Indistinguishable Elements Research Articles

Indistinguishable elements in the origins of quantum statistics. The case of Fermi–Dirac statistics

In this paper, we deal with the historical origins of Fermi–Dirac statistics, focusing on the contribution by Enrico Fermi of 1926. We argue that this statistics, as opposed to that of Bose–Einstein, has been somewhat overlooked in the usual accounts of the old quantum theory. Our main objective is to offer a critical analysis of Fermi’s seminal paper and its immediate impact. Secondly, we are also interested in assessing the status of the particle concept in the years 1926–1927, especially regarding the germ of quantum indistinguishability. We will see, for example, that the first applications of the Fermi–Dirac statistics to the study of metals or stellar matter had a technical nature, and that their main instigators barely touched upon interpretative matters. Finally, we will discuss the reflections and remarks made in these respects in two famous events in physics of 1927, the Como conference and the fifth Solvay congress.

Paramacrosystems with arrangement

Parastatistical macroscopic classical systems with distinguishable states cells and arranged state occupation IDA-systems are considered. In one-dimensional case IDA1-systems they are shown to be equivalent to the systems with indistinguishable elements and states II-systems. In two-dimensional case IDA2-systems they require special approach. Their microscopic states are associated with one-and two-dimensional Young diagrams, respectively. Macroscopic states are defined as neighbourhoods of microscopic ones. Explicit expressions for probabilities and entropies of macroscopic states are derived. Analysis of equilibrium states reveals shell-type mechanism of growth for IDA-systems.

Finite quantum models: constructive approach to description of quantum behavior

The universality of quantum mechanics — its applicability to physical systems of quite different nature and scales — indicates that quantum behavior can be a manifestation of general mathematical properties of systems containing indistinguishable elements, i.e., elements lying on the same orbit of some symmetry group. In this paper, we demonstrate that quantum behavior arises naturally in systems with finitely many elements connected by nontrivial symmetry groups. The “finite” approach allows one to see the peculiarities of a quantum description more distinctly without the need for concepts like “wave function collapse,” “Everett’s multiverses,” etc. In particular, under the finiteness assumption, any quantum dynamics is reduced to a simple permutation dynamics. The advantage of finite quantum models is that they can be studied constructively by means of computer algebra and computational group theory methods. Bibliography: 12 titles.

Analysis Based on Moment Vector Equation for Interacting Identical Elements with Nonlinear Dynamics

A method was developed for analyzing a system comprised of identical and indistinguishable elements with nonlinear dynamics. First, a moment vector equation (MVE) for the system was derived so as to avoid the curse of dimensionality by using the property that the elements are identical and indistinguishable.Next, an algorithm was developed to solve the MVE for deriving the moment vector in a steady state. It effectively uses eigen analysis on the basis of the property of the MVE. It can thus be used to clarify the structure of the solutions in the moment vector space and to derive multiple solutions by setting the initial value to the moment vector orthogonal to the solutions already obtained.Finally, the probability density function (pdf) for the state of the system was derived using the moment vectors in a steady state. Comparison of the pdfs thereby derived with those derived using numerical simulation showed that the method provided good approximations of the pdfs. Moreover, multiple solutions that are difficult to do using numerical simulation were derived.

Q-spaces and the Foundations of Quantum Mechanics

Our aim in this paper is to take quite seriously Heinz Post’s claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller’s words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. We build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg’s suggestion that new phenomena require the formation of a new “closed” (that is, axiomatic) theory, coping also with the physical theory’s underlying logic and mathematics.

The unlabelled speed of a hereditary graph property

A property of graphs is a collection P of graphs closed under isomorphism; we call P hereditary if it is closed under taking induced subgraphs. Given a property P , we write P n for the set of graphs in P with vertex set [ n ] = { 1 , … , n } , and P n for the isomorphism classes of graphs of order n that are in P . The cardinality | P n | is the labelled speed of P and | P n | is the unlabelled speed. In the last decade numerous results have been proved about the labelled speeds of hereditary properties, with emphasis on the striking phenomenon that only certain speeds are possible: there are various pairs of functions ( f ( n ) , F ( n ) ) , with F ( n ) much larger than f ( n ) , such that if the labelled speed is infinitely often larger than f ( n ) then it is also larger than F ( n ) for all sufficiently large values of n. Putting it concisely: the speed jumps from f ( n ) to F ( n ) . Recent work on hereditary graph properties has shown that “large” and “small” labelled speeds of hereditary graph properties do jump. The aim of this paper is to study the unlabelled speed of a hereditary property, with emphasis on jumps. Among other results, we shall show that the unlabelled speed of a hereditary graph property is either of polynomial order or at least S ( n ) , the number of ways of partitioning a set with n indistinguishable elements.

Probabilistic characteristics of paramacrosystems

Consideration is given to macrosystems called paramacrosystems with different elements, with stochastic behavior, and states of finite capacity. Thus is fulfilled the interval between two classical capacity characteristics of states: Fermi, with one element, and Einstein, with unlimited number of elements in each state. Using the apparatus of generating functions, we obtained expressions for probabilistic characteristics (functions of probability distribution and "physical" entropies) for four classes of paramacrosystems with corresponding combinations of distinguishable and indistinguishable elements and states with finite capacities.

A comment on Bermudez concerning the definability of identity

José Luis Bermúdez's article ‘Indistinguishable elements and mathematical structuralism’ (2007) is a contribution to the debate about the ‘identity problem for structuralism’ (see Burgess 1999 and Keränen 2001) and includes comments on Ladyman 2005 and Ketland 2006.1 Bermúdez writes that Ketland 2006‘contains some puzzling claims’ and, in particular, that one specific result (given without proof) ‘seems badly wrong’. My comment is simply that the result that Bermúdez disputes is correct (as are the others). Below, a proof is given. After all, results concerning the definability of identity in relational structures may be of independent philosophical and logical interest.2 Bermúdez writes, Ketland's paper contains some puzzling claims about the first-order definability of identity. He claims that ‘one can show that if the identity relation is first-order definable in a structure at all, then it is defined by the indiscernibility formula’ (Ketland 2006: 307), where the identity formula is essentially Quine's trick for eliminating the identity predicate. This seems badly wrong, since Quine's trick only applies to structures with finite signatures. ... any structure which contains the identity predicate and has an infinite signature is a counter-example to his claim. (Bermúdez 2007: 113, n. 2)

Balanced minimum covers of a finite set

In this paper a method of enumeration for n-balanced, labelled, minimum covers of a finite set of m indistinguishable elements is developed. The method is then used to tabulate the number of such covers for small values of m and n by means of a generating function.

Strong identities and fortification in transposition hypergroups

An element e in a hypergroup H is a strong identity if x ∈ ex = xe ⊂ x ⋃ e. The elements of H separate into two classes, the set A = {x ∈ H | ex = xe = x ⋃ e}, including e, of attractive elements and the set C = {x ∈ H – e | ex = xe = x} of canonical elements. If H is a transposition hypergroup then A is shown to be a closed subhypergroup of essentially indistinguishable elements. The structure of H is then determined, for A can be contracted into e leaving the “resulting” C ⋃ e, which is a polygroup under the relativized hyperoperation. Therefore, H can be reconstructed from A and C ⋃ e, as H is isomorphic to the expansion of the polygroup C ⋃ e by the transposition hypergroup A through e. The study of transposition hypergroups containing a strong identity separates into the study of polygroups and the study of transposition hypergroups of all attractive elements. A fortified transposition hypergroup H is defined and shown to contain a unique strong identity. Moreover, each nonidentity element is shown to have unique nonidentity left and right inverses that are identical. For H consisting of all attractive elements, the subhypergroups K that are symmetric, K = K –1, are studied. The double cosets of K, the sets K \\ (x/K) = (K \\ x)/K if x ∉ K, otherwise K, partition H. The resulting quotient space H : K of double cosets is proven to be a fortified transposition hypergroup in which K is the strong identity and every member is attractive.

Differences in the Occurrence of Two Base Pair Variants of Tn 1546 from Vancomycin-Resistant Enterococci from Humans, Pigs, and Poultry

In recent years the risk of transferring vancomycin-resistant enterococci (VRE) from animals to humans has caused great concern. Studies have shown that the glycopeptide growth promoter avoparcin selects for VRE (1, 3), and high numbers have been found in food animals and foodstuff in Europe (4, 7, 10). In one case, identical strains were isolated from both a turkey farmer and his turkeys (9). The finding of VRE in nonhospitalized humans and in meat eaters but not in vegetarians has further indicated a food-associated spread of vancomycin resistance from animals to humans (8). Most of the vancomycin-resistant Enterococcus faecium (VREF) strains isolated in Europe contain Tn1546. This transposon was first isolated from E. faecium BM4147 (2). Studies of Tn1546 have revealed that only the vanA, vanH, and vanX genes are essential for resistance (Fig. ​(Fig.1).1). Characterization of Tn1546-like elements in isolates of animal and human origin indicated only minor variations caused by inseration sequences outside the essential part of Tn1546 (Fig. ​(Fig.1),1), and a base pair variation in the vanX gene was discovered at position 8234 (5). In this position either a G (G type) or a T (T type) was found (Fig. ​(Fig.1).1). Based on all these variations, different types of Tn1546-like elements were defined, and indistinguishable elements were found in isolates of human and animal origin (5). FIG. 1 The Tn1546 encoding vancomycin resistance. The sizes and positions of the genes are indicated. The genes are grouped into the categories Mobility, Regulation, Essential, and Accessory according to their importance for vancomycin resistance. The positions ... In this study a total of 271 VREF isolates of animal (226) and human (45) origin were investigated for this base pair variation. All animal isolates originated from different herds, and all human isolates, except strains from Norway and Saudi Arabia, have previously been typed to independent clones (4–6). PCR amplification of a 424-bp amplicon of the vanX gene of Tn1546 (Fig. ​(Fig.1)1) was obtained from all isolates, confirming the presence of Tn1546-like elements. By digesting the amplicons with the restriction enzyme DdeI, two distinct patterns of fragments were obtained for the G and T types. Based on these results the distribution of the base pair variation could thus be ascertained. As evident from Table ​Table1,1, all isolates from poultry belonged to the G type, while 32 of the 33 porcine isolates belonged to the T type. In human isolates both variations were found but only one of the variants dominated locally. Only the G type was found in isolates from England, Norway, and Saudi Arabia, thus associating these isolates with poultry. For the human Nowegian isolates this could be expected, since they were isolated from poultry farmers. All Danish human VREF isolates belonged to the T type, thereby associating them with pigs. TABLE 1 Variations in Tn1546-like elements of VREF isolates of animal and human origin The present study showed that VREF isolates from pigs, poultry, and humans could be divided according to base pair variation in the vanX gene at position 8234 (G or T type). All poultry isolates belonged to the G type, whereas almost all porcine isolates, except one Danish isolate, belonged to the T type. This finding indicates, with particular reference to the Danish isolates, that horizontal exchange of VREF isolates or Tn1546-like elements between poultry and pigs does not occur frequently. On the other hand, both types were found among humans, indicating that humans may be infected from both sources. Based on this observation it may be hypothesized that the primary transmission is from animals to humans and not the other way around. However, further studies are required to test this assertion. The present results indicate that the base pair variation in the vanX gene may be a useful marker for epidemiological studies on the spread of VREF isolates and vancomycin resistance genes.

An asymptotic formula for the number of classes of sets of n indistinguishable elements

Letting X be a finite set and P the power set operation on a set, an approximation for the number of members of P d ( X), for d⩾2, is derived, where two members are considered identical if some permutation on X can transform one into the other. The approximation is asymptotic to the expected lower bound | P d ( X)|/| X|!

Enumerating $(\bf{2+2})$-free posets by indistinguishable elements

A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number of different strict down-sets of elements in the poset. Moreover, we show that (2+2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2+2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices.