What is wrong with this definition of ordered state?

Question

It is written in my book that a disorederd state is more probable than an ordered state and hence every system tends to move spontaneously to a state of higher disorder or higher probability.

But I think it depends on us since we can define any state as an ordered or disordered state.

Suppose we have 3 unbiased coins and all are tossed at once.

Now let's say that there is a man A and he defines an ordered state as having three heads at a time. For him the definition of entropy sets good since the probability of getting the ordered state (all head) is less than that for an disordered state.

Now say there is a man B and he defines an ordered state as a state with at least one head. Now , we know that the probability of getting an ordered state for that man is more than the probability of getting a disordered state.

How is this possible ? The definition of entropy is not favourable for B.

What is wrong with this intuition ? Do we need to change the definition of entropy ? Or am I wrong somewhere ?

Answer 1

The words "ordered" and "disordered", in relation to entropy, are the source of a lot of confusion and are not even always an accurate description.

In your coin example, a more typical use of the word "ordered" would be to say that all three coins are the same (either heads or tails). If you start in an ordered state, and each coin randomly flips, it is unlikely the system will remain in an ordered state, since there are 6 states with the 3 coins not all having the same face, and only 2 states where all 3 coins have the same face.

A more abstract, but also more correct, description of entropy is in terms of the microstates of the system. Entropy is the logarithm of the number of microstates that are consistent with the observed macroscopic properties of the system. In equilibrium, the macroscopic properties (energy, pressure, volume, chemical potential, etc) will be such that there are more microstates consistent with these properties than any other set of macroscopic properties.

Answer 2

There is another perspective about order/disorder.

Imagine that you have $10^{100}$ unbiased coins. What are the differences between a sequence (I used the word sequence because I care about the order of the elements) where all coins were sampled randomly (let us call this sequence , sequence $\mathbf{R}$) vs a sequence were all coins are heads (let us call this sequence, sequence $\mathbf{H}$)? Well, it will depend on the properties of sequences of $10^{100}$ coins in which you are interested. However, there are things that will not depend on this definition and are these "properties" that are used to describe what an ordered/disordered state is.

If you wanted to describe the sequence, how would you describe it (imagine that you want to tell me the configuration that you have)? For the sequence $\mathbf{H}$, this is rather trivial, you just tell me that all coins are heads. But how would you go about the other one? You would have to tell the state of every single one of the coins.

This implies that, you would take 5 seconds to give me all the information about the sequence $\mathbf{H}$, but you would not be able to tell the sequence $\mathbf{R}$ even if you had started at the beginning of time.

This is fundamentally the difference between an ordered configuration and a disordered configuration. Imagine that your physical system is a lattice with spins. If all spins are aligned, you say that the state is ordered (it requires very little information to describe completely the state). If the direction of the spins is random (it requires huge amounts of information to describe completely the sate).

You may then ask you the terms order/disorder are used to talk about this. The point is that an ordered system is a system in which you can recognize patterns, and when you recognize a pattern, you greatly reduce the amount of information required to describe your system.

Imagine another sequence of $10^{100}$ unbiased coins, but in which you notice that there are subsequences that can be used to generate the whole configuration (e.g. TTTHTTTHTTTHTTTH...), then you may say intuitively that this state is ordered, which would coincide with the fact that for you to tell me all about this sequence you would just say: "It starts with TTTH, and it then repeats until the end.".

This is also related with the probabilities mentioned by @GiorgioP. There are fewer ways (and fewer where is a euphemism) to generate configurations that contain patterns (ordered) than to generate configurations that contain no patterns (you can try for the example of the unbiased coins). Moreover, the bigger (bigger $\equiv$ contain more elements) the system is the less likely these ordered configurations are.

Answer 3

I think you should start to build your intuition only after having digested the definition your book is introducing.

In the present context, the sentence

a disordered state is more probable than an ordered state

should not be considered a simple observation, connecting two independent concepts of probability and order. It becomes a direct definition of what the author of your book means by order/disorder. The more probable is an event, the more disordered it is.

Therefore, your example with a man tossing coins cannot be used to challenge this concept of disorder. What can be done is to see whether such a definition of disorder agrees or not with our informal use of the same word.

Indeed, even though this definition does not leave too much to subjectivism, there is some situation where some conflict may exist between a definition of the disorder based on the probability and our daily life use of the term disorder. In a non-technical context, we tend to confuse the probability of an event (which in probability theory is a set of elementary samples) with an individual element of the event. In statistical mechanics, macrostates are events, while microstates are elementary samples. Therefore, while your book definition of disorder implies that everybody has to assign higher disorder to the macrostate characterized by having at least one head, it does not justify to say that the configuration (head, tail, tail) (in this order) is more disordered than the configuration (head, head, head). Here, the necessary intuition that has to be built is connected with the proper use of the concept of probability.

Probably, the most evident conflict between our use of the word disorder and probabilities is connected to the fact that most of the every-day uses of this term are connected with spatial disorder. In contrast, the equilibrium states' probabilities in Statistical Mechanics are based on the energy of the microstates and a choice of the macrostate compatible with our experimental possibility of control. Such a situation sometimes makes it possible to find conflicts between an intuitive approach based on the spatial order and probabilities. For example, Statistical Mechanics allows expressing the entropy of a classical perfect gas in terms of probabilities. The final result is an increasing function of the mass of the molecules. This result may look odd if we refer to the probability of spatial configurations (the same for any configuration and independent of the mass). It becomes understandable if we consider that the probability is the probability of events in the phase space and they do depend on the mass.

Answer 4

A somewhat similar question answered by me may be helpful in this case.
Concluding comments (to the aforementioned answer) -
Loosely speaking from equation (1), the entropy of a system is proportional to the number of accessible quantum states (this is what entropy is!) → more the amount of (co-ordinate/momentum) space available, higher the entropy, higher the disorder (because there will be more choices available for a particular particle to settle into - so, "order" does not mean a well-arranged/ordered outer appearance but refers to the number of accessible quantum states in 𝑉𝑟 & 𝑉𝑝, neither it is a relative term). Thus, a disordered state is simply more probable because by the very definition above there are more states available to be occupied - for eg., this essentially means a particular particle has higher probability to occupy that state which has higher frequency of occurence = eg. if a system has more spin-up (|↑⟩) states available than spin down states (|↓⟩), then the particle has higher probabilty to be spin-up rather than spin-down.

Answer 5

There is a unique definition of order and disorder because only one definition is compatible with thermodynamics. Thermodynamics, while being a more primitive theory is applicable in all situations, as long as the average statements of Clausius and Kelvin hold.

More disorder = more accessible micro states. Please note that the thermodynamic definition does NOT depend on vague words to define entropy. Entropy in thermodynamics is the state function that a system must have as a logical consequence of either the Kelvin or the Clausius statement of the second law. Only one definition of entropy is consistent with the bulk thermodynamic description. In a logical framework, it the first definition is the thermodynamic one and later definitions reduce to the thermodynamic one in a unique fashion.