In addition, we will consider two sets of macroscopic states, 'ordered' and 'disordered'. The particles are indistinguishable except by color. (Microscopic diffusion.) Let us consider a microscopic model of diffusion: a $10$-by-$10$ grid of particles, $50$ red and $50$ blue. The equilibrium state of a system is simply its most probable macrostate.Įxample. At this point, knowing also from statistical mechanics that the entropy is a monotonically increasing function of the number of microstates (specifically the logarithm of the number of microstates in an isolated system, but the functional form is unnecessary here), we can motivate the second law: the entropy of a macrostate of an isolated system is a measure of the number of its corresponding microstates the greater this number, the more likely this macrostate will be observed. This implies that the most likely macrostate for the system is that with the greatest number of corresponding microstates. In statistical mechanics we can go one level deeper, taking as a fundamental postulate the principle of equal a priori probabilities, the idea that the system has an equal probability of being in any given microstate. Many different microstates can correspond to the same macrostate, since a macrostate only deals with macroscopic properties. We talk of macrostates-macroscopic states of our system, characterized by (macroscopic) observables like pressure, temperature, volume, etc.-which may be realized by a number of microstates-a complete microscopic description of our system, consisting of the positions and momenta of each particle. Statistical mechanics brings a microscopic perspective into thermodynamics. In the framework of macroscopic thermodynamics, there is no deeper principle from which we can hope to derive the second law. This is taken as a fundamental postulate-we simply accept this statement as a fact regarding how the world works, and our justification is that no experiment has ever shown the second law incorrect. The second law of thermodynamics states that entropy always increases in an isolated system.
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