# Entropy / Structure Relations

I want to check on the validity of the following objective definition of order. Is it correct? Is there a more rigorous statement of this concept?

The further a system is from its maximum thermodynamic entropy, the more ordered it is and the lower its entropy. Specifically, if a system’s microstates are not of equal probability, which is usually the case because of interactions or correlations between the microstates, it is more ordered when it spends time in the less likely states. If the microstates are of equal probability, it can be constrained to remain in one of the states a disproportionate amount of time by imposing constraints on the system’s random fluctuations.

Thanks very much for the input.

• Aren't you just describing the property of entropy, or rather lack of it? Schroedinger already introduced the concept of negentropy. It seems like what you are describing is very similar. I.e. it is just $$\mathrm{order} = S_{max} - S,$$ with $S_{max}$ the thermodynamic entropy and $S$ the von Neumann/Shannon entropy of the state. Note also that by invoking probabilities you have already constrained your observer to an arbitrary, unspecified, limited set of observables or knowledge. Until this is defined your definition is subjective. – Mark Mitchison Jun 3 '14 at 23:29
• Thanks, @MarkMitchison. That does seem to be like what I'm after. I remain unclear about this though: Is the source of the subjectivity that I have to choose the microstate from which I'm measuring the distance to Smax, or are there other sources? I don't understand what the defining you refer to actually consists of. – johnhidley Jun 4 '14 at 13:34
• I was just pointing out that you haven't defined what properties of the system are assumed accessible, and which aren't. Technically this definition is called a thermodynamic state space, and it consists of a set of observables whose values determine the macrostate. Otherwise it is not clear what you mean by the concepts of "microstate", "probability", etc. – Mark Mitchison Jun 4 '14 at 18:56
• @MarkMitchison -- Thanks again for your helpful, much appreciated input. Is the following a coherent definition and a sensible characterization of order: Take a chemical system where the concentrations of component molecules oscillate such that some of the system's states have more total high-energy bonds. Work could drive the system toward more high energy bonds. The system configuration with maximum available energy in the bonds corresponds to maximum order and is least probable and lowest entropy. In this state space the observable is the % high-energy bonds and the macrostate is order. – johnhidley Jun 5 '14 at 19:42
• Order is tricky to define, and I'd suggest you're better off not trying to. Entropy is often said to measure disorder, but I find it better to think of it as measuring, well, entropy. We tend to find it correlates inversely with our intuitive notions of order, but this isn't always true. (For example, is an emulsion more ordered than phase-separated oil and water? My intuition says no, but entropy-as-disorder says yes.) For me the greater insight is attained by understanding entropy for what it is, rather than trying to map it to concepts like order. But that's just my opinion. – Nathaniel Sep 12 '14 at 6:50

This definition is not correct. A thermodynamic state $A$ of a particular system is said to be more ordered than another thermodynamic state $B$ of the same system if $A$ is invariant under less symmetry operations than $B$.

The entropy of a thermodynamic state of a particular system, on another hand, quantifies the number of microstates which are compatible with the said thermodynamic state.

It is clear from these two definitions that the definition of entropy does not entail that higher entropy equates higher disorder.

It so happens that in most cases in fact, higher number of symmetries allows for higher number of states and hence higher entropy but it is not always so.

Hard sphere systems (which do not have any energy scale) have been shown to have a fluid/solid transition above a certain packing fraction for instance. In this case, at high packing fraction, it is less constraining for the spheres to adopt an ordered structure than trying to satisfy invariance under any rotation and any translation; the obtained crystalline structures have then higher entropy than the disordered ones. The same goes for hard rods which, above a certain packing fraction display an entropy-driven phase transition from isotropic to nematic.

Morality, it is not wise to identify entropy with disorder as they do not assess the same features of a given thermodynamic state of a system.

• Thanks for your comment. I understand enough of what you have written to see that I cannot identify disorder with entropy although they are often related. I wonder whether the ordered lower entropy packing could be seen as an instance of what Jeremy England is talking about when he finds that energy moving though a system can create order in the system as the energy dissipates. See 'quantamagazine.org/20140122-a-new-physics-theory-of-life' for a popular account. There is a link to his paper in that article. – johnhidley Nov 10 '14 at 23:12
• I don't think this is related to Jeremy England's argument. I guess assuming a low entropy for an ordered system simply reflect our misconceptions (driven by an understanding of phase transitions guided by the idea of Enthalpy/Entropy balance such that crystals would emerge from enthalpy loss that overcompensates the entropy loss) about entropy nothing else... – gatsu Nov 11 '14 at 11:12

Before answering, I would ask where that definition is from. Is it a reference to a statement you found in a text? if so which one? And what are your criteria for 'rigorous' i.e are you asking for a METHOD by which to check, or are you asking for a consensus on if it rigorous/ not?

Of the bat though, without a context, the second part is suspect to me. I'd consider it to be self evident that if constraints are placed on 'random' fluctuations, the micro-states really are no longer of equal probability. ...BUT, perhaps they are viewing the constraints in regard to a system that usually has states of equal probabilitu interacting with an 'outside' influence. So, as I said, it would be useful to have a context. I suppose my criteria for 'rigourous' demands greater context, so my answer right now is 'to be determined.'

• Thanks for your comment. The definition is my own. I am trying to formulate a probabilistic view of structure or order/disorder analogous to Shannon's view of information and Boltzmann/Gibbs view of statistical entropy. My question about rigor is whether such a view makes sense in light of their views. I was indeed thinking of outside work being required to constrain the system to a particular microstate for a disproportionate period of time. – johnhidley Jun 3 '14 at 23:04