About thermometers and temperature

We discuss a class of mechanical models of thermometers and their minimal requirements to determine the temperature for systems out of the common scope of thermometry. In particular we consider: (1) anharmonic chains with long time of thermalization, such as the Fermi-Pasta-Ulam (FPU) model; (2) systems with long-range interactions where the equivalence of ensembles does not always hold; (3) systems featuring absolute negative temperatures. We show that for all the three classes of systems a mechanical thermometer model can be designed: a temporal average of a suitable mechanical observable of the thermometer is sufficient to get an estimate of the system’s temperature. Several interesting lessons are learnt from our numerical study: (1) the long thermalization times in FPU-like systems do not affect the thermometer, which is not coupled to normal modes but to a group of microscopic degrees of freedom; (2) a thermometer coupled to a long-range system measures its microcanonical temperature, even at values of the total energy where its canonical temperature would be very different; (3) a thermometer to read absolute negative temperatures must have a bounded total energy (as the system), otherwise it heavily perturbs the system changing the sign of its temperature. Our study shows that in order to also work in a correct way in ‘non standard’ cases, the proper model of thermometer must have a special functional form, e.g. the kinetic part cannot be quadratic.