Why is entropy favorable?

Question

I cannot seem to grasp the logic behind it. We say that more entropy (or more disordered system) is favorable over less entropy. But why? Why is randomness preferred over proper arrangement of atoms/molecules? The only thing I can think of is more degrees of freedom. Molecules in a more random system would have more degrees of freedom, and would thus be favorable. But, again, why? Why would more degrees of freedom be favorable? I've already asked my teachers and looked up internet, but haven't found any satisfactory answer. Also, is there any way you could also explain diffusion by the same concept (diffusion is entropically favorable, right?)?

PS: if this question is better suited for physics stack exchange, I'd be more than happy to have it migrated.

Answer 1

Thermodynamics. The second law of thermodynamics states that entropy always increases in an isolated system. 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. In the framework of macroscopic thermodynamics, there is no deeper principle from which we can hope to derive the second law.

Statistical mechanics. Statistical mechanics brings a microscopic perspective into thermodynamics. 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. Many different microstates can correspond to the same macrostate, since a macrostate only deals with macroscopic properties.

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. This implies that the most likely macrostate for the system is that with the greatest number of corresponding microstates. 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. The equilibrium state of a system is simply its most probable macrostate.

Example. (Microscopic diffusion.) Let us consider a microscopic model of diffusion: a $10$-by-$10$ grid of particles, $50$ red and $50$ blue. The particles are indistinguishable except by color. In addition, we will consider two sets of macroscopic states, 'ordered' and 'disordered'. Take as initial state the configuration with the two groups of particles separated by color into two equal halves by a horizontal boundary, the blue particles on the top and the red on the bottom. The diffusion process is akin to removing this artificial boundary.

There is one initial ordered microstate and $100!/(50!)^2$ final microstates. (Note that indistinguishable microstates should be considered the same microstate.) The entropy of our system has increased, because we have relaxed a constraint and allowed more microstates into our system. Most of these final states look disordered. Diffusion is therefore an entropically favorable process that brings an ordered system into a disordered one.

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Additional remark. I find statements like "entropy drives the system towards a particular state" to be somewhat misleading, because they imply that entropy behaves like a force. But entropy, as we've seen, is really just statistics.

Answer 2

It appears you're looking for an ELI5-style answer, not an elaborate definition.

Entropy just happens – as long as the universe isn't frozen solid, things will always be moving around, and that movement tends to introduce randomness more than it tends to introduce order.

Consider a deck of cards. Shuffle it. Is it perfectly sorted? No. Why? There are 10^67 ways to arrange a deck of cards, but only one way to sort it; chances are you didn't shuffle the deck back into perfect order. The number of different available states (e.g. card sequences) shows up in the various formal definitions of entropy.

The same principle applies everywhere else in the universe: in the face of random (microscopic) state changes, the odds are stacked in favour of mishmash and against (macroscopic) "order" appearing by mere chance – and overwhelmingly so.

Answer 3

Do not think of entropy as 'disorder' as this is misleading, better is that it is a 'measure of disorder' but this is equally vague. It is better to think of entropy as the number of ways that 'particles' or quanta (say vibrational or rotational quanta in a molecule) can be placed among the various energy levels available.

Thus at zero energy all the particles are in their lowest levels and there is only one way of doing this and so entropy is zero. As the energy is increased then more levels can become populated and so there are many different ways of populating all the energy level this and so the entropy is increased. (Think of the calculations, usually given as examples in calculating probability, where one has to find the number of ways of placing identical balls into bags.) The connection of entropy $S$ to counting the number of ways was discovered by Boltzmann with his famous equation $S=k_B\ln(\Omega)$ where $\Omega$ measures the number of ways of distributing particles among the energy levels. From $\Omega$ you can sort of see where the 'disorder' tag comes but using this word one misleads oneself.

Answer 4

If we look at why a chemical bond occurs maybe that might shed light on to this question. If we take diatomic hydrogen which has one electron from each hydrogen, the question arises why do two nuclei come together to form a bond since Coloumbic forces are increased. And the answer is because the electrons have a larger volume to exist in as they will be able to exchange between nuclei(I'm referring to the exchange integral). Thus we can summarize that a chemical bond occurs due to purely entropic reasons and as a result the energy of the system is decreased.