FAQ: What is a state? How does one determine what is a state and what is not?

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Sean Humbert, 7 Oct 02

Generally, one can think of the "state" of a system as the minimum number of variables (or pieces of information) required to predict the future. This is usually dictated by the type of mathematical model (finite state machines, difference equations, or differential equations) used to describe the evolution of the system. The choices for variables to include in the state is highly dependent on the fidelity of the model and the type of system. One can see immediately that the choice of variables to be included in the state is not unique. When we talk about physical systems modeled by differential equations, such as masses and springs, electric circuits or satellites (rigid bodies) rotating in space, we can attach some additional intuition: the variables in the state should be adequate to specify the energy of the system. (This is just a rule of thumb, not a strict principle!). For example, take a ball free-falling to earth: we can specify the position of the ball by specifying the height (h) above the ground, but we also need to include the velocity of the ball (dh/dt) to specify the total energy (E = 1/2*m*(dh/dt)^2 + mgh). Therefore, the state of the ball is (h,dh/dt).

In a second example, consider a cup of hot coffee. If we were interested only in a low fidelity model, we could take the average temperature of the liquid as the state, as the total energy of the system is thermal in nature. This model would be valid for answering questions like, "Is this coffee too hot to drink?". However, if we wanted to answer more complicated questions such as, "What happens to the vorticity in the coffee as the system cools down", we would need to include in the model a description of the motion of the fluid, and the state would need to be modified to include the position and velocity of the fluid particles in the coffee.