Friendly units for coldness


the information processing advantage
of environments cooler than 98.6 oF

Bruno Maddox's article in the June 2006 Discover Magazine on the Nightmare of Divided Loyalties between Fahrenheit and Centigrade is timely for this age of information. That's because thermodynamics texts following pre-1960 papers by Shannon and Jaynes have increasingly shown that the dog that wags temperature's tail is reciprocal-temperature, or what Claude Garrod has dubbed "coldness". After all, by serving as energy's uncertainty slope i.e. the rate of increase in state uncertainty per unit increase in thermal energy [cf. Am J Phys 71 (2003) 1142], coldness is precisely what makes temperature useful as a measure of equilibration. Thermal physics insights of the last century have also shown reciprocal temperature (1/kT) to have applications that temperature addresses less well. For example, it takes on negative absolute values under population inversion (e.g. of magnetic spins), and bits and bytes turn it into an informatic measure of the thermal ambient (to wit: the mutual information value of free energy) for developing correlations in all sorts of complex systems (e.g. within molecules, galaxies, and biological communities).

How does it work? Basically heat flows spontaneously (by chance maximizing uncertainty) from low to high uncertainty slopes that range between minus and plus infinity, at which end points reciprocal coldness (kT) asymptotically goes to absolute zero. Thus traditional treatments miss: (i) a great reason why absolute zero is hard to reach (nothing gains infinite bytes of state uncertainty per Calorie of heat added), and (ii) that negative absolute temperatures are routinely employed on invertable subsystems (say in the LASER on your DVD reader, or in a domino toppling contest) by moving up the temperature scale (thus avoiding absolute zero). Unlikely inversions (negative uncertainty slopes) even add something to the spectator value of gambling. Taken in reverse, it also means that new correlations between subsystems extract a price in the thermalization of available work. The price decreases if the heat created can be dumped into a colder reservoir, making this observation relevant to molecular engines as well as to humans faced with a warming environment.

If (inspired by Bruno's request for something everyone can relate to) you put coldness (1/kT) into everyday units (Calories and bytes) then you'll find that water freezes at almost precisely* 200 zettabytes per Calorie where zetta is the SI prefix for 1021. In other words, resetting a 200 gigabyte DNA string or computer memory to specified values will convert at least a picoCalorie of available work into ice-melting heat**. The same operation may generate less heat, if the heat can be dumped at lower temperature. Hence the bit depth of a digital camera can be made larger if its CCD is cooled, and when it's cold outside you in principle really can get more work done.

As a result one can say (cf. Figure below) that "chilly Europe" at 0oF is about 15 ZB/Cal colder than ice water, which at 0oC is about 15 ZB/Cal colder than room temperature. In turn room temperature is about 10 ZB/Cal colder than "hot Europe" at 100oF, which is approximately the temperature inside your body. Differences between ambient coldness and "burn" coldness (here 1/kTbasal for our bodies) in fact directly yield the Carnot availability of energy spent i.e. available work (in units of kTambient) per Calorie consumed. Thus our "long term survival zone" extends from about 0 to 40 zettabytes per Calorie colder than 98.6oF. Conductive heat loss gets out of hand for coldnesses greater than that, while runaway heating is a danger for coldnesses less than our basal value. Thus weather persistently outside this range is a fundamental problem for each of us, even though it might not seem so if you mainly see persistent heat or cold from within a temperature-controlled shell.

This coldness measure highlights an important asymmetry. When ambient is over 40 ZB/Cal colder than body temperature, your food-powered metabolism plus some insulation might keep you warm for a while. When coldness is negative with respect to body temperature, however, our metabolism generates waste heat that we must actively export to survive. Food seems less important than finding a way to catch some cooling breeze. Not only have present day humans relied during their evolution on a multi-million year ice age with periodic interglacials, but (regardless of our history) available work can inherently go further (more ZB/Cal of reversible thermalization) when it's cold outside. For example, modern technology will allow us to take advantage of the next glaciation with much more efficient home heating and nearly zero-energy ovens, while it can do much less with limited energy in a period of excess heat.

Other benchmarks: Boiling water's coldness is around 50 ZB/Cal below ice water's 200 ZB/Cal absolute. Even lower uncertainty slopes occur in the 9 ZB/Cal of our sun's surface, the 0 ZB/Cal of a spin system with equal up/down populations, and the -7 ZB/Cal of a He-Ne LASER's 99% excited-Ne inversion. Familiar things with really high coldness include dry ice at 35 ZB/Cal above freezing, liquid nitrogen at nearly 800 ZB/Cal absolute, liquid helium at over 13000 ZB/Cal, and our universe's blackbody ambient at around 20000 ZB/Cal. Thus local issues notwithstanding, the larger world around us is pretty cold! The strange behaviors of liquid nitrogen and helium also reflect high uncertainty slopes, and thus the extreme increases in state uncertainty that result from adding small amounts of heat.

In context of the Discovery article's request for some resolution to the nightmare, energy's uncertainty slope could therefore be the "hero, to misparaphrase Tina Turner" that Bruno was looking for, in the face of summer days with respect to which our blood literally must run cold. More to the point, telling students about physical in addition to historical units for the parameter that equilibrates on thermal contact could further prepare them to recognize other cross-disciplinary connections. Such connections (in cross-cutting fields like nanoscience, informatics, and astrobiology) will likely prove useful for tackling the multiscale challenges that our species faces in the days and millenia ahead.

Of course given our cultures' tenacity for repetition (plus the fact that kT is a useful measure of energy fluctuation in quadratic systems), it is unlikely that bytes per Calorie will become a routine part of the morning weather report anytime soon. In that context, perhaps YOU can instead find a mnemonic measure of kT*** to assist in the clarifying move from historical, to natural, perspectives on that continuum from cool to warm and beyond.


* To wit, energy's Lagrange multiplier dS/dE=1/kT is 200.208 zettabytes/kilocalorie when T = 273.15 Kelvin. Note that 1 kilocalorie equals one food Calorie.

** The state information return on thermal uncertainty (0oC heat) is ≤ 1/kTicewater ~ 200×1021 bytes/Calorie = 200×109 bytes/10-12 Cal = 200 GB/pCal.

*** How about "In Cal/YB, 0oC is 5 with 0oF down by 1/3 and 100oF up by 2/3"? Abiding challenge: Seek habitat within our 1 Calorie/yottabyte range.

Reality Check: Note that room temperature's 5.4 Cal/YB is nothing more than the kT = 1/40 eV/nat familiar for random energy fluctuations at STP.

Bits of reciprocal temperature: The widget below calculates the coldness of the environment relative to the human basal value in natural units (about 175 ZB/Cal absolute) given the temperature in historical units (degrees Fahrenheit). This quantity relates to the information value of available work, as well as to the thermodynamic value of a cool place to dump excess heat.

A citable PDF of this note, with more references, may be found here.
For more on heat capacity in bits, and correlation-based complexity, check out...
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