State of a system

The work of a curious fellow
   
A set of values of the system variables...

See the Quantum Mechanics notes for information on the state of very small (atom sized or smaller) objects.

In its broadest sense the state of a system at any instant is the information required to reproduce that system as it exists in that instant.

The set of numbers defining the state of a system, for systems made of large objects, is the position and momentum of all the moving parts at the same instant. Applying the laws of Nature to this initial state, allows us to calculate a future state. The accuracy of this prediction of the future depends on the precision with which the initial state was known and the correctness of our calculations.

I have found it convenient to think of two sorts of state changes, which I call active and passive. Imagine a baseball thrown by a pitcher and hit by a batter. Let's assume that we can neglect air resistance and the spin of the ball, clearly not realistic but good enough for this discussion. In this scenario the state of the ball at any instant is a set of six numbers defining its position and momentum in 3-dimensional space.

During the windup and pitch, the pitcher's hand is applying force to the baseball changing its state. I call the state changes from instant to instant during this time active state changes because there is an active participant, the pitcher, involved in the changes. Once the ball is released it follows a path determined by its state at the release point and the curvature of spacetime in the vicinity of the ball. That path will be a geodesic of spacetime. The state changes along the spacetime geodesic are passive state changes. The passive state changes continue until the bat makes contact with the ball. At that point active state changes take over until the bat and ball break contact. From there until the ball hits the fielder's glove or the ground or the bleachers, passive state changes take place.

For some systems the sensitivity to the initial state is so severe that it is not possible to calculate a future state with any useful accuracy even over a short time. These systems are called chaotic. Consider a system consisting of a simple pendulum with friction so small as to be negligible. It has one moving part, constrained to move in a fixed path. It is possible to predict its motion far into the future if we only know its starting position and velocity. Now consider a lottery machine system consisting of 40 numbered ping-pong balls in a box with air swirling through it. Such a system is chaotic so that useful predictions of its future state is not possible. We can confidently predict that when the gate is opened some ball will shortly find its way up the spout and leave the chamber.

Next consider a box filled with 1023 nitrogen molecules banging into each other and the walls of the container. In this case we can only predict that the molecules will be pretty equally distributed throughout the box. Both of lottery machine and box of nitrogen predictions are manifestations of the second law of thermodynamics. The second law just expresses the fact that of all possible future states, the one that occurs is usually the one that is most probable.

Macrostates and Microstates

Let's take a closer look at our box full of nitrogen molecules. The set of numbers identifying the position and momentum of each and every molecule in the box at a particular time is actually a microstate of the system. If I move a single molecule to another location or alter its momentum the new set of numbers is another microstate. With so many molecules and so much random motion among them we may consider every possible microstate equally probable, including the microstates where all the molecules reside in one half of the container.

Why then do we not observe all the molecules in one half of the container for however brief a period of time? Here we need to introduce the notion of a macrostate. Macrostates are collections of microstates that are indistinguishable from one another. If I take two molecules and swap their positions and momenta, since all the molecules are identical the two microstates must be in the same macrostate.

The situation where all the gas is in one half of the container is a macrostate with a huge number of microstates in it. The situation where the gas is uniformly distributed throughout the container is a macrostate with enormously more microstates in it. With all microstates being equally likely, the probability of the occurrence of a particular macrostate is equal to its number of microstates divided by the total number of possible microstates. That makes the probability of the half empty container macrostate so close to zero that the difference may be neglected.