 We've established that all mechanical processes are, in principle, microscopically reversible. The equations of motion can take us from past the future, but also from future to past, along precisely the same particle trajectories. A fundamental difference between the forward and reverse process is velocity. The rate of change of position represented here by the sloped red line. If a coordinate is increasing at some rate in the forward process, then it is decreasing at the same rate in the reverse process. The velocities have equal magnitude, but opposite sign. This gives us the recipe for reversing a mechanical process. Simply reverse all particle velocities. Let's return to the idea of a trick billiards shot. If we create any initial arrangement of the balls, and allow it to evolve for any length of time, then freeze the system and reverse all velocities. The system will then continue to evolve in a manner that precisely reverses its original evolution. Despite this theoretical possibility, we know that many complicated processes are never reversed in practice. No matter how long you wait, an egg that splatters on your floor is not going to spontaneously reassemble. To get a feeling for why this might be, let's go back to the point when we reverse the velocities and randomly tweak each velocity by one part in 10,000, one hundredth of one percent. Very small changes. Then let the system evolve. The balls do not return to their original nicely racked up state. This tells us that microscopically reversing a complex process requires the dynamical states of all the particles to be initialized to extremely precise values. For a large enough number of particles, this becomes practically impossible. This extreme sensitivity to initial conditions applies to the forward process as well. If we perturb the velocity of the cue ball by one hundredth of one percent, and allow the system to evolve for the same amount of time as before, the final state is very different than the final state of the unperturbed case. On the other hand, neither state is particularly special. In both cases, the balls are seemingly randomly distributed about the table with seemingly random velocities. The situation for the reverse process appears very different. For the unperturbed case, the system ends up in a distinctly special state. Six of the balls end up at rest in a nicely racked arrangement, and all kinetic energy is transferred to the single remaining ball. For the perturbed case, we do not end up with such a special state. We conclude that it's easy to convert a special state into a non-special state, but it's hard to convert a non-special state into a special state. In place of special and non-special, we might use the terms ordered and disordered. Then we could say that ordered states tend to evolve into disordered states. But disordered states do not tend to evolve into ordered states. The second law of thermodynamics is sometimes expressed using this terminology. Of course, to be rigorous, this requires a precise mathematical definition of order and disorder. We'll consider this in the next video in the context of statistical mechanics. The claim that all mechanical processes are, in theory, reversible certainly seems to be in conflict with our common experience. Maybe ideal elastic collisions are reversible, but real macroscopic processes are not perfectly elastic. They are characterized by frictional forces which dissipate energy. At any given time, the dynamic state of a particle is specified by its position and velocity. We've been assuming that forces between particles depend only on their positions, but dynamic frictional forces are characterized by a dependence on velocity. If we reverse a process, the velocities flip to their negatives. So the forces for the forward and reverse processes would not be equal. But our entire argument for microscopic reversibility depends on these being equal. Let's illustrate this with a simulation. We have a single ball on a billiard table that's ideal, except the side bumpers have friction. When the ball strikes a bumper, it experiences a force component that opposes its velocity. We give the ball an initial velocity and we observe that each time it bounces off a bumper, it slows down. Its kinetic energy is being dissipated. Reversing the ball's velocity, it continues to experience frictional dissipation on each bounce. Its trajectory is not precisely reversed. So for practical purposes, Max Planck has certainly justified in stating a version of the second law as, it is in no way possible to completely reverse any process in which heat has been produced by friction. But what is friction at the microscopic level? According to atomic theory, the bumper is composed of atoms. For simplicity, we treat the ball as a solid object, but of course it is also composed of atoms. When the ball strikes the bumper, the impact, which appears to be a single macroscopic collision, is actually countless microscopic collisions between the ball and the vast numbers of atoms making up the bumper and among the atoms themselves. So considered microscopically, the process is very complicated. When the ball rebounds, it leaves behind some of its kinetic energy as increased invisible kinetic energy of the atoms, which we call heat. Each of these microscopic collisions are actually elastic. The concept of friction is used as an effective macroscopic model to simplify the net result of this complicated microscopic process. Therefore, at a microscopic level, our reversibility argument still applies. But instead of only reversing the ball's velocity, we would also need to reverse the velocity of every atom in the bumper. So in theory, if in the forward process, each collision converts some of the ball's kinetic energy to heat in the bumper, then if at some point we reverse the ball's velocity and the velocities of all the bumper atoms, then each collision will convert heat in the bumper to increase kinetic energy of the ball, and the process will be precisely reversed. Of course, this never happens in the real world. Intuitively, if concentrated energy gets diffused as heat, it doesn't spontaneously undiffuse. Let's look at a simulation of energy diffusion. Here we have a lattice of atoms, half red and half blue. They are subject to elastic forces trying to return them to the equilibrium positions shown, and collisional forces when they strike another atom. Initially, the red atoms are given random velocities, while the blue atoms are all at rest. As time goes on, the kinetic energy that was concentrated in the red atoms diffuses to the blue atoms. Eventually, the energy is more or less equally distributed among all the atoms. Now, if we let this simulation run for a very long time, would we expect to ever see the energy undiffuse and re-concentrate in the red atoms? No. But is it impossible for this to happen? Well, no. Because as a mechanical system, this is microscopically reversible. If we reverse all particle velocities and let the system continue to evolve, we'll eventually see the blue atoms one by one freezing, until all kinetic energy is re-concentrated in the red atoms. So this is clearly theoretically possible, but maybe there is some way to show that without us stopping the system and manually reversing them, there is no way for this state with reverse velocities to ever occur spontaneously. Because if not, then we have to accept the possibility that if we stare at a glass of water long enough, we will eventually see an ice cube spontaneously form and begin to travel around the glass, until the water propels it upward for us to catch. Now, allow me to admit that I didn't actually observe this un-melting of an ice cube. Instead, I, obviously, took a video of an ice cube melting after being dropped into a glass of hot water, and then applied a video reversal filter. Still, I can't prove that this un-melting process is theoretically impossible. To appeal to a higher authority here, let me quote Richard Feynman. So far as we know, all the fundamental laws of physics, like Newton's equations, are reversible. Then where does irreversibility come from? In fact, the Poincare recurrence theorem tells us that bizarre, reversed processes, like the un-melting ice cube, are not only possible, given enough time they are inevitable. In the late 19th century, Henri Poincare was one of the mathematical physicists investigating, among other things, the theoretical foundations of thermodynamics. He pointed out that under a reasonable set of assumptions, if a closed system of particles has a given total energy and is limited to a finite region of space, then for any initial state the system will return arbitrarily close to that state after a finite length of time. So if we put an ice cube in a glass of hot water and perfectly sealed it inside a perfectly insulated box, it would initially melt, leaving a glass of warm water, but after a long enough time the ice cube would spontaneously unmelt to produce once again an ice cube in a glass of hot water. At least that's what the fundamental laws of physics imply. The basic argument is fairly simple. If the particles have a finite amount of space and energy available, then to with any given position and momentum uncertainties, there are a finite number of possible dynamical states available to them. As chaotic dynamics moves the system through these possible states, given enough time, the system will eventually reach any possible state. Since the initial state is obviously a possible state, the system will eventually return to it. Poincare wrote, according to this theory, to see heat pass from a cold body to a warm one, it will suffice to have a little patience. One would hope that someday the telescope will show us a world where the laws of thermodynamics are reversed. Actually, more than a little patience is required. For typical macroscopic systems, the Poincare recurrence time is unfathomably large, absolutely dwarfing the age of the universe, so long in fact that for practical purposes our unmelting ice cube is impossible. We'll return to some of these probabilistic concepts in the next video on statistical mechanics. For now let's adopt a more practical point of view. We know that for all practical purposes, irreversible processes exist. And we know irreversibility cannot be explained by conservation laws, expressed as inequality, such as the first law of thermodynamics. Because these remain true if we reverse the flow of time. To account for irreversibility, we need a law which takes the form of an inequality. The change in something denoted by s is greater than or equal to zero. As time goes on, this quantity can remain constant or increase but never decrease. If we reverse the flow of time where s was constant it will remain so, but where s was increasing it will now be decreasing. When we can flag a decrease as a violation of our law, then time periods where s is constant correspond to reversible processes, and time periods where s is increasing correspond to irreversible processes. It needs to be emphasized that these laws can only be expected to apply to closed systems or presumably the entire universe. Clausius named the something quantity s entropy. Based on our analysis of the Carnot and Sterling cycles, we'll join him in making the following statement. If heat Q is added to an object in thermal equilibrium at temperature T, the object's entropy increases by delta s equals Q over T. If we have two objects, one at temperature T hot and the other at temperature T cold, when heat Q flows from the hot object to the cold object, then the cold object's entropy changes by delta s cold equals Q over T cold, and the hot object's entropy changes by delta s hot equals minus Q over T hot, minus because we extract heat, which is mathematically equivalent to adding negative heat. The net entropy change is Q times the quantity 1 over T cold minus 1 over T hot. Since T cold is less than T hot, this quantity is positive. The entropy of the universe has increased, so this process is a go, consistent with the second law. If heat Q flows from the cold object to the hot object, the signs of the entropy changes are reversed, and the entropy of the universe would go down. This process is a no-go in violation of the second law. We see that this definition of entropy in the second law leads to the equivalence of the statement, the entropy of a closed system can never decrease, and Clausius's statement that heat cannot of itself pass from a colder to a hotter body.