A question about temperature in the concept of a macroscopic statistical-mechanical system

Question

I recently have come across a question while working on Statistical Mechanics. The question itself was quite straight forward (no this is not a "do my homework" question in case you were wondering) and the answer was quite intuitive which agreed to the given solution in this book page 248, however I have a question regarding assumed definitions/concepts. The question reads as follows:

A one dimensional chain is hung on a ceiling. One of its extremes is fixed, while the other holds a mass $M$. Gravity is acting along the negative $z$ direction. The chain is formed by two kinds of distinguishable rings: they are ellipses with the major axis oriented vertically or horizontally. The major and minor axes have lengths $l + a$ and $l − a$ respectively. The number of rings is fixed to $N$ and the chain is in thermal equilibrium at temperature $T$. Find the average energy and the average length of the chain. Comment on both low and high temperature results.

Now my question is this, is the temperature $T$ the real physical temperature that we talk about when considering the rest of thermodynamics or is it something else that is acting like a "temperature"?

The reason I ask this is because it seems to be a bit out of the ordinary for temperature to interact in such a manner with what is essentially a mechanical system. For example at high temperatures one even obtains an analogue of Curie's Law $\frac{\partial \langle L\rangle}{\partial w}=\frac{Na^2}{k_b T}$ where $w=Mg$. Am I wrong in interpreting this as saying that increasing the temperature lowers the weight's contribution to the average length?

Answer 1

Yes the quantity in the question is the real physical temperature but for a macroscopic chain you would not expect to observe this type of fluctuation as they occur so rarely. It would require the bottom link to "just happen" to obtain that much energy from its interactions with the environment, which for an everyday chain is very unlikely to happen.

To illustrate lets say that a link in the chain has a mass on the order of say $10\;\mathrm{g}$ and and $a$ is around $1\;\mathrm{mm}$. Then the energy needed to rotate the bottom link in the chain is $2amg \sim 2\times 10^{-4}\;\mathrm{J}$. This corresponds to a temperature $T = \frac{2amg}{k_B} \sim 10^{19}\;\mathrm{k}$ at which you expect this type of fluctuation to occur frequently. Below this temperature fluctuations are exponentially suppressed as $\exp\left(-\frac{2amg}{k_BT}\right)$.

In the high temperature all states are equally probable, regardless of energy, so you expect about half the links to be orientated one way and half the other. This means that you would expect the weight to have a smaller and smaller impact at very high temperatures.