entropy Article Ensemble Equivalence for Distinguishable Particles Antonio Fernández-Peralta and Raúl Toral * Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC), Consejo Superior de Investigaciones Científicas (CSIC)–Universitat de les Illes Balears (UIB), Palma de Mallorca 07122, Spain; afperalta@ifisc.uib-csic.es * Correspondance: raul@ifisc.uib-csic.es; Tel.: +34-971-17-32-35; Fax: +34-971-17-32-48 Academic Editor: Antonio M. Scarfone Received: 29 April 2016; Accepted: 8 July 2016; Published: 13 July 2016 Abstract: Statistics of distinguishable particles has become relevant in systems of colloidal particles and in the context of applications of statistical mechanics to complex networks. In this paper, we present evidence that a commonly used expression for the partition function of a system of distinguishable particles leads to huge fluctuations of the number of particles in the grand canonical ensemble and, consequently, to nonequivalence of statistical ensembles. We will show that the alternative definition of the partition function including, naturally, Boltzmann’s correct counting factor for distinguishable particles solves the problem and restores ensemble equivalence. Finally, we also show that this choice for the partition function does not produce any inconsistency for a system of distinguishable localized particles, where the monoparticular partition function is not extensive. Keywords: entropy; distinguishable particles; ensemble equivalence 1. Introduction The Gibbs paradox, namely, the entropy not being extensive for a classical ideal gas, is commonly solved by adding an ad hoc term to the entropy, −k log(N!) or, using Stirling formula, −kN log(N/e), where N is the number of particles and k is Boltzmann’s constant. This term appears if one divides the number of available states by N!, the so-called “correct Boltzmann counting”. Its physical justification is usually attributed by several authors and textbooks [1,2] to quantum mechanics and the indistinguishable nature of identical particles. The inclusion or not of Boltzmann’s correct counting and the correct definition of entropy for distinguishable particles has been widely discussed in the literature since the earlier works by Ehrenfest and Trkal [3] and van Kampen [4] and the more modern contributions, see, e.g., [5–14]. This topic turns out not to be just of academic interest, but appears to be relevant in current research, where the importance of distinguishable statistics has increased to a large degree. For example, when studying colloids one has to take into account that no two colloidal particles are exactly alike, with differences in the mass and shape of each particle, and they are, consequently, distinguishable between them. In such a case, the indistinguishability explanation of the Gibbs paradox fails resoundingly and we have to consider other possible alternatives [15,16] to explain experimental results with colloids. Another example is that of statistical mechanics of networks [17–19], where edges/links of the network are considered as particles, and pairs of vertices/nodes are considered as energy states, establishing a straightforward analogy with quantum physical systems. In this case, links correspond to individual identifiable actions and it seems very forced to regard them as indistinguishable [20]. This distinguishability of links becomes important for multi-edged networks (where a pair of nodes can have more than one link) when making statistics and entropy measures [21]. Entropy 2016, 18, 259; doi:10.3390/e18070259 www.mdpi.com/journal/entropy Entropy 2016, 18, 259 2 of 13 In this paper we want to contribute to this topic by discussing the issue of ensemble equivalence. The problem appeared to us [22] when studying the grand canonical formalism of distinguishable particles (or network links). When using the common textbook expression of the partition function [1], anomalous fluctuations of the number of particles are obtained, leading to ensemble nonequivalence between the microcanonical/canonical and the grand canonical ensembles. The nonequivalence of ensembles casts doubts on the suitability of the statistical description. We will prove that the inclusion of the N! term for distinguishable particles solves the problem and restores ensemble equivalence. Although it would seem that this inclusion should give non-extensive thermodynamic potentials for localized particles, we will show that this is not the case and the definition including N! is consistent in all cases. The paper is organized as follows: in Section2 we introduce the standard definitions of distinguishability, identical, etc., and other notation used in Statistical Mechanics, together with some simple Hamiltonian functions that will clarify those concepts. Section3 presents the problem of ensemble equivalence for distinguishable particles that arises when using the common textbook expression for the partition function and entropy. Finally, in Section4 we explain how the definition of entropy including Boltzmann’s correct counting solves this problem for all the examples given in Section2. We end with some conclusions in Section5. 2. Preliminary Concepts Indistinguishable particles are those described by a particle-exchange invariant set of microscopic states. If this invariance is not fulfilled, particles are called distinguishable. In classical mechanics, a microstate for a system of N particles corresponds to a point in phase space (q, p) ≡ (~q1,~q2,...,~qN,~p1,~p2,...,~pN), (1) of generalized coordinates f~qig and momenta f~pig. If the generalized coordinates and momenta of any two particles are exchanged (say 1 and 2) (~q1,~q2,...,~qN,~p1,~p2,...,~pN) ! (~q2,~q1,...,~qN,~p2,~p1,...,~pN), (2) we obtain a different point in phase space (except for a set of points of zero measure). Consequently, in classical mechanics particle-exchange invariance never holds and particles are always distinguishable. This is independent on whether particles are identical or not. Identical particles are those whose Hamiltonian is invariant under the exchange of any two particles (say 1 and 2), H(~q1,~q2,...,~qN,~p1,~p2,...,~pN) = H(~q2,~q1,...,~qN,~p2,~p1,...,~pN). (3) This definition of identical particles also holds in quantum mechanics if we consider that the quantum Hamiltonian depends on operators associated to the generalized coordinates and momenta as well as on operators associated to internal degrees of freedom such as spin. In quantum mechanics, the N particle microstate is a vector in the Hilbert space jyi which, in position representation and not considering internal degrees of freedom, is represented by a complex function y(~r1,~r2,...,~rN). A state of N identical particles must be symmetric (+, bosons) or antisymmetric (−, fermions) under particle exchange y(~r2,~r1,...,~rN) = ±y(~r1,~r2,...,~rN). (4) Furthermore, according to the quantum mechanics postulates, particles are indistinguishable if and only if they are identical, otherwise they are distinguishable. Particles are called non-interacting or ideal when it is possible to define monoparticular (i) Hamiltonians h (~qi,~pi), depending only on the generalized coordinates and momenta of one Entropy 2016, 18, 259 3 of 13 N (i) particle, such that the N-particle Hamiltonian is H = ∑i=1 h . In the quantum case, the monoparticular Hamiltonian might include spin and other quantum internal degrees of freedom. For quantum systems (independently on whether particles are ideal or not), it is possible to construct the state jyi as a linear superposition of the eigenfunctions jlii of monoparticular (i) (i)j i = j i Hamiltonians h , defined as h li eli li : • For distinguishable particles any state can be written as an unrestricted linear combination of elements of the product basis jl1ijl2i ... jlNi ≡ jl1, l2,..., lNi. For an ideal system it is: Hj i = ( + ··· + )j i l1, l2,..., lN el1 elN l1, l2,..., lN . (5) • For indistinguishable particles one can use instead the second-quantization basis jjn0, n1, n2,... ii in terms of the occupation numbers n` of individual levels `. In this representation, the wave function is always invariant under particle exchange, and one avoids an explicit symmetrization or anti-symmetrization process. For an ideal system it is: Hjjn0, n1,... ii = (n0e0 + n1e1 + ... )jjn0, n1,... ii. (6) (i) In the ideal case, we can define the one-particle partition function Z1 associated to particle i. In the context of classical mechanics the definition is (i) Z d~q d~p (i) Z = i i e−bh (~qi,~pi), b = 1/kT, (7) 1 h fi where T is the temperature and fi the number of degrees of freedom of particle i. The quantum counterpart is (i) −be Z = li 1 ∑ e . (8) li Non-interacting particles are said to be non-localized if the monoparticular partition function Z1 fulfills Z1(V, T) = V f (T), (9) and localized if it depends only on temperature Z1(V, T) = f(T). (10) Intuitively, localized particles correspond to those for which the eigenfunctions of the monoparticular Hamiltonian are bounded in space. Examples being the infinitely-confining harmonic or infinite square well potentials in a finite region. In order to fix ideas and to understand the concepts and definitions presented above, we will categorize the particles of four different Hamiltonians, whose statistics will be considered later. We restrict ourselves to the simplest examples (non-interacting particles) addressed in common textbooks of statistical mechanics. The results can be generalized to interacting particles as the nature of the problem addressed in this paper does not concern interactions. (i) In the first example, we consider a non-relativistic gas of non-interacting identical particles without any internal or rotational degrees of freedom and not subject to any external field. The Hamiltonian can be written as N ~p 2 H = ∑ i . (11) i=1 2m We will see later on that those kind of particles are non-localized. Classically, this is a system of identical, distinguishable, non-localized particles. The quantum version represents a system of identical, hence indistinguishable, non-localized particles. Entropy 2016, 18, 259 4 of 13 (ii) The second example is the previous ideal gas but particles have different masses.