# Does the microcanonical ensemble have no thermal fluctuations?

After studying statistical mechanics, I understood that thermal fluctuations arise when the system of interest is in contact with a reservoir at some temperature $T$ exchanging energy. Because of this exchange, the energy of the system can fluctuate and its deviation is proportional to $T$.

If this is so, then if a given system is in the microcanonical ensemble, it doesn't have thermal fluctuations (because the microcanonical ensemble has fixed energy and is completely isolated). I find this idea confusing. For example, it is said that magnetization in a magnetic material above the Curie temperature isn't possible because of thermal fluctuations but... that means that if we isolate the system then it can get magnetized at any temperature? This can't be right.

• If the system is isolated, there is no notion of temperature (at least not in the same way). – Norbert Schuch Nov 6 '16 at 12:54

## 2 Answers

The mere presence of thermal fluctuations (in a canonical ensemble) doesn't eliminate the possibility of spontaneous magnetization. The spontaneous magnetization is an average quantity but the actual state of the system will fluctuate about this average. In particular, there are small fluctuations at low but non-zero temperature. You can see this in simulations of the Ising model at low temperature (for instance, see the second half of this video).

Moreover, given the right boundary conditions, you can have a non-zero spontaneous magnetization at sufficiently low temperatures. This is because of the strong correlations between far away spins. At high temperatures, the correlations decay exponentially and so you get zero spontaneous magnetization. This competition between energy and entropy and its dependence on temperature is well illustrated by the Peierls argument.

• I) About the first paragraph: I know that the mere presence of thermal fluctuations doesn't eliminate the possibility of spontaneous magnetization. I specifically said "magnetization in a magnetic material above the Curie temperature isn't possible because of thermal fluctuations". II) About the second paragraph: I don't know where're you going with this, the question was "Can an isolated system get magnetized at any temperature?" You are talking about competition between interactions and thermal fluctuations but the question was precisely if there were any thermal fluctuations at all. – P. C. Spaniel Nov 7 '16 at 5:25
• This answer was meant to elaborate on the comment by Norbert Schuch above. This comment correctly answers your question about the microcanonical ensemble (there is no notion of temperature). The second paragraph is a response to your statement about spontaneous magnetization being impossible above the Curie temperature. The explanation is that this is due to entropic (thermal) effects. For a microcanonical ensemble, the magnetization is fixed (you can think of magnetization as playing the role of "particle number" in a grand canonical ensemble). – Ben Nov 7 '16 at 5:34

That means that if we isolate the system then it can get magnetized at any temperature?

The answer is yes, in some way.

Take a ferromagnet at a temperature $T<T_c$ and take it to a magnetization state $M$ by using an external magnetic field. Now, isolate the system completely. Since $E=E(M)$, will stay in the magnetization state $M$ forever.

However, can we still say that the temperature is $T$ once the system is isolated? Sure, we can formally define temperature as

$$T = \left( \frac{\partial S}{\partial E} \right)^{-1}$$

however it is clear that the state of our system would have been exactly the same if we started the whole process from another temperature $T'$. All the system knows once it's isolated is its magnetization $M$ (and energy, which I assume is a function of magnetization only), which by hypothesis is the same, no matter the temperature where we started from.

Therefore, you can see that the concept of temperature is a bit tricky in an isolated system.