The Second Law

Second Law of Thermodynamics: The entropy of a thermally isolated system never decreases. Thermally isolated systems are also called closed systems: There is no exchange of energy or matter with the environment. The above statement of the second law can be shown to be equivalent to other statements found in books (e.g. the Kelvin and Clausius statements) that summarize some empirical facts about nature. Processes in closed systems for which the entropy remains constant are called reversible while those for which the entropy increases are called irreversible. To see how the above terms correspond to our intuitive use of them, consider the following example: An isolated box contains an equal number of two types of molecules, say 'white' and 'black'. Suppose one had the extreme case where all the white molecules were on the left and all the black molecules on the right. Of course this is an unnatural situation and soon, due to collisions, the molecules will totally mix. The first situation corresponds to a state of maximum order and minimum uncertainty about the microstate while the second situation corresponds to large disorder (randomness) and maximum ignorance on our part about the microstate of the system. One can compute the entropy of the ordered and disorderd states: Clearly there are many more ways to form the disordered state than the ordered state and thus from the definition of entropy, the entropy of the disordered state is much higher than the entropy of the ordered sate. Since we know that the gases will mix, the system goes from the ordered state to the disordered state, increasing its entropy in accordance with the second law. We also know that if the number of molecules is very large, it is extremely unlikely that the gases will revert to the totally separated state at some future time. Thus the increase in entropy and disorder in the system is for all practical purposes irreversible, in agreement with the definition above. Note that the irreversibility is not due to the underlying fundamental laws (e.g. Newtons laws are reversible) but a result of the system going from an unlikely ordered state to a more probable disordered state, and the fact that for large systems (large number of molecules), the probability of the system reverting to the ordered state being negligible. In the above example, the probability of any one molecule being on the left half of the box is $1/2$. If there are $N$ molecules, the probability that all of them are on the left is $(1/2)^N$. Even for $N$ as small as $100$ this works out to be about $10^{-30}$, an infinitesimal quantity ! For macroscopic materials, $N$ is of the order of $10^{23}$, and so the resulting probability is even lower. Thus the Second Law is actually a statement about average behaviour that becomes overwhelmingly likely in a very large system, meaning that exceptions will be unobservable in all practical situations. What is the largest isolated system? The universe of course! Thus the second law of thermodynamics states that the entropy of the universe never decreases. In fact looking around we see that most of the changes in the universe lead to an increase in randomness and are irreversible. Thus the entropy of the universe seems to be increasing! Thermodynamically, one says that time flows in the direction of increasing entropy, or that the arrow of time is in the direction of increasing entropy of the universe.