Physics Asked on April 12, 2021
While studying Statistical Mechanics, I learnt about Microcanonical Ensembles, and how they have a uniform probability distribution for the microstates. While I understood the rationale for it (the Principle of Indifference), what I didn’t understand was that why couldn’t this same logic be applied to either the Canonical or Grand Canonical Ensemble?
In other words –
What’s so special about a Microcanonical Ensemble, as compared to a Canonical or Grand Canonical Ensemble, that we can apply the principle of indifference to get a constant probability distribution for the microstates for it (and only it).
The Principle of Indifference can be considered a particular case of the Maximum Entropy principle (maxent). From the maxent principle, it is immediate to obtain other ensembles, with a non-uniform probability distribution, just adding additional constraints on the ensemble.
I'll sketch the derivation in the case of a discrete and finite set of states. The generalization to continuous distribution is quite straightforward, although technically less simple.
Let start from the information theory entropy of a probability distribution ${p_i}$ ($p_i$ is the probability of the $i$-th state and ${p_i}$ indicates the set of probabilities of all the states of the system.
Without additional constraints, the maxent method requires to minimize the following expression for the average information (Shannon's entropy): $$ H({p_i})= -sum_i p_i log p_itag{1} $$ under the obvious constraint that $$sum_i p_i=1tag{2}$$ which can be implemented with the technique of Lagrange multipliers.
The probability distribution corresponding to the maximum is $p_i=constant$, i.e. we have the same result as the Principle of indifference.
If, in addition to the normalization we assure that the ensemble is made by micro-states that can exchange energy with the constraint that the average energy is fixed, we have an additional constraint: $$ sum_i p_i E_i = E,tag{3} $$ that can be included with an additional Lagrange multiplier. The result of minimizing the function $(1)$ with the two constraints $(2)$ and $(3)$ is the canonical distribution function and the Lagrange multiplier can be written as $1/k_BT$ once the proper connection with thermodynamics has been done.
The Gran Canonical ensemble can be derived by adding the third constraint on the average number of particles.
Therefore, within such an approach to the ensembles, there is nothing special with the microcanonical ensemble.
Correct answer by GiorgioP on April 12, 2021
A canonical ensemble is not an isolated system - it is in contact with a heat bath kept at temperature $T$. It can exchange energy with the bath, and so the system energy is not actually fixed but can fluctuate a little. The mean energy is governed by the bath temperature $T$ of course. Now, the system can be in different energy states since it can fluctuate, so the question is what probabilities should we assign to each energy? A simple though experiment about an 'extreme' case where the system can in principle have many very high energies but the bath is kept at a low temperature suggests that we should not assign every state equal probability but it must depend on the energy somehow.
The same reasoning goes for every quantity that is not held fixed. If your system is exchanging particles with a particle reservoir (eg in quantum field theories your particle number is not fixed since the vacuum can create particle-anti-particle pairsand so acts as a bath) then you must assign different probabilities to states with different particle numbers.
A microcanonical ensemble is suited to the case where nothing is fluctuating, and you have perfect precision over each variable defining your microstate. As such, there is simply no variation in energies to be assigning probabilities on the basis of.
Answered by jacob1729 on April 12, 2021
In the microcanonical ensemble, you select all states with a given energy $E$. Assume you get $N$ states with that energy. Because you need to conserve energy, only the states with such an energy can be visited by the system.
To this, you add the "principle of indifference", saying that the system spends the same amount of time in each state, and you get a probability distribution $$rho sim 1/N$$ So far so good.
In a way, the universe is a microcanonical ensemble, although pretty big. The total energy is conserved. However, what happens if take a big micro-canonical ensemble with energy $E_0$, we cut a small parte of the universe, we allow it to exchange heat (and only heat) at constant temperature with the universe and see what happens? Imagine the system we have considering has energy $E_1$ inside it. The rest of the universe has energy $E_2$. The only thing we know for sure, is that $$E_1+E_2=E_0$$ at all times, but because the small system can exchange energy with the outside, it can "borrow" some of the energy or it can give some of its energy, so that $E_1$ and $E_2$ can both change (but only keeping $E_0$ constant!).
The constraint that the total energy is conserved now only holds for the universe, but the small space we are considering can change its energy by exchanges with the universe. So it can fluctuate.
It can, but does it? In principle it could be that the small system is sort of micro-microcanonical: it stays at constant energy $E_1$ so that principle of indifference holds. But in the event that the energy fluctuates, indifference can not hold anymore as low-energy states will have to be favored. Hence the form of the canonical probability distribution allowing for states with different energy.
So now you ask yourself, can I quantify this? You basically take the global probability distribution. You marginalise it taking into account only the small sub-system you considered. You do some math under the assumption $E_1ll E_2$ and find the Boltzmann distribution allowing for fluctuations. You will probably do it in detail in your course. Similar reasoning goes for the Grancanonical where you can exchange particles.
Summing up, what is special in the canonical ensemble is that as it can exchange energy with the outside, it can populate states of higher energy than expected by borrowing energy from the outside, so the states are not indifferent anymore. Temperature gives you a measure of how much you can fluctuate. Everytime you allow you system to vary something (with some constraints) you get a different system which (if allowed by the constraints) can fluctuate and violate "indifference" (or rather, you need to "weight" indifference with some extra variable)
Answered by JalfredP on April 12, 2021
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