Temperature and Pressure

Note to self. I have to think harder about the simple idea that Temperature and Pressure are emergent properties of the interaction of elementary particles as their number approaches the thermodynamic limit.

On the microscopic scale, temperature is defined as the average energy of microscopic motions of a single particle in the system per degree of freedom. On the macroscopic scale, temperature is the unique physical property that determines the direction of heat flow between two objects placed in thermal contact. If no heat flow occurs, the two objects have the same temperature; otherwise heat flows from the hotter object to the colder object. These two basic principles are stated in the zeroth law and second law of thermodynamics, respectively.

Pressure (symbol: 'p') is the force acting on an area applied to an object in a direction perpendicular to the surface.

Pressure seems like it doesn't necessarily need different definitions for the micro and macro scales. And the two definitions for temperature seem like they would be better classified as technical/intuitive, rather than micro and macro.

So Pressure is only an emergent property in the most trivial sense, because and individual particle can exert pressure on a container and the pressure of a gas scales linearly with the number of particles (The linear scaling, in my own nomenclature, is why is say that the emergence is trivial).

The triviality of the scaling holds with temperature, but not the first criterion. Individual particles don't have temperature for the same reason that individual people don't have a public opinion. Public opinion is the average opinion of a population of people. It is defined in terms of a group of people, and only has meaning when talking about a group of people.

Temperature is an emergent property in the sense that it is only meaningful in the context of a collection of particles. This is a stronger emergence than pressure exhibits, but for some reason, yet to be determined, it is weaker than most examples of what people typical cite as prototypical cases of emergence.