same general method as to derive any ensemble, as given by E.T. Jaynes in his 1956 paper Information Theory and Statistical Mechanics. Let  be a function with expectation value

where  is the probability of , which is not known a priori. The probabilities  obey normalization

To find , the Shannon entropy  is maximized, where the Shannon entropy goes as

The method of Lagrange multipliers is used to maximize  under the constraints  and the normalization condition, using Lagrange multipliers  and  to find

 is found via normalization

and  can be written as

where  is the partition function

This is easily generalized to any number of equations  via the incorporation of more Lagrange multipliers.

Now investigating the Gibbs rotational ensemble, the method of Lagrange multipliers is again used to maximize the Shannon entropy , but this time under the constraints of energy expectation value  and angular momentum expectation value , which gives  as

Via normalization,  is found to be

Like before,  and  are given by

The entropy  of the system is given by

such that

where  is the Boltzmann constant. The system is assumed to be in equilibrium, follow the laws of thermodynamics, and have fixed uniform temperature  and angular velocity . The first law of thermodynamics as applied to this system is

Recalling the entropy differential

Combining the first law of thermodynamics with the entropy differential gives

Comparing this result with the entropy differential given by entropy maximization allows determination of  and 

which allows the probability of a given state  to be written as

which is recognized as the probability of some microstate given a prescribed macrostate using the Gibbs rotational ensemble. The term  can be recognized as the effective Hamiltonian  for the system, which then simplifies the Gibbs rotational partition function to that of a normal canonical system

Applicability
The Gibbs rotational ensemble is useful for calculations regarding rotating systems. It is commonly used for describing particle distribution in centrifuges. For example, take a rotating cylinder (height , radius ) with fixed particle number , fixed volume , fixed average energy , and average angular momentum . The expectation value of number density of particles  at radius  can be written as

Using the Gibbs rotational partition function,  can be calculated to be

Density of a particle at a given point can be thought of as unity divided by an infinitesimal volume, which can be represented as a delta function.

which finally gives  as

which is the expected result.

Difference between Grand canonical ensemble and Gibbs canonical ensemble 
The Grand canonical ensemble and the Gibbs canonical ensemble are two different statistical ensembles used in statistical mechanics to describe systems with different constraints.

The grand canonical ensemble describes a system that can exchange both energy and particles with a reservoir. It is characterized by three variables: the temperature (T), chemical potential (μ), and volume (V) of the system. The chemical potential determines the average particle number in this ensemble, which allows for some variation in the number of particles. The grand canonical ensemble is commonly used to study systems with a fixed temperature and chemical potential, but a variable particle number, such as gases in contact with a particle reservoir.

On the other hand, the Gibbs canonical ensemble describes a system that can exchange energy but has a fixed number of particles. It is characterized by two variables: the temperature (T) and volume (V) of the system. In this ensemble, the energy of the system can fluctuate, but the number of particles remains fixed. The Gibbs canonical ensemble is commonly used to study systems with a fixed temperature and particle number, but variable energy, such as systems in thermal equilibrium.

References 

Statistical mechanics