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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 k B ln 2 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 attribute, if they have the same underlying probability distribution. This implies that two identical mixtures do not need to have identical finite-size realisations of their compositions. By averaging over composition realisations, it is found that (1) fixed composition mixtures behave as homogeneous single-component substances and (2) in the limit of a large system size, the entropy of mixing per particle shows a continuous variation from k B ln 2 to 0, as two different substances are made more similar, thereby resolving the "real" Gibbs paradox. [ABSTRACT FROM AUTHOR]

In science, so-called "problem tasks" sometimes arise, the solution of which has been sought for a long time by several generations of scientists. Possessing deep knowledge and developed intuition, they sometimes predict the result of a problem, but they cannot find a solution. Among such problems of science can be attributed the problem that entered physics under the name "Gibbs paradox". Gibbs predicted all the properties of the entropy of a mixture. He showed that when two different gases are mixed, the entropy of the mixture exceeds the sum of the initial entropies of the components by a value called the jump of entropy. However, he was unable to explain the physical nature of this jump. The fact is that its value does not depend on the nature of gases and has the same value for a mixture of gases of any arbitrary chemical composition. For more than a century, there has been the Gibbs paradox associated with the fundamental parameter of physical systems — entropy, and so far no one has been able to clarify this paradox or at least find out what causes it to form. [ABSTRACT FROM AUTHOR]

Distinguishability plays a major role in quantum and statistical physics. When particles are identical their wave function must be either symmetric or antisymmetric under permutations and the number of microscopic states, which determines entropy, is counted up to permutations. When the particles are distinguishable, wavefunctions have no symmetry and each permutation is a different microstate. This binary and discontinuous classification raises a few questions: one may wonder what happens if particles are almost identical, or when the property that distinguishes between them is irrelevant to the physical interactions in a given system. Here I sketch a general answer to these questions. For any pair of non-identical particles there is a timescale, τ d , required for a measurement to resolve the differences between them. Below τ d , particles seem identical, above it - different, and the uncertainty principle provides a lower bound for τ d . Thermal systems admit a conjugate temperature scale, T d . Above this temperature the system appears to equilibrate before it resolves the differences between particles, below this temperature the system identifies these differences before equilibration. As the physical differences between particles decline towards zero, τ d → ∞ and T d → 0 . [ABSTRACT FROM AUTHOR]

This paper is dedicated to overcoming the main contradictions attributed to entropy, so that this parameter could take its undoubtedly deserved place in the arsenals of modern science. [ABSTRACT FROM AUTHOR]

The concept of indistinguishable particles in quantum theory is fundamental to questions of ontology. All ordinary matter is made of electrons, protons, neutrons, and photons and they are all indistinguishable particles. Yet the concept itself has proved elusive, in part because of the interpretational difficulties that afflict quantum theory quite generally, and in part because the concept was so central to the discovery of the quantum itself, by Planck in 1900; it came encumbered with revolution. I offer a deflationary reading of the concept 'indistinguishable' that is identical to Gibbs' concept 'generic phase', save that it is defined for state spaces with only finitely -many states of bounded volume and energy (finitely-many orthogonal states, in quantum mechanics). That, and that alone, makes for the difference between the quantum and Gibbsean concepts of indistinguishability. This claim is heretical on several counts, but here we consider only the content of the claim itself, and its bearing on the early history of quantum theory rather than in relation to contemporary debates about particle indistinguishability and permutation symmetry. It powerfully illuminates that history. • Indistinguishable particles have the same state-independent properties, but may differ wildly in state-dependent properties. • Gibbs' concept of indistinguishability is the same as that in quantum theory, when carried over to a finite state space. • Not even classical indistinguishable particles are statistically independent, in Einstein's sense. • Einstein's 1905 light quantum argument assumed rather than derived statistical independence. • The statistical dependencies of indistinguishable particles can be explained classically, without appeal to entanglement. [ABSTRACT FROM AUTHOR]

The postulate of nonequilibrium is at the heart of the second law of thermodynamics. According to this postulate, there is a real property of matter -- "nonequilibrium," which characterizes the uneven distribution of matter and motion in space. All processes (reversible and irreversible) can occur only in nonequilibrium systems. As a quantitative characteristic of the nonequilibrium of the system, the maximum work that can be performed during the transition of the nonequilibrium system to the equilibrium state is considered. The only formulation of the second law is given. When real (irreversible) processes occur, the nonequilibrium of the isolated system decreases, and in reversible processes, the nonequilibrium in the system of locally equilibrium subsystems does not change (the increment of one kind of the nonequilibrium entirely compensated by a decrease in the disequilibrium of some other kind). When the system reaches an equilibrium state, the disequilibrium disappears, and all processes cease. The article provides a calculated confirmation of the theoretical provisions of the concept of nonequilibrium and its mathematical apparatus by examples of determining the loss of the nonequilibrium of system when an isothermal mixing of dissimilar gases, and changes of nonequilibrium of system "pure solvent -- solution" in the transition of part of the solvent in the solution. The mixing of the same gases leads to the Gibbs paradox, which is also considered in this paper. The concept of nonequilibrium was developed and the quantitative characteristics (measures) of nonequilibrium of the system were introduced allow to study nonequilibrium systems consisting of locally equilibrium subsystems in the sections of classical thermodynamics as simply as individual equilibrium systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The link between statistical mechanics and information theory is examined with reference to the classical ideal gas. In particular, Jaynes' assertion [E. T. Jaynes, Phys. Rev. 106(4), 620 (1957)] that a clear distinction between statistical and physical aspects, the latter consisting of the correct enumeration of the system's states and their properties, is questioned in relation to the information entropy. Jaynes suggested that statistical mechanics need not be considered a physical theory, as the techniques of statistical inference lead to physically sensible results. However, we show that a physical theory of the statistics of the ideal gas leads to expressions for the information entropy that differ from those found in statistical mechanics because the probability distributions underlying the entropy differ from the canonical distribution. In particular, we show that the distribution of energy states accessible to a classical ideal gas connected to a thermal reservoir is given by the Gamma distribution and in open systems, in which the number of particles fluctuates, by a weighted sum of Gamma distributions. Computer simulations using a hard-sphere model of a classical ideal gas demonstrate the ideas. We compare the Shannon information entropy with the thermodynamic entropy and argue on the basis of the work presented here that information theoretic entropy and thermodynamic entropy, whilst seemingly related, are not necessarily identical. [ABSTRACT FROM AUTHOR]

The suggestion that particles of the same kind may be indistinguishable in a fundamental sense, even so that challenges to traditional notions of individuality and identity may arise, has first come up in the context of classical statistical mechanics. In particular, the Gibbs paradox has sometimes been interpreted as a sign of the untenability of the classical concept of a particle and as a premonition that quantum theory is needed. This idea of a 'quantum connection' stubbornly persists in the literature, even though it has also been criticized frequently. Here we shall argue that although this criticism is justified, the proposed alternative solutions have often been wrong and have not put the paradox in its right perspective. In fact, the Gibbs paradox is unrelated to fundamental issues of particle identity; only distinguishability in a pragmatic sense plays a role (in this we develop ideas of van Kampen []), and in principle the paradox always is there as long as the concept of a particle applies at all. In line with this we show that the paradox survives even in quantum mechanics, in spite of the quantum mechanical (anti-)symmetrization postulates. [ABSTRACT FROM AUTHOR]

The link; between statistical mechanics and information theory is examined with reference to the classical ideal gas. In particular, Jaynes' assertion [E. T. Jaynes, Phys. Rev. 106(4), 620 (1957)] that a clear distinction between statistical and physical aspects, the latter consisting of the correct enumeration of the system's states and their properties, is questioned in relation to the information entropy. Jaynes suggested that statistical mechanics need not be considered a physical theory, as the techniques of statistical inference lead to physically sensible results. However, we show that a physical theory of the statistics of the ideal gas leads to expressions for the information entropy that differ from those found in statistical mechanics because the probability distributions underlying the entropy differ from the canonical distribution. In particular, we show that the distribution of energy states accessible to a classical ideal gas connected to a thermal reservoir is given by the Gamma distribution and in open systems, in which the number of particles fluctuates, by a weighted sum of Gamma distributions. Computer simulations using a hard-sphere model of a classical ideal gas demonstrate the ideas. We compare the Shannon information entropy with the thermodynamic entropy and argue on the basis of the work presented here that information theoretic entropy and thermodynamic entropy, whilst seemingly related, are not necessarily identical. [ABSTRACT FROM AUTHOR]

Colloidal particles are distinguishable. Moreover, their thermodynamic properties are extensive. Statistical mechanics predicts such behaviour if one accepts that the configurational integral of a system ofNcolloids must be divided byN!. In many textbooks, it is argued that the factorN! corrects for the fact that identical particles (in the quantum mechanical sense) are indistinguishable. Clearly, this argument does not apply to colloids. This article explains why, nevertheless, all is well. The point has been made before, but has not yet sunk in. I also discuss the effect of polydispersity. [ABSTRACT FROM AUTHOR]

The idea that, in the microscopic world, particles are indistinguishable, interchangeable and without identity has been central in quantum physics. The same idea has been enrolled in statistical thermodynamics even in a classical framework of analysis to make theoretical results agree with experience. In thermodynamics of gases, this hypothesis is associated with several problems, logical and technical. For this case, an alternative theoretical framework is provided, replacing the indistinguishability hypothesis with standard probability and statistics. In this framework, entropy is a probabilistic notion applied to thermodynamic systems and is not extensive per se. Rather, the extensive entropy used in thermodynamics is the difference of two probabilistic entropies. According to this simple view, no paradoxical behaviors, such as the Gibbs paradox, appear. Such a simple probabilistic view within a classical physical framework, in which entropy is none other than uncertainty applicable irrespective of particle size, enables generalization of mathematical descriptions of processes across any type and scale of systems ruled by uncertainty. [ABSTRACT FROM AUTHOR]

The ideal-gas barometric and pressure laws are derived from the Democritian concept of independent corpuscles moving in vacuum, plus a principle of simplicity, namely that these laws are independent of the kinetic part of the Hamiltonian. A single corpuscle in contact with a heat bath in a cylinder and submitted to a constant force (weight) is considered. The paper importantly supplements a previously published paper: First, the stability of ideal gases is established. Second, we show that when walls separate the cylinder into parts and are later removed, the entropy is unaffected. We obtain full agreement with Landsberg's and others' (1994) classical thermodynamic result for the entropy of a column of gas submitted to gravity. [ABSTRACT FROM AUTHOR]

The solution of the Gibbs paradox in the 'Mathematical Encyclopedia' [1] contained an error. It is also contained in many physics textbooks. A correct solution is given in this paper. New thermodynamic quantities taking into account the occurrence of dimers and clusters are introduced. [ABSTRACT FROM AUTHOR]

Based on the property of extensivity (mathematically, homogeneity of first degree), we derive in a mathematically consistent manner the explicit expressions of the chemical potential μ and the Clausius entropy S for the case of monoatomic ideal gases in open systems within phenomenological thermodynamics. Neither information theoretic nor quantum mechanical statistical concepts are invoked in this derivation. Considering a specific expression of the constant term of S, the derived entropy coincides with the Sackur-Tetrode entropy in the thermodynamic limit. We demonstrate, however, that the former limit is not contained in the classical thermodynamic relations, implying that the usual resolutions of Gibbs paradox do not succeed in bridging the gap between the thermo-dynamics and statistical mechanics. We finally consider the volume of the phase space as an entropic measure, albeit, without invoking the thermodynamic limit to investigate its relation to the thermo-dynamic equation of state and observables. [ABSTRACT FROM AUTHOR]

By analyzing different examples of practical entropy calculations and using concepts such as conformational and residual entropies, I show herein that experimental calorimetric entropies of single molecules can be theoretically reproduced considering chemically identical atoms either as distinguishable or indistinguishable particles. The broadly used correction in entropy calculations due to the symmetry number and particle indistinguishability is not mandatory, as an ad hoc correction, to obtain accurate values of absolute and relative entropies. It is shown that, for any chemical reaction of any kind, considering distinguishability or indistinguishability among identical atoms is irrelevant as long as we act consistently in the calculation of all the required entropy contributions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it. [ABSTRACT FROM AUTHOR]

The author constructs a one-particle distribution generalizing the Maxwell-Boltzmann and Bose-Einstein distributions for classical particles, without incurring the Gibbs paradox, and, simultaneously, constructs the thermodynamics of fluids and phase transitions on the basis of the concept of creation of dimers and clusters and obtains well-known laws and relations of thermodynamics as well as explains some recently, discovered effects (such as the jamming effect). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the relationship between probability theory and the economics of assets. We show that the notion of Bose condensate can be applied in economics. [ABSTRACT FROM AUTHOR]

Abstract: A series of pyridinium-based cationic surfactants has been synthesised and their amphiphilic properties have been studied by conductivity and surface tension measurements. The modification of the substitution pattern on the pyridinium ring by hydrophobic moieties (methyl vs. hydrogen and presence or not of condensed benzene ring) gave the opportunity to investigate structure–activity relationships. Characterization by conductivity and surface tension measurements shed light on the behaviour at the air/water interface and in the micellar environment. In particular, the tendency to form ion pairs at very low concentration was evidenced for all the surfactants substituted on the ring, but not for the simple pyridinium ones. The formation of ion pairs affects both the conductivity and the surface tension plots, showing that a series of steps is involved during the adsorption to the air/water surface. An attempt was made to qualify the single steps in the adsorption at the surface layer. Those steps were attributed to different chemical species (free surfactant ions or ion pairs) and to different arrangements of the surfactant. This work also represents a contribution of investigation at very low surfactant concentrations and high surface tension values. [Copyright &y& Elsevier]

The issue of the thermodynamics of a system of distinguishable particles is discussed in this paper. In constructing the statistical mechanics of distinguishable particles from the definition of Boltzmann entropy, it is found that the entropy is not extensive. The inextensivity leads to the so-called Gibbs paradox in which the mixing entropy of two identical classical gases increases. Lots of literature from different points of view were created to resolve the paradox. In this paper, starting from the Boltzmann entropy, we present the thermodynamics of the system of distinguishable particles. A straightforward way to get the corrected Boltzmann counting is shown. The corrected Boltzmann counting factor can be justified in classical statistical mechanics. [ABSTRACT FROM AUTHOR]

We obtain a Bose-Einstein-type distribution for the classical vapor. We show that the analogue of the Bose condensate is the formation of clusters. We write a new PV diagram for interaction with the form of the Lennard-Jones potential using scattering theory and a strict constraint. We compare our results with experimental data. [ABSTRACT FROM AUTHOR]

The author presents a study on derived formulas which indicate how the number of gas particles are distributed within a given interval of velocities. He cites that if the number of particles are much less than the maximal number of particles, the modified formula coincide with the Gibbs distribution. He pointed out that the solution of the Gibbs paradox can be derived once Gibbs formula in its classical form is realized as invalid.

In this paper we analyze the relationship between entropy and information in the context of the mixing process of two identical ideal gases. We will argue that entropy has a special information-based feature that is enfolded in the statistical entropy, but the second law does not include it directly. Therefore, in some given processes in thermodynamics where there is no matter and energy interaction between the system and the environment, the state of the system may go toward a situation of lower probability to increase the observer's information about the environment. This is a kind of information-based interaction in which the total entropy is not constrained by the second law. [ABSTRACT FROM AUTHOR]

Abstract: The concept of classical indistinguishability is analysed and defended against a number of well-known criticisms, with particular attention to the Gibbs’ paradox. Granted that it is as much at home in classical as in quantum statistical mechanics, the question arises as to why indistinguishability, in quantum mechanics but not in classical mechanics, forces a change in statistics. The answer, illustrated with simple examples, is that the equilibrium measure on classical phase space is continuous, whilst on Hilbert space it is discrete. The relevance of names, or equivalently, properties stable in time that can be used as names, is also discussed. [Copyright &y& Elsevier]

We consider an elementary collision model of a molecular reservoir upon which an external field is applied and the work is dissipated into heat. To realize macroscopic irreversibility at the microscopic level, we introduce a "graceful" irreversible map which randomly mixes the identities of the molecules. This map is expected to generate informatic entropy exactly equal to the independently calculable irreversible thermodynamic entropy. [ABSTRACT FROM AUTHOR]

The conventional way of explaining Gibbs paradox as due to the distinguishability of particles has been challenged recently and a new fundamental definition for the entropy has been proposed that gives the same entropy for distinguishable particles as for indistinguishable particles. In this paper arguments are presented that strongly favor the conventional definition of entropy and its resolution of Gibbs paradox. [ABSTRACT FROM AUTHOR]

In contrast to the supercritical region where the isotherms ideally coincide with the isotherms of the van der Waals model, we introduce an additional condition for the Hartley entropy to attain maximum on the spinodal in the subcritical region. To determine the parameters of the distribution and isotherms, we use spinodal equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article is a detailed history of the Gibbs paradox, with philosophical morals. It purports to explain the origins of the paradox, to describe and criticize solutions of the paradox from the early times to the present, to use the history of statistical mechanics as a reservoir of ideas for clarifying foundations and removing prejudices, and to relate the paradox to broad misunderstandings of the nature of physical theory. [ABSTRACT FROM AUTHOR]