What conditions do a bunch of atoms need to satisfy to have a temperature?
Suppose that we have a beam of helium atoms travelling in a common straight line, equally spaced with the same velocity. If we place a thermometer in the path of this beam, the atoms will impinge on it and heat it up, to register a temperature. However, if we move the thermometer along with the beam, the indicated temperature can be reduced, ultimately to zero.
Now add a second collimated beam, exactly opposite to the first one, on a very close parallel path. In this case, whichever way we move the thermometer, a positive temperature will register, and it seems reasonable to assign the minimal such reading to be the temperature of this "beam dipole". Is this the simplest possible, or least-entropic, system that can be said to have a temperature?
Added in response to replies: I am troubled by the need for entropy to be maximized as a condition for temperature to be meaningful. The above scenario has the unrealistic quality of an idealized micro-experiment (as Maxwell's demon might carry out), which puts it out of range of classical statistical mechanics. So let's consider a macro-scale (even industrial-scale) alternative, at the risk of introducing features that may distract from the essential idea of defining temperature.
Suppose that we have a gas-phase chemical reactor. The aim is to combine the primary reactants and cool the product as quickly as possible so as to minimize secondary reactions among the products. This is done by injecting two symmetrically opposed, narrow, very-high-speed jets of cooled gaseous reactants under a bath of inert liquid coolant. The reaction rates at any given pressure, temperature, and concentration of reactants and products are well known. (For convenience, we may assume that the reaction is neither exothermic nor endothermic, but the change in entropy favours the reaction at the designed parameter settings.) When the reaction rate is measured, there is an implied temperature profile in the reaction chamber, assuming a classical chemical-thermodynamic--fluid-dynamic model of the process (which would surely be very complicated, but whose details need not concern us!).
New question: Is this implied temperature profile fictitious?