section_path: Measurement/Continuous or discrete
section_name: Continuous or discrete
paragraph_text: Continuous or discrete
  
  When one measures the variation of temperature across a region of space or time, do the temperature measurements turn out to be continuous or discrete? There is a widely held misconception that such temperature measurements must always be continuous. This misconception partly originates from the historical view associated with the continuity of classical physical quantities, which states that physical quantities must assume every intermediate value between a starting value and a final value. However, the classical picture is only true in the cases where temperature is measured in a system that is in equilibrium, that is, temperature may not be continuous outside these conditions. For systems outside equilibrium, such as at interfaces between materials (e.g., a metal/non-metal interface or a liquid-vapour interface) temperature measurements may show steep discontinuities in time and space. For instance, Fang and Ward were some of the first authors to successfully report temperature discontinuities of as much as 7.8 K at the surface of evaporating water droplets. This was reported at inter-molecular scales, or at the scale of the mean free path of molecules which is typically of the order of a few micrometers in gases at room temperature. Generally speaking, temperature discontinuities are considered to be norms rather than exceptions in cases of interfacial heat transfer. This is due to the abrupt change in the vibrational or thermal properties of the materials across such interfaces which prevent instantaneous transfer of heat and the establishment of thermal equilibrium (a prerequisite for having a uniform equilibrium temperature across the interface). Further, temperature measurements at the macro-scale (typical observational scale) may be too coarse-grained as they average out the microscopic thermal information based on the scale of the representative sample volume of the control system, and thus it is likely that temperature discontinuities at the micro-scale may be overlooked in such averages. Such an averaging may even produce incorrect or misleading results in many cases of temperature measurements, even at macro-scales, and thus it is prudent that one examines the micro-physical information carefully before averaging out or smoothing out any potential temperature discontinuities in a system as such discontinuities cannot always be averaged or smoothed out. Temperature discontiuities, rather than merely being anomalies, have actually substantially improved our understanding and predictive abilities pertaining to heat transfer at small scales.