Does the Clausius statement of the second law apply to microscopic phenomena?

If we have two interacting gases of different temperatures, then it may be possible that a packet of particles(*) which move at high speed go from the hot gas into the cold one (by chance) and raise the temperature of the cold gas by transferring heat. However, it is said by the Clausius statement of the second law the heat only flows from a hot body to cold, so the situation outlined must be a violation of the second law. However, if we think of the bulk picture, then of course it is obvious that the heat is transferred from a hot gas into the cold gas.

Before people tell me that temperature is an emergent property, consider a packet of particles large enough for the temperature to be definable, if this

Does this mean that the second law of thermodynamics holds only when considering the macroscopic picture? Or is the second law saying is that it is not possible for such a 'packet' transfer to occur?

I seek a reference for this if it is discussed in a textbook/paper

*: Large enough for the temperature to be definable

• Right off the bat, temperature is a macroscopic property. Individual particle collusions do not change temperature. Oct 4, 2020 at 19:26
• Consider a packet of particles large enough for the temperature to be definable Oct 4, 2020 at 19:28
• The larger the packet the less the deviation between individual particle kinetic energy and the average Oct 4, 2020 at 19:33
• Have to leave, can follow up later Oct 4, 2020 at 19:34

If we have two interacting gases of different temperatures, then it may be possible that a packet of particles(*) which move at high speed...

In a "packet of particles" the individual particles will not all move at a "high speed". If the packet of particles is large enough, then the speeds of the individual particles will vary about the average speed of the packet of particles according to the Stephan-Boltzmann distribution of the speeds about the average.

...go from the hot gas into the cold one (by chance) and raise the temperature of the cold gas by transferring heat.

Not sure what you mean by "go from the hot gas into the cold one (by chance)". But individual particles of the high temperature packet of gas will collide with individual particles of the low temperature gas. On average in the collisions there will be a net transfer of kinetic energy from the high temperature particles to the low temperature particles simply because the average kinetic energy of the particles of the high temperature gas is greater than the average kinetic energy of the particles of the low temperature gas.

However, it is said by the Clausius statement of the second law the heat only flows from a hot body to cold, so the situation outlined must be a violation of the second law.

There is no violation of the second law. Although there may be energy transfer from collisions of higher speed particles of the lower temperature body to lower speed particles of the higher temperature body, the net transfer involving collisions of all the particles will be from the higher temperature body to the lower temperature body.

Before people tell me that temperature is an emergent property, consider a packet of particles large enough for the temperature to be definable, if this does mean that the second law of thermodynamics holds only when considering the macroscopic picture? Or is the second law saying is that it is not possible for such a 'packet' transfer to occur?

If the packet of particles is large enough for the temperature to be definable, that is, for the Stephan-Boltzmann distribution to apply, then the net transfer of energy is from the packet having the higher average kinetic energy (i.e.,higher temperature) to the packet having the lower average kinetic energy. That, however, does not preclude the possibility of energy transfer from some high energy particles of the low temperature packet to some low energy particles of the high temperature packet. That does not violate the second law.

Hope this helps.

• You keep saying 'net' when talking about heat transfer but this point was never explicitly said in any source I've seen from my memory. In the bulk picture, I do agree with you but I'm not convinced on the microscopic picture Oct 5, 2020 at 10:12
• I think I have consistently said net energy transfer and net kinetic energy transfer and not net heat transfer in my descriptions of what is going on at the microscopic level. If somewhere I said net heat transfer I meant energy because heat transfer normally refers to a macroscopic process. If you can’t visualize that in a collection of collisions some particles of the higher temperature body can gain kinetic energy then there is nothing more I can do to convince you on the microscopic picture Oct 5, 2020 at 10:58