Academia.eduAcademia.edu

Outline

Title
Abstract
Figures
Introduction
Conclusion and Discussion
References
All Topics
Chemistry
Theoretical Chemistry

Gibbs’ paradox according to Gibbs and slightly beyond

Profile image of Fabien PaillussonFabien Paillusson

Molecular Physics

https://doi.org/10.1080/00268976.2018.1463467
visibility

…

description

16 pages

link

1 file

Abstract

The so-called Gibbs paradox is a paradigmatic narrative illustrating the necessity to account for the N! ways of permuting N identical particles when summing over microstates. Yet, there exist some mixing scenarios for which the expected thermodynamic outcome depends on the viewpoint one chooses to justify this combinatorial term. After a brief summary on the Gibbs' paradox and what is the standard rationale used to justify its resolution, we will allow ourself to question from a historical standpoint whether the Gibbs paradox has actually anything to do with Gibbs' work. In so doing, we also aim at shedding a new light with regards to some of the theoretical claims surrounding its resolution. We will then turn to the statistical thermodynamics of discrete and continuous mixtures and introduce the notion of composition entropy to characterise these systems. This will enable us to address, in a certain sense, a "curiosity" pointed out by Gibbs in a paper published in 1876. Finally, we will finish by proposing a connexion between the results we propose and a recent extension of the Landauer bound regarding the minimum amount of heat to be dissipated to reset one bit of memory.

Figures (10)
Figure 1. Mixing of gases. Schematic representation of the mixing by diffusion between two gases. Case A: mixing of two gases each made of different particle species. Case B: mixing of two gases made of the same particle species. From Eq. (3) one can find an anomaly in the paradig- matic case of the mixing of two ideal gases in the same conditions of pressure and temperature.
Figure 1. Mixing of gases. Schematic representation of the mixing by diffusion between two gases. Case A: mixing of two gases each made of different particle species. Case B: mixing of two gases made of the same particle species. From Eq. (3) one can find an anomaly in the paradig- matic case of the mixing of two ideal gases in the same conditions of pressure and temperature.
This passage outlines what we have called above the epis- temic view. It means that, as far as entropy and its varia- tions are concerned, in a statistical sense, what matters is whether the substances are the same or not and not nec- essarily whether the particles comprising the substances could be distinguished or not if one were to use, say, a sufficiently good magnifying glass. This is emphasised further in the fact that Gibbs refers to acting on ‘exter- nal bodies’ to trigger separation; which bears similari- ties with interpretations of thermodynamics as a control theory [33]. If such an action results in changing an ex- Figure 2. Mixing of two identical binary miatures. Schematic representation of the mixing of two identical binary mixtures. Left panel: particle representation with 3 particles of type 1 and 4 particles of type 2 in each compartment. Right panel: composition representation with molecular fraction of 3/7 for type 1 and 4/7 for type 2. We note that the composition is un- changed upon mixing illustrating the fact that two substance can be identical while comprising multiple types of particles.
This passage outlines what we have called above the epis- temic view. It means that, as far as entropy and its varia- tions are concerned, in a statistical sense, what matters is whether the substances are the same or not and not nec- essarily whether the particles comprising the substances could be distinguished or not if one were to use, say, a sufficiently good magnifying glass. This is emphasised further in the fact that Gibbs refers to acting on ‘exter- nal bodies’ to trigger separation; which bears similari- ties with interpretations of thermodynamics as a control theory [33]. If such an action results in changing an ex- Figure 2. Mixing of two identical binary miatures. Schematic representation of the mixing of two identical binary mixtures. Left panel: particle representation with 3 particles of type 1 and 4 particles of type 2 in each compartment. Right panel: composition representation with molecular fraction of 3/7 for type 1 and 4/7 for type 2. We note that the composition is un- changed upon mixing illustrating the fact that two substance can be identical while comprising multiple types of particles.
Figure 3. Another Gibbs’ paradoz. Graph illustrating the absence of dependence of the entropy of mixing in the nature and similarity of the two mixed substances except when the latter are exactly identical.
Figure 3. Another Gibbs’ paradoz. Graph illustrating the absence of dependence of the entropy of mixing in the nature and similarity of the two mixed substances except when the latter are exactly identical.
We now consider a mixture whose composition is char- acterised by an attribute X of the constituants that takes on real values. Figure 5. Composition representation. Schematic representa- tion of the characterisation of a composition by the probabil- ity distribution of an attribute X of its constituents modelled as a random variable. Left panel: discrete mixture with a dis- crete probability distribution. Right panel: continuous mix- ture with a continuous probability distribution. Both panels illustrate that upon mixing two mixtures with the same com- position, all other things being equal, the composition of the final mixture is unchanged.
We now consider a mixture whose composition is char- acterised by an attribute X of the constituants that takes on real values. Figure 5. Composition representation. Schematic representa- tion of the characterisation of a composition by the probabil- ity distribution of an attribute X of its constituents modelled as a random variable. Left panel: discrete mixture with a dis- crete probability distribution. Right panel: continuous mix- ture with a continuous probability distribution. Both panels illustrate that upon mixing two mixtures with the same com- position, all other things being equal, the composition of the final mixture is unchanged.
Figure 6. WN! rationales chart flow for continuous mixtures Chart flow representation of the two viewpoints often opposed to justify the use of 1/N! in Eq. (5), namely the quantum indistinguishability viewpoint and the epistemic viewpoint. Propositions highlighted in green are “ True ” within the fol- lowed viewpoint and are propagated to further elements, if there are any. “ Truth ” propagation is represented by a green line in the diagram. Likewise, propositions highlighted in red are “ False ” within the followed viewpoint and do not propagate to further elements. We notice that the quantum and epistemic viewpoints disagree on the “ Truth ” character of the proposition that one must divide Eq. (1) by N!.
Figure 6. WN! rationales chart flow for continuous mixtures Chart flow representation of the two viewpoints often opposed to justify the use of 1/N! in Eq. (5), namely the quantum indistinguishability viewpoint and the epistemic viewpoint. Propositions highlighted in green are “ True ” within the fol- lowed viewpoint and are propagated to further elements, if there are any. “ Truth ” propagation is represented by a green line in the diagram. Likewise, propositions highlighted in red are “ False ” within the followed viewpoint and do not propagate to further elements. We notice that the quantum and epistemic viewpoints disagree on the “ Truth ” character of the proposition that one must divide Eq. (1) by N!.
Figure 7. Mixing of two polydisperse gases. Composition rep- resentation of the mixing of two continuous mixtures 1 and 2 of composition C; and C2 respectively. At this stage, one may want to determine the entropy of mixing of such substances as well as the final composition. This may shed some light on the initial paradox of similarity independence of the entropy of mixing pointed out by Gibbs in Ref. [31].
Figure 7. Mixing of two polydisperse gases. Composition rep- resentation of the mixing of two continuous mixtures 1 and 2 of composition C; and C2 respectively. At this stage, one may want to determine the entropy of mixing of such substances as well as the final composition. This may shed some light on the initial paradox of similarity independence of the entropy of mixing pointed out by Gibbs in Ref. [31].
Mixing entropy vs substance dissimilarity Figure 8. A solution to the “real ” Gibbs’ paradox. Plot of the dimensionless mixing entropy per particle as a function of the dissimilarity 612 between two polydisperse substances with compositions C; and C2 both normally distributed with the same standard deviation but with different means ju; and p2 respectively. Contrary to Fig. 3 illustrating an expected sharp drop from In 2 to zero as two substances become strictly iden- tical, we observe here that AS;,i2/Nkg goes continuously to zero as the substances become more and more similar i.e. as the overlap between their probability distributions increases.
Mixing entropy vs substance dissimilarity Figure 8. A solution to the “real ” Gibbs’ paradox. Plot of the dimensionless mixing entropy per particle as a function of the dissimilarity 612 between two polydisperse substances with compositions C; and C2 both normally distributed with the same standard deviation but with different means ju; and p2 respectively. Contrary to Fig. 3 illustrating an expected sharp drop from In 2 to zero as two substances become strictly iden- tical, we observe here that AS;,i2/Nkg goes continuously to zero as the substances become more and more similar i.e. as the overlap between their probability distributions increases.
Mapping from 1-particle system to 1-bit memory Figure 9. Mapping a 1-particle system onto a 1-bit memory system. Schematic representation on how a single particle in a box can be mapped onto a 1-bit memory state depending on where it is located with respect to a virtual dividing sur- face splitting the box in two equal parts. The bit memory states are denoted |0) and |1) depending on whether the par- ticle in the left hand side of the box or the right hand side respectively.
Mapping from 1-particle system to 1-bit memory Figure 9. Mapping a 1-particle system onto a 1-bit memory system. Schematic representation on how a single particle in a box can be mapped onto a 1-bit memory state depending on where it is located with respect to a virtual dividing sur- face splitting the box in two equal parts. The bit memory states are denoted |0) and |1) depending on whether the par- ticle in the left hand side of the box or the right hand side respectively.
Figure 10. Dissipated energy conditioned on particle type. Plot of the dissipated energy surface as a function particle type and a measure of how dissimilar are the compositions C1 and C2. The semi-transparent surface above the energy sur- face represents the distribution pc, which is used to compute the weighted average of Euiss:r(2)/kBI over the values of x to get the curve of (Egiss;r(z))2/keT coinciding with that of ASmic/Nkp in Fig. 7. Since we have mapped our demixing problem onto a memory resetting problem, we can determine the energy dissipated by applying the separation protocol P to a single equilibrated particle in a box with attribute X € [x,x + da] by applying Eq. (25) and replacing r with r(x).
Figure 10. Dissipated energy conditioned on particle type. Plot of the dissipated energy surface as a function particle type and a measure of how dissimilar are the compositions C1 and C2. The semi-transparent surface above the energy sur- face represents the distribution pc, which is used to compute the weighted average of Euiss:r(2)/kBI over the values of x to get the curve of (Egiss;r(z))2/keT coinciding with that of ASmic/Nkp in Fig. 7. Since we have mapped our demixing problem onto a memory resetting problem, we can determine the energy dissipated by applying the separation protocol P to a single equilibrated particle in a box with attribute X € [x,x + da] by applying Eq. (25) and replacing r with r(x).

Related papers

The Gibbs Paradox
Simon Saunders

Entropy, 2018

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs' notion of 'generic phase'). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. 'Distinguishability', in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

View PDFchevron_right
Fundamental Theoretical Corrections that Solve Gibbs' Paradox.
James A Putnam

Clausius first defined thermodynamic entropy. His definition included straight-forward thermodynamic properties of heat and temperature. Heat is energy in transit. As simple as his definition was, its meaning is not explained to this day. Descriptions of entropy mention but deliberately avoid directly addressing Clausius' entropy in favor of directing the readers' attention to Boltzmann's definition and the others that followed in its path. Boltzmann's entropy, easily explained, is not Clausius' entropy. This paper explains Clausius' entropy. The indirect, elusive "A measure of this..." or "A measure of that..." where "this" and "that" are not thermodynamic entropy, is replaced with Thermodynamic Entropy Itself Is...!! Theoretical physics cannot continue to not understand Clausius' Real Thermodynamic Entropy.

View PDFchevron_right
Comment on “The Gibbs paradox and the distinguishability of identical particles,” by M. A. M. Versteegh and D. Dieks [Am. J. Phys. 79, 741–746 (2011)]
David Corti

American Journal of Physics, 2012

We revisit recent discussions concerning the Gibbs paradox-the apparent discrepancy between the entropy change upon mixing identical gases as evaluated from the statistical mechanics of classical distinguishable particles and macroscopic thermodynamics. Contrary to what is often stated, we show that thermodynamics does not require this entropy of mixing to be zero. A zero value follows from the implicit assumption that the identical gas particles are indistinguishable. If the identical particles are explicitly assumed to be distinguishable, thermodynamics yields the same entropy of mixing as classical statistical mechanics.

View PDFchevron_right
The Gibbs Paradox and Particle Individuality
Dennis Dieks

Entropy, 2018

A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from "yes" to "no" is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S = k ln W or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.

View PDFchevron_right
Gibbs’ Paradox and the Definition of Entropy
Robert Swendsen

Entropy, 2008

Gibbs' Paradox is shown to arise from an incorrect traditional definition of the entropy that has unfortunately become entrenched in physics textbooks. Among its flaws, the traditional definition predicts a violation of the second law of thermodynamics when applied to colloids. By adopting Boltzmann's definition of the entropy, the violation of the second law is eliminated, the properties of colloids are correctly predicted, and Gibbs' Paradox vanishes.

View PDFchevron_right
Gibbs paradox: Mixing and non mixing potentials
Udayanandan k m

Entropy of mixing leading to Gibbs paradox is studied for different physical systems with and without potentials. The effect of potentials on the mixing and non mixing character of physical systems is discussed. We hope this article will encourage students to search for new problems which will help them understand statistical mechanics better.

View PDFchevron_right
The Gibbs Paradox Revisited
Dennis Dieks

Explanation, Prediction, and Confirmation, 2011

The Gibbs paradox has frequently been interpreted as a sign that particles of the same kind are fundamentally indistinguishable; and that quantum mechanics, with its identical fermions and bosons, is indispensable for making sense of this. In this article we shall argue, on the contrary, that analysis of the paradox supports the idea that classical particles are always distinguishable. Perhaps surprisingly, this analysis extends to quantum mechanics: even according to quantum mechanics there can be distinguishable particles of the same kind. Our most important general conclusion will accordingly be that the universally accepted notion that quantum particles of the same kind are necessarily indistinguishable rests on a confusion about how particles are represented in quantum theory.

View PDFchevron_right
Statistical and kinematic arrangements and their thermodynamic and informational interpretations: Gibbs paradox
Hejna Bohdan

He264591@athos, 2025

A new way to deal with the given topic is the approach of finding the connection between the mathematical definitions of statistical and information entropy, as well as their mutual relations within the system of stochastic quantities. Especially with thermodynamic entropies expressed on an isolated system in which our repeatable observations are made and where it is a cyclic conversion of thermal energy within the observed/measured system. At the same time, we use the description of information to analyze the Gibbs paradox, which is also a current property of such observation/measurement of an equilibrium thermodynamic system. Although not immediately obvious in the first connection with the thermal analysis, it indicates a whole series of questions not yet solved but greedy like this one.

View PDFchevron_right
Planck's experiment on gas mixing, Gibbs' paradox and the definition of entropy in statistical mechanics
Andrea Paglietti

Physics Letters A, 2018

The interpretation of entropy provided by statistical thermodynamics is not adequate to represent the thermodynamic entropy of the gas of noninteracting particles considered in this theory. Planck's thought experiment on reversible mixing and Gibbs' paradox provide perhaps the best-known evidence of this. The assumption that the internal energy of an ideal gas depends only on its temperature is introduced both in the kinetic theory of gases and in the classical thermodynamics. Such an assumption is no doubt adequate to deal with real gases at appropriately low pressures and high temperatures. However, the present paper shows that the same assumption implies that the entropy of an ideal gas, like its internal energy, must also depend only on temperature. The paper calculates the expression of the entropy function that is consistent with the internal energy function of the gas. From this expression, the thermodynamic entropy of the ideal gas-as distinct from its statistical entropy-is finally expressed in terms of statistical mechanics variables.

View PDFchevron_right
The Gibbs paradox and the distinguishability of identical particles
Dennis Dieks

American Journal of Physics, 2011

Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the micro-level, and from the relation S = k ln W leads to a non-extensive 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 after all; and even that quantum mechanics is indispensable for making sense of this. In contrast, we argue that the classical statistics of distinguishable particles and the resulting non-extensive 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.

View PDFchevron_right

References (55)

  1. D. C. Mattis, A Guide for Students and Researchers: Sta- tistical Mechanics Made Simple, 1st ed. (World Scientific, 2003).
  2. W. Greiner, L. Neise, and H. Stocker, Thermodynamics and statistical mechanics, 1st ed. (Springer, 1995).
  3. M. D. and D. Allan, Statistical Mechanics, 1st ed. (Uni- versity Science Books, 2000).
  4. M. E. Tuckerman, Statistical mechanics: theory and molecular simulation, 1st ed. (Oxford university press, 2010).
  5. C. Kittel, Elementary statistical physics, 5th ed. (Dover publications, 2004).
  6. D. Chandler, Introduction to Modern Statistical Mechan- ics, 1st ed. (Oxford University Press, 1987).
  7. M. Glazer and J. Wark, Statistical mechanics a survival guide, 1st ed. (Oxford University Press, 2001).
  8. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, 1st ed. (John Wiley & Sons, 1975).
  9. J. Honerkamp, Statistical Physics, 2nd ed. (Springer, 2002).
  10. K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, 1987).
  11. G. Morandi, F. Napoli, and E. Ercolessi, Statistical Me- chanics an intermediate course, 2nd ed. (World Scientific, 2001).
  12. R. Balian, From Microphysics to Macrophysics: Methods and applications of statistical physics (Springer, 2003).
  13. O. Penrose, Fundation of Statistical Mechanics: A deduc- tive Treatment (Dover Publications, 2005).
  14. D. S. Betts and R. E. Turner, Introductory Statistical Mechanics (Addison Wesley, 1992).
  15. Y. M. Guttman, The concept of probability in statistical mechanics (Cambridge University Press, 2000).
  16. G. V. Rosser, An introduction to statistical physics (Ellis Horwood publishers, 1982).
  17. B. Diu, D. Guthmann, C.and Lederer, and B. Roulet, Physique Statistique (Hermann, 2001).
  18. D. S. Lemons, A student's guide to entropy (Cambridge university press, 2013).
  19. A. Ben-Naim, Entropy and the Second Law (World Sci- entific, 2012).
  20. G. H. Wannier, Statistical Physics (Dover, 1966).
  21. B. H. Lavenda, Statistical Physics: A probabilistic Ap- proach (Wiley Interscience, 1991).
  22. C. B. P. Finn, Thermal Physics, 2nd ed. (CRC Press, 1993).
  23. J. M. Seddon and J. D. Gale, Thermodynamics and sta- tistical mechanics, 1st ed. (RSC publishing, 2001).
  24. D. Kondepudi and I. Prigogine, Modern thermodynamics, 2nd ed. (Wiley, 2015).
  25. N. Sator and N. Pavloff, Physique Statistique, 1st ed. (Vuibert, 2016).
  26. D. Goodstein, Thermal physics, Energy and Entropy, 1st ed. (Cambridge University Press, 2015).
  27. S. J. Blundell and K. M. Blundell, Concepts in thermal physics, 1st ed. (Oxford University Press, 2010).
  28. L. Landau and E. M. Lifshitz, Statistical Physics part 1, 3rd ed. (Elsevier, 1980).
  29. J. W. Gibbs, Elementary Principles in Statistical Me- chanics (Ox Bow Press, 1981).
  30. Some disagreement on where to put exactly the N ! per- sists but as far as mathematics is concerned it is equiva- lent.
  31. J. W. Gibbs, Connecticut Acad. Sci. , 108 (1876).
  32. E. T. Jaynes, in Maximum Entropy and Bayesian Meth- ods, edited by C. Smith, G. Erickson, and P. Neudorfer (Kluwer Academic, 1992) pp. 1-22.
  33. D. Wallace, "Thermodynamics as control theory," (2013).
  34. A. M. L. Messiah and O. W. Greenberg, Phys. Rev. 136, B248 (1964).
  35. O. W. Greenberg and A. M. L. Messiah, Phys. Rev. 138, B1155 (1965).
  36. E. T. Jaynes, "Clearing up mysteries -the origi- nal goal," in Maximum Entropy and Bayesian Meth- ods: Cambridge, England, 1988 , edited by J. Skilling (Springer Netherlands, Dordrecht, 1989) pp. 1-27.
  37. J. J. Salacuse, J.Chem.Phys. 81, 2468 (1984).
  38. J. Zhu, M. Li, R. Rogers, W. Meyer, R. H. Ottewill, W. B. STS-73 Space Shuttle Crew, Russel, and P. M. Chaikin, Nature 387, 883 (1997).
  39. D. Frenkel, Nat. Mat. 14, 9 (2015).
  40. R. H. Swendsen, Am. J. Phys. 74, 187 (2006).
  41. D. Frenkel, Mol. Phys. 112, 2325 (2014).
  42. M. Ozawa and L. Berthier, J. Chem. Phys. 146, 014502 (2017).
  43. M. E. Cates and V. N. Manoharan, Soft. Matt. 15, 6538 (2015).
  44. F. Paillusson and I. Pagonabarraga, J. Stat. Mech. 2014, P10038 (2014).
  45. In more technical terms, the density ρ ∆ C (x) can be in- terpreted as the Radon-Nikodym derivative of the prob- ability measure of a random variable X ∆ defined as the conditional expectation of X with respect to the sub-σ- algebra (made of the set of the ∆x covering the domain of integration of X) of the Borel σ-algebra.
  46. This limit must be interpreted in a careful way for it does not exist in actuality. One must interpret ∆x as compris- ing two contributions Λ(x) and such that ∆x = Λ(x) and interpret the limit when ∆x tends to zero as the limit when tends to zero at fixed Λ(x). A diverging term in -ln appears which makes the limit non-existent. To by- pass this issue, one may regularise by subtracting -ln from the limit.
  47. P. Maynar and E. Trizac, Phys. Rev. Lett. 106, 160603 (2011).
  48. D. M. Endres and J. E. Schindelin, IEEE Trans. Inf. The- ory. 49, 1858 (2003).
  49. P. Sollich, J. Phys. Cond. Matt. 14, 79 (2002).
  50. L. Szilar, Z. Phys. 53, 840 (1929).
  51. R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
  52. C. H. Bennett, Stud. Hist. Philos. Sci. B 34, 501 (2003).
  53. A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature 5, 183 (2012).
  54. P. Enders, Progr. Phys. 3, 85 (2007).
  55. P. Enders, Entropy 11, 454 (2009).

Related papers

Probability, Entropy, and Gibbs’ Paradox(es)
Robert Swendsen

Entropy

Two distinct puzzles, which are both known as Gibbs' paradox, have interested physicists since they were first identified in the 1870s. They each have significance for the foundations of statistical mechanics and have led to lively discussions with a wide variety of suggested resolutions. Most proposed resolutions had involved quantum mechanics, although the original puzzles were entirely classical and were posed before quantum mechanics was invented. In this paper, I show that contrary to what has often been suggested, quantum mechanics is not essential for resolving the paradoxes. I present a resolution of the paradoxes that does not depend on quantum mechanics and includes the case of colloidal solutions, for which quantum mechanics is not relevant.

View PDFchevron_right
The “Real” Gibbs Paradox and a Composition-Based Resolution
Fabien Paillusson

Entropy

There is no documented evidence to suggest that J. W. Gibbs did not recognize the indistinguishable nature of states involving the permutation of identical particles or that he did not know how to justify on a priori grounds that the mixing entropy of two identical substances must be zero. However, there is documented evidence to suggest that Gibbs was puzzled by one of his theoretical findings, namely that the entropy change per particle would amount to kBln2 when equal amounts of any two different substances are mixed, no matter how similar these substances may be, and would drop straight to zero as soon as they become exactly identical. The present paper is concerned with this latter version of the Gibbs paradox and, to this end, develops a theory characterising real finite-size mixtures as realisations sampled from a probability distribution over a measurable attribute of the constituents of the substances. In this view, two substances are identical, relative to this measurable ...

View PDFchevron_right
Yet another resolution of the Gibbs paradox: an information theory approach
Chih-Yuan Tseng

2002

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.

View PDFchevron_right
On the Occurrence of Gibbs Paradox
Davide Giusti

Journal of Computational and Theoretical Transport, 2019

This work focuses on the conceptual problem of the intermixing of two samples of the same ideal gas under identical conditions (p,V,T) giving rise to the riddle known as Gibbs paradox. Starting from the very general entropy definition of the kinetic theory, for clearly focusing the question, we found that a careful analysis both from the points of view of classical kinetic theory and thermodynamics (considering also the proper constraints) drives once more to the conclusion that they are in mutual agreement and that, actually, the paradox does not subsist. Considering also the approach of Statistical Mechanics, it is easy to conclude that it also is formally coherent with the more straightforward solution provided by kinetic theory.

View PDFchevron_right
A note on Gibbs paradox
Radhakrishnamurty Padyala

We show in this note that Gibbs paradox arises not due to application of thermodynamic principles, whether classical or statistical or even quantum mechanical, but due to incorrect

View PDFchevron_right

Related topics

  • Molecular Physics
  • THEORETICAL AND COMPUTATIONAL CH...

    • Cited by

    On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures
    Fabien Paillusson

    Entropy

    Most undergraduate students who have followed a thermodynamics course would have been asked to evaluate the volume occupied by one mole of air under standard conditions of pressure and temperature. However, what is this task exactly referring to? If air is to be regarded as a mixture, under what circumstances can this mixture be considered as comprising only one component called “air” in classical statistical mechanics? Furthermore, following the paradigmatic Gibbs’ mixing thought experiment, if one mixes air from a container with air from another container, all other things being equal, should there be a change in entropy? The present paper addresses these questions by developing a prior-based statistical mechanics framework to characterise binary mixtures’ composition realisations and their effect on thermodynamic free energies and entropies. It is found that (a) there exist circumstances for which an ideal binary mixture is thermodynamically equivalent to a single component ideal...

    View PDFchevron_right
    The “Real” Gibbs Paradox and a Composition-Based Resolution
    Fabien Paillusson

    Entropy

    There is no documented evidence to suggest that J. W. Gibbs did not recognize the indistinguishable nature of states involving the permutation of identical particles or that he did not know how to justify on a priori grounds that the mixing entropy of two identical substances must be zero. However, there is documented evidence to suggest that Gibbs was puzzled by one of his theoretical findings, namely that the entropy change per particle would amount to kBln2 when equal amounts of any two different substances are mixed, no matter how similar these substances may be, and would drop straight to zero as soon as they become exactly identical. The present paper is concerned with this latter version of the Gibbs paradox and, to this end, develops a theory characterising real finite-size mixtures as realisations sampled from a probability distribution over a measurable attribute of the constituents of the substances. In this view, two substances are identical, relative to this measurable ...

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved