# In Defense of Gibbs and the Traditional Definition of the Entropy of Distinguishable Particles

## Abstract

**:**

## 1. Introduction

_{i}P

_{i}ln P

_{i}

_{i}is the probability of the i

^{th}state and the sum goes over all the Ω states accessible to the system. Equation (3) has its origins with Gibbs [24]. It is not limited to the microcanonical ensemble. However, when it is applied to a microcanonical ensemble, P

_{i}= 1/ Ω for each state by Gibbs equal a priori postulate, and Equation (3) reduces to Equation (1). Of course, Gibbs predated quantum mechanics, so his writing involved S = −k <ln P> and continuous variables in phase space. When applied to classical distinguishable ideal monatomic gases, Gibbs’ entropy has the well known form

_{K}/N)]+ Σ

_{i}Y

_{i}

_{G}(V,N) = Nk ln V

_{S}(V,N) = Nk ln (V/N)

_{G}in Equation (5) is correct for classical, distinguishable particles. If my argument is accepted, then it dispels the objection [1,2] to the conventional Equation (1), even though it abandons extensitivity, which many feel should be postulational. My argument would not, of course, preclude an alternative fundamental definition of entropy in a microcanonical ensemble, especially for non-equilibrium processes [27].

_{S}in Equation (6) is incorrect when one takes distinguishability to its logical limit. Since the proposal [1] to use probability W along the lines of Equation (2) leads to Equation (6), my argument therefore implies that the recent attempt to follow Boltzmann’s writings to provide a fundamental definition of entropy [2] is also untenable.

_{BCD}that corresponded to Equation (5) above carried the subscripts BCD for Boltzmann Classical Distinguishable and those subscripts should be changed to GCD for Gibbs Classical Distinguishable. In [25] the entropy that corresponded to Equation (6) above carried the subscripts SCD for Swendsen Classical Distinguishable and now Boltzmann’s name could be attached to that, although that may be taking too great liberty as Boltzmann apparently never wrote that equation [2]. In this paper I use S

_{S}instead of S

_{B}in Equation (6).

_{n}H

_{2n+2}in which each molecule with the same value of n has the same mass but the locations of the multiple branches distinguish the different isomers; furthermore, the number of molecules in this class exceeds Avogadro’s number for values of n of order 100. Because only one counter-example can disprove conventional wisdom, it is therefore appropriate to spend some thought provoking effort to try to understand the statistical mechanics of distinguishable particles.

## 2. Definitions

_{i}for each particle i such that M

_{i}provides a barrier to particle i but is transparent to all other particles. In my third sample below only two semi-permeable membranes will be required. The difficulty of physically realizing these semi-permeable membranes for a real colloidal system such as homogenized milk [2] should not be allowed to preclude gedanken experiments any more than the impossibility of performing completely reversible processes should not be allowed to prevent us from using thermodynamics.

_{K}term in Equation (4). Each Y

_{i}term in Equation (4) depends upon the mass of the ith particle as well as contributions to the entropy from internal degrees of freedom of the particles. For particles to be truly distinguishable, Y

_{i}would generally be different for each particle. The example of hydrocarbon isomers noted above shows that this difference is not necessarily due to a difference in mass, but different Y

_{i}could be due to vibrational frequencies, although if the difference only involves modes with energies large compared to kT, the differences in Y

_{i}may be made as small as one pleases. Most importantly, any such differences in Y

_{i}will remain constant for the examples given below because they involve the same particles and no changes in temperature or average energies. Therefore, it is appropriate to focus on the translational volume terms, as in Equations (5) and (6), instead of the total entropy in Equation (4).

## 3. Example 1

_{i}semi-permeable membrane. This takes external work w done on the system. To maintain the same energy, heat q = w would have to flow out of the system isothermally and that would decrease the thermodynamic entropy of the system by minus q/T. (iii) Starting from state α, one could use the M

_{i}semi-permeable membranes to extract work w during quasistatic expansions for each particle i [25]. For these three reasons, the entropy should increase for process αβ. The traditional Gibbs Equation (5) does this nicely,

_{G}

_{α}= Nk ln V

_{G}

_{β}= Nk ln 2V

_{G}

_{αβ}= Nk ln 2.

_{S}

_{α}= Nk ln(2V/N)

_{S}

_{β}= Nk ln(2V/N)

_{S}

_{αβ}= 0

## 4. Example 2

_{S}

_{βγ}is obviously zero. However, if state γ is assumed to be the same as state α, then it would appear that ΔS

_{G}

_{βγ}= −ΔS

_{G}

_{αβ}< 0. The fallacy behind this incorrect result is the assumption that state γ is the same as state α. State α was prepared knowing precisely which distinguishable particles were in each subvolume, but there is no such knowledge about state γ, so from the information point of view it is clear that states α and γ are not equivalent [26,27,28]. Independent of the information perspective, sorting the particles in state γ back to state α can be accomplished with the semi-permeable membranes M

_{i}, but this requires the expenditure of free energy by the sorter, which should indeed reduce the entropy of the particles.

_{G}= k ln (volume) in Equation (5), where the volume in state α is V and it is 2V in state β. What is the volume available to a particular distinguishable particle in state γ? Clearly, it is V if it is known to be in either subvolume, but it is not known to the experimenter who inserted the partition or to any other observer which subvolume a particular distinguishable particle is in, so the missing information is still k ln 2V. This suggests that application of Equation (5) to state γ should use the volume 2V. This answer will likely be uncomfortable to many, as it was to me, because it implies that the definition of entropy depends upon observers. Surely, the system ‘knows’ which subvolume the particle is in after the partition is inserted! However, the concept of entropy has always been divorced from what the system ‘knows’. In general, any system of classical gases, indistinguishable or distinguishable, ‘knows’ precisely the position of all its particles in phase space, so entropy would never increase even in a free expansion if it is defined by what the system ‘knows’. The idea that the microscopic state of the system determines the entropy is therefore not a valid counterargument to the argument presented here, namely, that the effective volume to use for state γ in Equation (5) is 2V, not V. This is consistent with the perspective that entropy is the missing information about the system that observers do not know, but could in principle obtain [27,28]. It may also be noted that the information interpretation of entropy is often faulted as being subjective [30] (but see [27] for a different perspective on the objectivity/subjectivity issue). In our case the information is objective in the sense it could not be obtained for observers as a class without the expenditure of free energy to obtain it.

_{2}ʹ in one subvolume and particles N

_{2}+1ʹ, …, Nʹ in the other subvolume, where the primes indicate a permutation of the original numbering scheme to account for the probable outcome that it is a different set of particles in each subvolume. Although this state, which we will call state δ, is different than state α, the entropy of states δ and α should clearly be the same. The idea for identifying the locations of the particles is to use the M

_{i}membranes. If sweeping the M

_{i}membrane through one of the subvolumes encounters no pressure, then particle i is in the other subvolume. If the sweep encounters pressure, one stops the sweep and returns M

_{i}to its original position quasistatically, so no work is done on the system and particle i has been identified to be in that subvolume. Unlike the use of the semi-permeable membranes to reproduce state α precisely, this process requires no net work on the system of particles. However, it does produce information to the observer. Supposing that there is a connection of entropy to missing information [26,27,28], this gives a decrease in the observer’s evaluation of the entropy of the system. How entropy of this system can be reduced with no exchange of heat or work is reminiscent of the nontrivial Maxwell demon paradox [31]. Generally, the demon (observer) is supposed to have to expend free energy to accomplish the determination of the subvolume locations of the particles in state δ and this pays for the increase in the free energy (decrease in entropy) of the system.

## 5. Example 3

_{A}A particles and N

_{B}B particles in the same volume V, the AB blue walled container is moved quasistatically as indicated in Figure 1(ii). Negligible work is done (i.e., the same amount as slowly moving any container), so w = 0. The containers are maintained at constant temperature, so dU = 0. Therefore, TdS = dU + w = 0 and the final separated state A + B, which has N

_{A}A particles in volume V and N

_{B}B particles also in a separate volume V, has the same entropy S

_{A+B}as the initial entropy S

_{A∪Β}of the A∪Β state.

_{A∪Β}(Ν

_{Α}+ Ν

_{Β},V) = S

_{A+Β}= S

_{A}(Ν

_{Α},V) + S

_{B}(Ν

_{Β},V)

_{A}= N

_{B}= N/2 keeps the math simple. Of course, with N distinguishable particles, there are many more ways to designate the A and B types, but only one way will suffice for our example as all others with N

_{A}=N

_{B}are equivalent. As with the usual case where particles within each type are indistinguishable, for distinguishable particles we also need semi-permeable walls. Let us consider an AB (AB, respectively) wall that is impermeable to distinguishable particles of type A (B, respectively) and permeable (impermeable, respectively) to particles of type B (A, respectively). If one has all N M

_{i}membranes from examples 1 and 2, then an AB wall can be constructed using M

_{i}with i = 1, …, N/2 in series, but having all M

_{i}membranes, while sufficient, is not necessary for having AB and AB walls.

**Figure 1.**(i) The union of A type particles (red squares) and B type particles (blue diamonds) in a common volume V. Within each type, the particles are indistinguishable in traditional discussions; they are distinguishable in the novel discussion in this paper. (ii) Procedure to separate the A and B particles using red (blue) walls that are impermeable to A (B) particles and permeable to B (A) particles (respectively). (iii) The sum of separate gases of A and B particles, each contained in the same volume V in all three panels.

_{G}in Equation (5) into the two sides of Equation (9).

_{G,A∪Β}(Ν,V) = Νk ln V

_{G,A+B}= 2(Nk/2) ln V

_{S}in Equation (6) into the two sides of Equation (9).

_{S,A∪Β}(Ν,V) = Νk ln (V/N)

_{S,A+Β}= 2(Νk/2) ln (V/(N/2))

_{S}does not satisfy Equation (9), having a discrepancy of Nk ln 2. I attribute the root cause to Equation (6) being the proper formula only when all the particles are indistinguishable. Then there can be no Gibbs separation process so it is not surprising that Equation (9) is not valid. In case there is any distinguishability between subsets of the particles, the right hand side of Equation (9) should be used to calculate S

_{A∪Β}of the mixture. For example, when there are M mutually distinguishable subsets, each consisting of N/M indistinguishable particles, Equation (9) gives S

_{A∪Β}= Nk ln(MV/N). In the limit of complete indistinguishability, S

_{A∪Β}= Nk ln(V/N) as required. In the limit of complete distinguishability, M = N and S

_{A∪Β}= Nk ln V and S

_{G}is recovered.

## 6. General Summary

## Acknowledgment

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Nagle, J.F.
In Defense of Gibbs and the Traditional Definition of the Entropy of Distinguishable Particles. *Entropy* **2010**, *12*, 1936-1945.
https://doi.org/10.3390/e12081936

**AMA Style**

Nagle JF.
In Defense of Gibbs and the Traditional Definition of the Entropy of Distinguishable Particles. *Entropy*. 2010; 12(8):1936-1945.
https://doi.org/10.3390/e12081936

**Chicago/Turabian Style**

Nagle, John F.
2010. "In Defense of Gibbs and the Traditional Definition of the Entropy of Distinguishable Particles" *Entropy* 12, no. 8: 1936-1945.
https://doi.org/10.3390/e12081936