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The Gibbs Paradox

Simon W. Saunders

Published Jul 25, 2018• 0 citations
OpenAlex

Abstract

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.

Related papers

The Gibbs Paradox

Simon Saunders

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.

arXiv:1808.01953•Entropy•Jul 25, 2018•20 citations

The Gibbs paradox and the physical criteria for indistinguishability of identical particles

C. S. Unnikrishnan

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.

arXiv:1811.03967•International Journal of Quantum Information•Aug 18, 2016•1 citation

The Gibbs Paradox. About the paradox and the indistinguishability of identical particles

S.C. van Berkum

The Gibbs Paradox is an important question in thermodynamics. In this thesis I search for a solution for the paradox from a classical, philosophical and quantum mechanical point of view. A common way to solve the Gibbs Paradox is by using the reduced entropy. However, this definition of entropy should not be interpreted as an argument for the indistinguishability of identical particles, because all other particles which have unique trajectories through space-time are distinguishable in principle. The Gibbs paradox could be solved in a few ways, but each solution has certain constrains.

•May 26, 2014•0 citations

The Gibbs Paradox: Lessons from Thermodynamics

Janneke van Lith

The Gibbs paradox in statistical mechanics is often taken to indicate that already in the classical domain particles should be treated as fundamentally indistinguishable. This paper shows, on the contrary, how one can recover the thermodynamical account of the entropy of mixing, while treating states that only differ by permutations of similar particles as distinct. By reference to the orthodox theory of thermodynamics, it is argued that entropy differences are only meaningful if they are related to reversible processes connecting the initial and final state. For mixing processes, this means that processes should be considered in which particle number is allowed to vary. Within the context of statistical mechanics, the Gibbsian grandcanonical ensemble is a suitable device for describing such processes. It is shown how the grandcanonical entropy relates in the appropriate way to changes of other thermodynamical quantities in reversible processes, and how the thermodynamical account of the entropy of mixing is recovered even when treating the particles as distinguishable.

Entropy•Apr 30, 2018•8 citations

The Gibbs paradox and the distinguishability of identical particles

Marijn A. M. Versteegh, Dennis Dieks

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.

arXiv:1012.4111•American Journal of Physics•Jun 17, 2011•44 citations

The Gibbs Paradox Revisited

Dennis Dieks

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.

arXiv:1003.0179•Feb 28, 2010•0 citations
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.
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.
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.
The Gibbs Paradox is an important question in thermodynamics. In this thesis I search for a solution for the paradox from a classical, philosophical and quantum mechanical point of view. A common way to solve the Gibbs Paradox is by using the reduced entropy. However, this definition of entropy should not be interpreted as an argument for the indistinguishability of identical particles, because all other particles which have unique trajectories through space-time are distinguishable in principle. The Gibbs paradox could be solved in a few ways, but each solution has certain constrains.
The Gibbs paradox in statistical mechanics is often taken to indicate that already in the classical domain particles should be treated as fundamentally indistinguishable. This paper shows, on the contrary, how one can recover the thermodynamical account of the entropy of mixing, while treating states that only differ by permutations of similar particles as distinct. By reference to the orthodox theory of thermodynamics, it is argued that entropy differences are only meaningful if they are related to reversible processes connecting the initial and final state. For mixing processes, this means that processes should be considered in which particle number is allowed to vary. Within the context of statistical mechanics, the Gibbsian grandcanonical ensemble is a suitable device for describing such processes. It is shown how the grandcanonical entropy relates in the appropriate way to changes of other thermodynamical quantities in reversible processes, and how the thermodynamical account of the entropy of mixing is recovered even when treating the particles as distinguishable.
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.
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.