Heat death paradox
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Formulated in 1862 by Lord Kelvin, Hermann von Helmholtz and William John Macquorn Rankine,  the heat death paradox, also known as Clausius' paradox and thermodynamic paradox, is a reductio ad absurdum argument that uses thermodynamics to show the impossibility of an infinitely old universe.
Assuming that the universe is eternal, a question arises: How is it that thermodynamic equilibrium has not already been achieved?
This paradox is based upon the classical model of the universe in which the universe is eternal. Clausius’ paradox is a paradox of paradigm. It was necessary to amend the fundamental ideas about the universe, which brought about the change of the paradigm. The paradox was solved when the paradigm was changed. The paradox which is valid in the classical stationary model of the universe is not so in Fridman’s nonstationary relativistic model. This is a solved paradox.
The paradox was based upon the rigid mechanical point of view of the Second principle of thermodynamics postulated by Rudolf Clausius according to which heat can only be transferred from a warmer to a colder object. If the universe was eternal, as claimed in the classical stationary model of the universe, it should already be cold. 
Any hot object transfers heat to its cooler surroundings, until everything is at the same temperature. For two objects at the same temperature as much heat flows from one body as flows from the other, and the net effect is no change. If the universe were infinitely old, there must have been enough time for the stars to cool and warm their surroundings. Everywhere should therefore be at the same temperature and there should either be no stars, or everything should be as hot as stars.
The paradox does not arise in Big Bang, steady state or chaotic inflationary cosmologies. In Big Bang cosmology, the universe is not old enough to have reached equilibrium. Steady state and chaotic inflation escape the paradox by expanding. Radiation is continually being red-shifted by the expansion, causing background cooling. Thus, they balance entropy production, and are in eternal equilibrium.
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