Rigorous definition of Hysteresis I have been studying thermodynamics and have come across this concept of hysteresis but am struggling to find a rigorous way to define it. Here are the types of definitions I have come across


*

*Say a process takes a system from one point in a state space to another and then back. Upon return, if some physical quantities have changed, there was hysteresis. 


This doesn't make sense to me because why not extend the state space to include the changed factor? If energy changed, add that as a dimension to the state space and in fact you did not return. It's not hysteresis then. In other words, how does one define the state space rigorously then?


*The same as before but instead of saying physical quantities of the system changed, the surroundings have changed.


I feel like this is just wrong because, often, hysteresis does refer to changes in the system itself like the energy change due to friction or something.


*Take some process p and conduct it on the system. Now conduct "p^-1" which is basically do it backwards i.e. pull the piston up and then down. If the system's state does not return to the same point, then there is hysteresis. 


My problem with this is that I'm not sure how well-defined "p^-1" actually is or if, in general, it makes sense to think about it. What if there existed a process where there was no obvious way to do it backwards but there still existed friction and other hysteresis-creating entities. Does it have hysteresis or no? I find it hard to believe that the existence of hysteresis is dependent on the process's ability to be done in reverse but this definition I think is closest.


*Hysteresis is the dependence of the state of the system on its history.


To be honest, not even sure what this means. This could mean that either a) now there are two entities that uniquely describe the system's current condition: its position in state space AND its history (which I don't think it's saying) or b) the state it is in is determined by it history but this means the history doesn't actually add information about the system i.e. if the magnetic moment is different based on how it got there, simply adding the magnetic moment as another dimension to the state space should make the history irrelevant.
If someone could please help me see what I am missing or let me know if hysteresis is simply ill-define in a general sense, that would be greatly appreciated!
 A: 
This doesn't make sense to me because why not extend the state space to include the changed factor?

Because the hysteresis is defined relative to set of variables that are directly controllable by experimenter. There are usually additional state variables that are necessary to determine thermodynamic state, but are not directly controllable. For example, H field in an iron torus wound with current-carrying wire is directly controllable, but B field or magnetization M is not. When H field is manipulated to get from 0 to maximum value and back, the system's magnetization does not get to its initial value. That is hysteresis.
A: 
Hysteresis is the dependence of the state of the system on its history.

What this should really say is that the time evolution of the system depends on its history. This is to contrast things like classical mechanics, where all you need to know is the current state of the system to know how it will evolve. 
For hysteresis, it's not enough to know the current state to know how it will evolve over time. You have to know how you got to that state (its history) to know where it will go next.
I think this one and the first one are the best definitions/descriptions, but another answer already covers the first one so I thought I'd talk about this one.
