Ensemble Theory(classical)-VI (Equivalence of Ensembles and Fluctuations)
P.K. Ahluwalia
LEARNING OUTCOMES
After studying this module, you shall be able to:-
- See how the types of ensembles viz micro-canonical, canonical and grand canonical ensemble are equivalent.
- Statistically calculate these fluctuations in canonical and grand canonical ensemble without invoking thermodynamics by calculating mean square fluctuations, <(x-<x>)2 > , also known as variance of a physical quantity x, which is a measure of spread or dispersion of probability distribution about the average value.
- Appreciate that a system in equilibrium keep on going from one microstate to another, without going very far, leading to very small fluctuations in close vicinity of equilibrium state.
- Calculate variance of physical quantities such as total energy and total number of particles in different ensembles and see that these lead to same result for relative fluctuation establishing their equivalence.
- Establish the fact that in equilibrium statistical mechanics fluctuations are so small that probability of observing any value other than mean value is very very rare.
- Know under what circumstances fluctuations become important.
- Further establish this equivalence by deriving Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann distributions from micro-canonical ensemble.
- Derive a generic equation of state for an ideal gas for each of the three statistics.
1. INTRODUCTION
In modules XII, XIII and XIV and XV we dwelt in detail about micro-canonical, canonical and grand canonical ensembles and laid down a systematic procedure to derive thermodynamic properties and applied theses to study a variety of interesting physical macroscopic systems such as Classical ideal gas. We also saw that how this procedure helps us to calculate average values of physical parameters of the studied macroscopic physical systems. The most interesting aspect of the study has been that these three different ensembles took us from completely isolated system to closed system to a completely open system yet gave same expressions for entropy, free energy and equation of state, for example for our prototype system called classical ideal gas. In this module we proceed to seek an explanation for this interesting convergence to the same results though three ensembles representing three different physical situations. We can also seek this equivalence by calculating variance or mean fluctuations of energy in the case of canonical ensemble and number of particles in the case of grand canonical ensemble from their respective mean values leading to a result that relative fluctuations from the mean values are extremely small for macroscopic systems. We shall build on the general considerations about fluctuations which we had studied earlier in module XI. In the end we shall also look at some examples where these may become important and should not be ignored. We further establish the equivalence of the three types of ensembles by deriving Fermi-Dirac, Bose- Einstein and Maxwell distribution functions by using micro-canonical ensemble to see whether we get the same result as we got for these distributions by grand canonical ensemble.
2. Equivalence of Different Ensembles
We approach this equivalence from point of view of fluctuations. Let us recall the results which we derived in module XIV on canonical ensemble for calculating average energy (Table 1) and module XV in grand canonical ensemble for calculating average energy and average number of particles (Table 2).
References:-
1. Pathria R.K. and Beale P. D., “Statistical Mechanics”, 3rd Edition, Oxford: Elsiever Ltd., 2011
2. Kubo R., “Statistical Mechanics: An Advanced Course with problems and Solutions,” Amsterdam: North Holland Publishing Company, 1965.
3. Panat P.V., “Thermodynamics and Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008
4. Pal B.P., “ An Introductory Course of Statistical Mechanics, New Delhi: Narosa Publishing House Pvt. Ltd., 2008
5. Brush S.G., “Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager”, New Jersey: Princeton University Press, 1983.
6. Fermi E, “Thermodynamics,” Dover Publications, 1956.
7. Fermi E., “Notes on Thermodynamics and Statistics,”The University of Chicago Press, 1966.
Web Links:-
Demo of Critical opalescence resulting from fluctuations near critical point http://www.doitpoms.ac.uk/tlplib/solid-solutions/demo.php
A site devoted to teaching Statistical and thermal physics It has many links to simulations, research articles, demos and talks http://stp.darku.edu/