Photon non conservation leads to the transfer of heat from low temperature to high temperature without consuming external energy.
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Container A contains 2mol/L of CO2, while container B contains 1mol/L of CO2.
The photon density at point A is greater than that at point B.
Radiation energy ranges from A to B.
Photons are generated at point A and annihilated at point B.
B to A can also reflux energy through thermal conduction.
This forms an energy cycle, with a temperature difference and no need for external energy consumption.
dear Miao,
a reconsideration of the setup might be necessary.
in principle, a heat conduction required a contact.
and a heat radiation is neither a transfer of photons nor any other particles. for evidence, vacuum can contain heat.
moreover, temperature is an intensive property. meaning that the molecular density does not affect the its value at thermal equilibrium.
in other words, once a thermal equilibrium is reached in an adiabatic realm by whatever mean like conduction, radiation, ... a part of the system with less molecular density will have the same temperature as the part with a higher molecular density.
regards,
There are several mistakes in the argument.
You do not simply transfer heat by generating photons in system A and absorbing them in system B. If you do that, for example, using a laser in A, then the energy that you transfer is not heat, as it is directed and not thermal radiation.
On the other hand, if you use thermal radiation from A to B, then the photon density will not be larger at A, if A is at lower temperature. If the radiation can go from A to B, radiation can also go from B to A, and if B is hotter, more energy will be transferred from B to A than from A to B by radiation. Add the conduction in, then you know where the energy will go.
So your main mistake is to not carefully distinguish between heat and other forms of energy (with less entropy). There may be no "mechanical" work done, if all the energy exchange between the two reservoirs is via photons. But there are other forms of energy exchanged than heat, and these must also be taken into account in a proper formulation of the second law. One statement of the second law is that there is no process that allows the transfer of heat from a cooler to a hotter reservoir without doing anything else. If you send photons that are not thermal from the cool to the hot reservoir, you are not only transferring heat. You are, in fact, not using reservoir A as a reservoir at all (except for the reception of photons from B; the non-thermal photons do not constitute part of a heat reservoir, albeit the laser may be embedded in the reservoir).
Anyway, there is a well-developed quantum theory of radiation with many interesting results (for example that the pressure of a photon gas depends on its temperature only, not its volume), and I assure you that none of these results implies a violation of the second law of thermodynamics.
Bo Miao "Gas radiation has very simple and direct knowledge: gas density is high, radiation is strong, which is easy to imagine."
Easy to imagine but still sometimes wrong. The power of thermal radiation is proportional to the fourth power of the temperature, the area of the (black) radiator and the Stefan-Boltzmann constant. So if you increase the density of the gas but decrease its temperature, the radiation goes down, not up. It does not depend on the density directly, but only in the way that on increasing density you often increase the temperature as well (if you compress a gas adiabatically, if will of course heat up; but if you cool it strongly enough during the compression it may have a higher density and lower temperature after compression).
"The photon density inside the gas is directly proportional to the number of radiative structures."
No. The total energy density of the thermal photons is simply proportional to T4 (and independent of how many radiative structures were necessary to get to that temperature), the spectral energy density is given by the Planck distribution (the integral of which over frequency is proportional to T4), and the number density of photons of a given frequency is given by dividing the spectral energy density by h times the frequency.
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