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

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?

• Hi Jepsilon. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. – Qmechanic Jul 17 '18 at 15:29

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 $\exp\left(-\frac{2amg}{k_BT}\right)$.