Entropic signatures of advanced civilizations

Time to conclude my short study on the entropic limits to advanced civilizations. 

In the first part I argued that life, including consciousness, is neither a state of extremely high entropy nor a state of extremely low entropy. Rather, it exists in between, somewhere in the middle of a big stream of entropy from low to high. In the second part, I theorized that even advanced beings that consist only of information, i.e. disembodied consciousnesses, must create entropy purely by the act of thinking and must dump this excess entropy into the big stream.

Without temperature differences, there is no net current of heat - in other words, to have an entropy stream requires a temperature gradient. For us, the source of the stream is the star around which we live, a high temperature object. Deep space is our cold reservoir, at the chilling temperature of the cosmic background radiation, barely 2.7 degrees above absolute minimum. 

It is the temperature at which the civilization operates that sets the physical limits, rather than the temperature of the star. The maximum "bits per second" rate at which entropy can be dumped (and thus thoughts generated) is proportional to the operating temperature squared. The minimal power (energy consumed per second) that is needed for that is proportional to the operating temperature cubed.

The trick to keep the total number of thoughts infinite, while keeping the total amount of energy consumed finite is to let the operating temperature decrease with time in a very specific manner: it should go down faster than the inverse square root of time, but slower than the inverse cube root of time. 

In addition, the operating temperature must be higher than the cold reservoir. So, that means that it should be higher than the temperature of the cosmic microwave background radiation, and that could pose a problem at some point. But, I speculate that if that temperature is not low enough, a highly advanced civilization can turn around the heat flow, and use its own star as the cold reservoir. An (artificial) black hole will do just fine - its characteristic temperature can be controlled by its radius, and can be brought extremely close to absolute zero. Excess entropy can then be dumped into the black hole. Manipulating spacetime and creating artificial black holes is something the Talocan could do - perhaps not (yet) the Jove. 

So, one can hypothesize that the hallmark of a sedentary hyper-advanced civilization would be a cold star or black hole with a very specific and rather unnatural power law dependence of its temperature over time. 

PS. I am writing down the calculations in a more orderly manner, and will link them here later for the mathematically inclined.

[[ooc attribution: this builds on Freeman Dyson's speculations. In Science 131, p.1303 (1960), he considers a Dyson sphere's infrared radiation as a signature. That is the radiation that corresponds to its operating temperature.]]