 We've seen that when a ball moving at an angle theta-0 strikes a fixed ball, it bounces off at an angle of about theta-1 equals 2L over D times theta-0, where D is the ball diameter and L is the horizontal distance traveled. After n collisions, the ball travels with angle theta-n equals 2L over D to the nth power times theta-0. We solve this for the theta-0 value that results in the moving ball undergoing n collisions before flying off into space. The result is theta-0 equals D over L times the nth power of D over 2L. Let's examine this process using our numerical solution of the equations of motion. Here we run four cases. At lower right, we have theta-0 equals 0, and the red ball should collide with the blue balls an infinite number of times. For the other three cases, we use our formula to set an initial angle theta-0 that will produce n equals 2, 4, and 6 collisions. Here is one collision, two, three, four, five, and six. Our theta-0 formula works as intended. Let's watch that again. Notice how small the vertical deviation of the red ball is until near the final collision. Here is one collision, two, three, four, five, and six. We threw a wrinkle into the n equals 4 case. The tiny theta-0 tilt of the initial velocity was pointed downward instead of upward. As a result, the red ball eventually flew off in the downward direction. This highlights how an extremely small change in the initial state can produce a radically different future state. Our formula tells us how small theta-0 needs to be for the moving ball to undergo n collisions. Taking the diameter of molecules to be three angstroms and the mean free path to be 1,000 angstroms, inside the parentheses we get 3 over 2,000 after canceling the common factor of 10 to the minus 10. Let's round this to 1 over 1,000 and neglect the leading d over l factor. Then theta-0 is roughly 1 over 1,000 to the nth power. If molecules undergo 4 billion collisions per second, then for 1 microsecond n will be 4,000. Since 1,000 is 10 to the 3, 1,000 to the n is about 10 to the 3 times 4,000 or 10 to the 12,000. Let's round this to 10 to the 10,000. This suggests that to predict the motion of molecules for one microsecond, we need to know their direction of motion to within an angle of about 1 over 10 to the 10,000. How small of an angle is this? Imagine that the molecule is traveling in the horizontal direction with velocity v and theta-0 exactly zero. But it's subject to a small acceleration in the vertical direction. After a time t, this will produce a vertical velocity component of A t. And the molecule will end up traveling at an angle theta-0 approximately equal to A t over v. If we set this equal to 1 over 10 to the 10,000, substitute 1 microsecond for t and 400 meters per second for v, then we can solve for the acceleration A that will undermine our ability to predict the motion for more than about one microsecond into the future. Because 10 to the 10,000 is such a huge number, the resulting acceleration is so small that it's dwarfed by the gravitational field of an electron at the edge of the observable universe. We conclude that it is impossible to specify the microstate of a real gas and to predict its evolution using equations of motion in any physically meaningful sense. That's the bad news. The good news is we don't need to. We've already seen that to determine important macroscopic quantities, such as temperature, we only need to know a single statistic about molecular motions, the mean square velocity, the square velocity averaged over all molecules. This is a simple example of what came to be known as statistical mechanics, using the laws of physics not to precisely describe the individual motion of each molecule, but only to determine their overall statistics. We will consider statistical mechanics in a future video. The chaotic nature of molecular motion raises a foundational, even philosophical question. Is classical physics deterministic? Max Born considered this question. Born played a central role in developing the probabilistic interpretation of quantum mechanics. As we discuss in the quantum mechanics video series, quantum mechanics does not make exact predictions about the evolution of atomic scale systems, but only gives the probabilities of certain events occurring. Many physicists objected strenuously to this view of nature. Einstein famously objected that God does not play dice. In his Nobel Prize lecture, Born considers this issue. He notes that during the development of classical physics, mechanical determinism gradually became a kind of article of faith. A famous expression of the kind of faith in the determinism of classical physics that Born is referring to was given by Laplace in 1814. He wrote, An intellect which at a certain moment would know all forces that set nature in motion and all positions of all items of which nature is composed, if this intellect were all so vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom. For such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. However, in light of results such as those we've just considered, Born points out that determinism lapses completely into indeterminism as soon as the slightest inaccuracy in the data on velocity is permitted. Is there any sense, and I mean any physical sense, not metaphysical sense, in which one can speak of absolute data? His conclusion is, the determinism of classical physics turns out to be an illusion created by overrating mathematical logical concepts. It is an idol, not an ideal in scientific research. In light of this, the probabilistic character of quantum mechanics is, arguably, not such a radical departure from classical physics as it actually functions in practice at the atomic scale. Although we can't know the precise microstate of a gas, we can certainly measure many of a gas's properties, such as pressure and temperature, which we will call state variables. To develop a theory of thermodynamics, it's enough to know that the macroscopic state of a system in equilibrium is described by various state variables, such as pressure, volume, number of molecules, temperature, and internal energy. These are also sometimes called state functions. For an ideal monatomic gas of n atoms, the state is fully specified by pressure and volume alone. This is because any other state variable can be expressed as a function of p and v. For example, the ideal gas law tells us that temperature equals pv over nk, and likewise for other state variables. Therefore, the state of a system is completely represented by a point on a pv diagram. For example, all properties of, let's say, state one of the system are fixed by the corresponding pressure p1 and volume v1. The system can evolve from one state to another along different paths on the pv diagram, but the final state, say state two, is independent of the path. It depends only on the final pressure p2 and volume v2. We will denote very small differential changes in state variables with a d prefix. For example, dp, dv, and dt represent very small changes in pressure, volume, and temperature. Other quantities of interest are associated with the process that takes a system from one state to another. We call these process functions. To distinguish them from state variables, we use the Greek letter delta as a prefix to denote very small quantities. For example, delta q might represent a tiny amount of heat added to a gas, and delta w, a tiny amount of work done on the gas. Associated with the process, this slightly changes the state of the gas. So starting at a state with pressure, volume, and temperature, pv and t, we might move to a nearby state with values p plus dp, v plus dv, and t plus dt. And the process that caused this change of state might have involved adding heat delta q to the gas and doing work delta w on it. A different process could move us between these two states along a different path, characterized by different values of the process functions. In the next video, we will begin to explore the relations between the various state variables and process functions associated with an ideal gas and other types of systems. This will lead us to, among other things, the derivation of the laws of thermodynamics.