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 .[1] The Jaynes procedure can be used to obtain this ensemble.[2] 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 .[1][3]

where is the partition function

Derivation

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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.[3] 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.[3]

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  ,[3] 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.[1][3][2] 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

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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

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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.[4] 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.[5]

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.[6]

References

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  1. ^ a b c Gibbs, Josiah Willard (2010) [1902]. Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511686948. ISBN 9781108017022.
  2. ^ a b Thomson, Mitchell; Dyer, Charles C. (2012-03-29). "Black Hole Statistical Mechanics and The Angular Velocity Ensemble". arXiv:1203.6542 [gr-qc].
  3. ^ a b c d e Jaynes, Edwin Thompson; Heims, S.P. (1962). "Theory of Gyromagnetic Effects and Some Related Magnetic Phenomena". Reviews of Modern Physics. 34 (2): 143–165. Bibcode:1962RvMP...34..143H. doi:10.1103/RevModPhys.34.143.
  4. ^ "Grand Canonical Ensemble - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2023-05-15.
  5. ^ "LECTURE 9 Statistical Mechanics". ps.uci.edu. Retrieved 2023-05-15.
  6. ^ Emch, Gérard G.; Liu, Chuang (2002). "The Gibbs Canonical Ensembles". In Emch, Gérard G.; Liu, Chuang (eds.). The Logic of Thermostatistical Physics. Berlin, Heidelberg: Springer. pp. 331–372. doi:10.1007/978-3-662-04886-3_10. ISBN 978-3-662-04886-3. Retrieved 2023-05-15.