# What does the value of entropy say?

I understand that entropy is just the natural logarithm of the multiplicity times the Boltzmann constant, but I'd like to understand what the value of entropy actually says, does it just say that when it's large, that the multiplicity of the system is large or?

So for example what is the difference between a system having $$S=0.5$$ J/K or a system having $$S=100$$ J/K, or is entropy only really meaningful if we look at the change in entropy rather than entropy itself?

Entropy in statistical mechanics is defined as: $$S = -k_{b}\sum_i P_i ln(P_i)$$ where i is an index running through all the microstates in which your macrostate can be, $$P_i$$ is the probability that your system configure in one of those and $$k_b$$ is the Boltzmann constant. If the entropy increases, the number of $$P_i$$ different from zero increases, meaning that your macrostate has more possibile configuration. If the entropy of your system is $$0 \frac{J}{K}$$ then there must exist a microstate j such that $$P_j = 1$$ and for every other microstate i different from j $$P_i=0$$, meaning that there's just one microstate that build your macrostate.

The Boltzmann definition of entropy is $$k_B \ln(W)$$, where $$W$$ is the number of different micro states consistent with a particular macro state (basically, the measured observables like temperature, pressure, etc.). So if you have a system with entropy greater than 0, it means that there are multiple different micro states consistent with that macro state. The absolute value of entropy, unlike the enthalpy, energy, or several other quantities, actually does have a physical meaning, even if the counting of the micro states is physically inaccessible (or infinite). What is fascinating about this definition is that the second law tells us that there is a spontaneity condition governed by the entropy, which is just related to how many ways there are to configure the system.

Now, I want to be clear that there are other definitions of entropy, such as Rudolf Clausius’ definition in terms of heat and temperature, $$dS= \frac{\delta q}{T}$$ There is also the perfectly general Gibbs definition, which only involves the probabilities of occupying particular micro states, $$S = -k_B\sum_j p_j \ln(p_j)$$ But I think that by far the most accessible definition is that of Boltzmann as far as gaining intuition for what the entropy actually means.

I understand that entropy is just the natural logarithm of the multiplicity times the Boltzmann constant, but I'd like to understand what the value of entropy actually says, does it just say that when it's large, that the multiplicity of the system is large or?

Such a definition should include the restriction to an isolated system (fixed energy and the other extensive quantities). For systems with different external conditions, one should somewhat modify this definition. In the following, I'll provide an answer within such constraint.

Due to the monotonic behavior of the logarithm, the order relation between multiplicities is preserved by the passage to the logarithm, i.e., to entropies. Therefore, if the multiplicities are in the relation $$W_1>W_2$$, the corresponding entropies will be in the relation $$S_1>S_2$$.

However, there are two important things to note in these relations.

The first is that entropy is an extensive quantity. Then, a larger entropy may just be the effect of a larger number of degrees of freedom. In the case of the example in the question, a system having $$𝑆=0.5$$ J/K has a smaller entropy than a system having $$𝑆=100$$ J/K, and, consequently, fewer microstates. Whether such a difference in the available microstates is there because the first system is much smaller than the second or has much smaller energy is not within the information contained in the bare entropy values.

The second relevant concept is that the usual thermodynamics cannot be obtained from statistical mechanics without the thermodynamic limit. Said in another way, the correct connection between Boltzmann's formula and entropy (for a simple, one-component system) is the following: $$s={\mathrm {thermodynamic ~limit}} \frac{S}{N}=k_B \lim_ {\substack{E \rightarrow +\infty\\V \rightarrow +\infty \\N \rightarrow +\infty\\ \frac{E}{N}=cost \\ \frac{E}{N}=cost}}\frac{\log W(E,V,N)}{N}.$$ The main consequence of such specification is that we can consider the values of entropy as providing information about $$\log W$$, only within arbitrary functions growing slower than linearly with the system size. In particular, zero entropy is consistent with a non-degenerate macrostate but also with any macrostate whose degeneracy grows slower than linearly with $$N$$.