Gibb’s paradox seems to arise when one considers an ideal gas of N particles, energy E, temperature T in a box of volume V and compares its entropy to two subsystems created by inserting a partition into the middle of the container without doing work. Each subsystem has N/2 particles, occupies V/2 and has E/2 and T. The issue is that one wishes to have 2 S subsystem = S full system, i.e. to have entropy be extensive. To do so, a function of N is inserted into the entropy formula and it is argued that this has to do with identical particles and ultimately with quantum mechanics. We argue that there seems to be a problem with this picture, namely the assignment of N/2 particles to each subsystem as this involves a measurement and hence changes entropy. To simplify matters, we consider a full system of 2 particles. Boltzmann’s notion of entropy is - k ln( number of states). Here the issue primarily seems to be with volume states, i.e. a particle can be in any tiny volume cell with equal probability. In the full V case, each of the two particles can be in any volume cell in V, but the same is true if one inserts a partition separating V into two V/2 cells. One cannot possibly know if one particle is in each cell unless one makes a measurement before inserting the partition and making a measurement changes entropy. When writing the number of states, these are probabilistic scenarios. It is possible that 2 particles are on one side of the partition and 0 on the other or 1 on each. The number of possible states for the partitioned system is the same as for the full volume. As a result, there is no paradox. There is only a paradox if one assumes that there is 1 particle on each side which is an assumption which cannot be made without making a measurement.
Gibb’s paradox seems to arise when one considers an ideal gas of N particles, energy E, temperature T in a box of volume V and compares its entropy to two subsystems created by inserting a partition into the middle of the container without doing work. Each subsystem has N/2 particles, occupies V/2 and has E/2 and T. The issue is that one wishes to have 2 S subsystem = S full system, i.e. to have entropy be extensive. To do so, a function of N is inserted into the entropy formula and it is argued that this has to do with identical particles and ultimately with quantum mechanics. We argue that there seems to be a problem with this picture, namely the assignment of N/2 particles to each subsystem as this involves a measurement and hence changes entropy. To simplify matters, we consider a full system of 2 particles. Boltzmann’s notion of entropy is - k ln( number of states). Here the issue primarily seems to be with volume states, i.e. a particle can be in any tiny volume cell with equal probability. In the full V case, each of the two particles can be in any volume cell in V, but the same is true if one inserts a partition separating V into two V/2 cells. One cannot possibly know if one particle is in each cell unless one makes a measurement before inserting the partition and making a measurement changes entropy. When writing the number of states, these are probabilistic scenarios. It is possible that 2 particles are on one side of the partition and 0 on the other or 1 on each. The number of possible states for the partitioned system is the same as for the full volume. As a result, there is no paradox. There is only a paradox if one assumes that there is 1 particle on each side which is an assumption which cannot be made without making a measurement.
Gibb’s Paradox is associated with introducing a 1/N! factor in the partition function= 1/N! {Sum over i exp(-ei/T)} power N (1). It is often stated that N! is required to deal with identical particles. It is further stated that in classical physics, particles have an identity, but in quantum mechanics, they are identical and so the N! is really a quantum mechanical feature. In (2), we argued that dE=TdS - PdV is entirely focused on interactions and so if S = -k ln(number of states), one should consider interaction states, not identity states. We argued that a green particle differs in colour from a red one, but this has nothing to do with interactions and one would completely ignore colour when calculating the number of states. The same reasoning applies to two identifiably different particles with the same e,p being at x1, and x2 versus x2, x1. In terms of identity states, there are two, but they both represent the same physical interaction and are counted once. Even though classical particles have an identity, and quantum ones also because they have a center-of-mass, from the point of view of an interaction, a particle at (x,y,z) with a certain momentum and energy is all that one needs to know for interaction purposes. It is irrelevant which particle is at this (x,y,z) and so N! Is introduced for this reason, as argued in (2). Here we consider the quantum mechanical case. We have argued in previous notes that quantum mechanics follows from special relativity. In particular, a particle still has a center-of-mass and so an identity. If it moves with constant speed, one has the classical equation x=vt for the center-of-mass. This begs the question: Where are the quantum mechanical features? We argue they are in the interaction. The fact that the center-of-mass is at some x,t does not mean that a p (momentum) impulse hit is rendered at that point. In fact, the Lorentz invariant -Et+px, which is more general than x=vt, may be considered to be a kind of free particle potential with x,t and x+hbar/p, t+hbar/E having the same potential values and rendering the same hit, regardless of whether x=vt. -Et+px is already created using the notion x=vt for the center-of-mass and so -Et+px holds for more x, t points than simply x=vt. As a result, the quantum free particle wavefunction exp(-iEt+ipx) represents a probability to interact, not a center-of-mass identity of the particle. In other words, quantum mechanics is a statistical theory of interaction and as such the probability bypasses the actual position of the center-of-mass which identifies the particle. In a 2-slit experiment, a particle has a specific center-of-mass and moves as x=vt, but at the 2-slit apparatus (with slits about hbar/p apart), it may probabilistically interact with both slits. This is a probabilistic interaction calculation which has little to do with the center-of-mass position and hence any position identity of the particle. As a result, a more general quantum wavefunction W(x)= Sum over p a(p)exp(ipx) (bound state) is also an interaction probability. Thus, quantum mechanics automatically only considers interactions and this leads some to argue that there is no particle identity in quantum mechanics. We argue that just like classical physics, there is a particle identity (center-of-mass), but the interaction is the main focus and does not depend on this identity position. In other words, one only deals with interaction probabilities, just like classical statistical mechanical problems should only deal with interaction states (as argued in (2)), and not with any physical state which may be identified, but does not change the interaction. The difference between quantum mechanics and classical mechanics, is that quantum mechanics is formulated on the notion of interaction (so the particle identity is removed from the calculation right away), while in the classical case one retains the identity and has to manually correct for identity states in order to convert the number to the number of interaction states.
Historically statistical mechanics was developed for gasses for which there are very large numbers of molecules (i.e. on the order of 10 to 23rd power). As a result factorial schemes were developed to count the number of arrangements possible, the ln taken of this result and Stirling’s approximation for large N applied. The Shannon’s entropy form S/(-k) = Sum over i P(ei)ln(P(ei)) is consistent with this large N approach. Furthermore, large N statistical mechanics was linked to thermodynamics, in particular the first law dE=TdS - PdV. Surprisingly then, statistical mechanical results were applied to systems with small numbers of particles (e.g. a nucleus with about 100 nucleons etc). In Parts I and II we argued that the first law of thermodynamics describes two separate kinds of changes associated with energy changes. The second, work, is linked to the change of a mechanical parameter such as the volume of a box etc. The energy of a bound quantum particle in a box with an infinite potential is proportional to nn/LL. Thus we argued that a pure quantum bound state may also be associated with the first law of thermodynamics with changes in L with n, the energy level, held constant representing pure work. We further argued that for a bound state two classical like probabilities exists P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx) with P(x) being proportional to 1/L and P(p) to L. For a quantum adiabatic transformation (i.e. changing L with n staying constant) the term dS=0. If one wishes to create S from P(x) and P(p) and desires linearity i.e. S=Sp(n)+Sx, then Sp=Integral dp P(p)ln(P(p)) and Sx= Integral dx P(x) ln(P(x)) is a solution. If one next considers a quantum wavefunction for two levels W(x,t) = a(1)exp(-i E1 t) W1(x) + a(2) exp(-iE2 t) W2(x) with Eave = Sum over n en P(en) where Wn=a(n)a(n), one may again utilize the first law of thermodynamics. In Parts I and II we carried over the entropy form P(ei)ln(P(ei)). In this note we wish to argue that this form follows at the two energy level picture by considerations of a function which consists entirely of probabilities i.e. P(ei) and is linear in the sense that such a function for two identical two state systems should yield S1+S2 as the first law should apply to each system. Furthermore P(ei) should be a function of P(ei/T) where T is a parameter which behaves as 1/LL because ei is proportional to 1/LL. In this way any function of P(ei) remains unchanged under a quantum adiabatic transformation for which i’s stay constant, but L changes. As a result, the entropy at the quantum level seems to follow from such arguments and not from large N approximations of factorial counting approaches. In large N situations it is argued that P(ei)=C(T)exp(-ei/T) follows from maximizing S with respect to the form of P(ei) subject to constant average energy. For small N situations (e.g. the 2 level example) one may solve Sum over i ei d(a(i)a(i)) = T dS directly without maximization. The maximization approach follows directly from dE=TdS (work=0) because dE= Sum ei dP(ei) in such scenarios. This is of the form of a Lagrange constraint This has been pointed out in Part II. In this note, we argue that the maximization of entropy is not needed at all. For dE=Sum over i ei dP(ei) and the ln form of entropy P(ei) is the inverse function of ln. Thus we argue that linearity and not large N arguments are enough to obtain Shannon’s form of entropy and that this ln form suggests P(ei/T) = C(T)exp(-ei/T) if dE=Sum over ei dP(ei)= TdS (for dW=0). Thus maximization of entropy subject to constant average energy is also not needed (although it does follow mathematically from dE=TdS.)
Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the microlevel, and from the relation S=k ln W leads to a nonextensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable and that quantum mechanics is indispensable for making sense of this inconsistency. In contrast, we argue that the classical statistics of distinguishable particles and the resulting nonextensive entropy function are perfectly acceptable from both a theoretical and an experimental perspective. The inconsistency with thermodynamics can be removed by taking into account that the entropy concept in statistical mechanics is not completely identical to the thermodynamical one. We observe that even identical quantum particles are in some cases distinguishable, and conclude that quantum mechanics is irrelevant to the Gibbs paradox.
In physics, there are two distinct paradoxes, which are both known as the “Gibbs paradox”. This article is concerned with only one of them: the false increase in entropy, which is calculated from the process of combining two gases of the same kind consisting of distinguishable particles. In the following, this paradox will be referred to as the Gibbs paradox of the first kind (GP1). (Two particles are said to be distinguishable if they are either non-identical, that is, if they have different properties, or if they are identical and there are microstates which change under transposition of the two particles.) The GP1 is demonstrated and subsequently analyzed. The analysis shows that, for (quantum or classical) systems of distinguishable particles, it is generally uncertain of which particles they consist. The neglect of this uncertainty is the root of the GP1. For the statistical description of a system of distinguishable particles, an underlying set of particles, containing all particles that in principle qualify for being part of the system, is assumed to be known. Of which elements of this underlying particle set the system is composed, differs from microstate to microstate. Thus, the system is described by an ensemble of possible particle compositions. The uncertainty about the particle composition contributes to the entropy of the system. Systems for which all possible particle compositions are equiprobable will be called harmonic . Classical systems of distinguishable identical particles are harmonic as a matter of principle; quantum or classical systems of non-identical particles are not necessarily harmonic, since for them the composition probabilities depend individually on the preparation of the system. Harmonic systems with the same underlying particle set are always correlated; hence, for harmonic systems, the entropy is no longer additive and loses its thermodynamic meaning. A quantity derived from entropy is introduced, the reduced entropy , which, for harmonic systems, replaces the entropy as thermodynamic potential. For identical classical particles, the equivalence (in particular with respect to the second law of thermodynamics) between distinguishability and indistinguishability is proved. The resolution of the GP1 is demonstrated applying the previously found results.
The “Gibbs Paradox” refers to several related questions concerning entropy in thermodynamics and statistical mechanics: whether it is an extensive quantity or not, how it changes when identical particles are mixed, and the proper way to count states in systems of identical particles. Several authors have recognized that the paradox is resolved once it is realized that there is no such thing as the entropy of a system, that there are many entropies, and that the choice between treating particles as being distinguishable or not depends on the resolution of the experiment. The purpose of this note is essentially pedagogical; we add to their analysis by examining the paradox from the point of view of information theory. Our argument is based on that ‘grouping’ property of entropy that Shannon recognized, by including it among his axioms, as an essential requirement on any measure of information. Not only does it provide the right connection between different entropies but, in addition, it draws our attention to the obvious fact that addressing issues of distinguishability and of counting states requires a clear idea about what precisely do we mean by a state.
Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistinguishability that is crucial for the treatment of the Gibbs’ problem and the consistency of its solution with conventional thermodynamics. Quantum mechanics provides a quantitative criterion, not possible in the classical picture, for the degree of indistinguishability in terms of visibility of quantum interference, or overlap of the states as pointed out by von Neumann, thereby endowing the entropy expression with mathematical continuity and physical reasonableness.