Particle Permutation Research Articles

The "Real" Gibbs Paradox and a Composition-Based Resolution.

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 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 kBln2 to 0, as two different substances are made more similar, thereby resolving the "real" Gibbs paradox.

The concept ‘indistinguishable’

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.

Path integral molecular dynamics for bosons

Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.

The two-loop five-particle amplitude in mathcal{N} = 8 supergravity

We compute for the first time the two-loop five-particle amplitude in mathcal{N} = 8 supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. We observe that one of the symbol letters is absent from the amplitude. The latter satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate. We extract the finite remainder function, which depends on fewer rational factors. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as a supplementary file.

Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis.

The characteristics of the early fault signal of the rolling bearing are weak and this leads to difficulties in feature extraction. In order to diagnose and identify the fault feature from the bearing vibration signal, an adaptive local iterative filter decomposition method based on permutation entropy is proposed in this paper. As a new time-frequency analysis method, the adaptive local iterative filtering overcomes two main problems of mode decomposition, comparing traditional methods: modal aliasing and the number of components is uncertain. However, there are still some problems in adaptive local iterative filtering, mainly the selection of threshold parameters and the number of components. In this paper, an improved adaptive local iterative filtering algorithm based on particle swarm optimization and permutation entropy is proposed. Firstly, particle swarm optimization is applied to select threshold parameters and the number of components in ALIF. Then, permutation entropy is used to evaluate the mode components we desire. In order to verify the effectiveness of the proposed method, the numerical simulation and experimental data of bearing failure are analyzed.

The Extension of the Monte Carlo Method for Neutron Transfer Problems Calculating to the Problems of Quantum Mechanics

There are several methods of numerical solution of eigenvalue problems by the Monte Carlo method, which are used in the calculation of nuclear reactors. This paper is devoted to the investigation of the possibility of using such methods for solving the stationary Schrodinger equation. The latter equation can easily be transformed into the form of an integral equation of the first kind, very similar to those integral equations that arise in problems of nuclear power. The Monte Carlo method for this form of the stationary Schrodinger equation looks very attractive, since it naturally parallels and is very convenient for calculations on multiprocessor systems. In addition, in this case it is necessary to operate with functions defined on a large-dimensional space. This is also natural for the Monte Carlo method. It is described how the methods long used for the calculation of nuclear reactors are transformed for this case. The main problem is that the wave function of fermions changes its sign under a permutation of identical particles, and can not be nonnegative. The proposed approach is significantly different from the known methods of applying the Monte Carlo method to quantum mechanical problems. In this paper, several examples of the successful application of the proposed new method are given.

The Gibbs Paradox: Lessons from Thermodynamics.

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.

A novel discrete particle swarm algorithm for assembly line balancing problems

PurposeThe purpose of this paper is to develop a feasible sequence-oriented new discrete particle swarm optimization (NDPSO) algorithm with novel particles’ updating mechanism for solving simple assembly line balancing problems (SALBPs).Design/methodology/approachIn the NDPSO, a task-oriented representation is adopted to solve type I and type II SALBPs, and a particle directly represents a feasible task sequence (FTS) as a permutation. Then, the particle (permutation) is updated as a whole using the geometric crossover based on the edit distance with swaps for two permutations. Furthermore, the fragment mutation with adaptive mutation probability is incorporated into the NDPSO to improve exploration ability.FindingsCase study illustrates the effectiveness of the NDPSO. Comparative results between the NDPSO and existing real-encoded PSO (CPSO) and direct discrete PSO (DDPSO) against benchmark instances of type I SALBP and type II SALBP show promising higher performance of the proposed NDPSO.Originality/valueA novel particles’ updating mechanism for FTS-encoded particle is proposed to solve the SALBPs. The comparative results indicate that updating of FTS as a whole seems superior to existing updating of FTS by fragment with respect to exploration ability for solving SALBPs. The novel particles’ updating mechanism is also applicable to generalized assembly line balancing problems.

Discrete phase-space mappings, tomographic condition and permutation invariance

We analyze various families of discrete maps in N-qubit systems in the context of permutation invariance. We prove that the tomographic condition imposed on the self-dual (Wigner) map is incompatible with the requirement of the invariance under particle permutations (except for the two-qubit case), which makes it impossible to project the Wootters-like Wigner function into the space of symmetric measurements. We also provide several explicit forms of the self-dual mappings: (a) tomographic and (b) permutation invariant, and analyze the symmetric projection in the latter case.

Protocols for quantum binary voting

Two new protocols for quantum binary voting are proposed. One of the proposed protocols is designed using a standard scheme for controlled deterministic secure quantum communication (CDSQC), and the other one is designed using the idea of quantum cryptographic switch, which uses a technique known as permutation of particles. A few possible alternative approaches to accomplish the same task (quantum binary voting) have also been discussed. Security of the proposed protocols is analyzed. Further, the efficiencies of the proposed protocols are computed, and are compared with that of the existing protocols. The comparison has established that the proposed protocols are more efficient than the existing protocols.

Identical Wells, Symmetry Breaking, and the Near-Unitary Limit

Energy level splitting from the unitary limit of contact interactions to the near unitary limit for a few identical atoms in an effectively one-dimensional well can be understood as an example of symmetry breaking. At the unitary limit in addition to particle permutation symmetry there is a larger symmetry corresponding to exchanging the $N!$ possible orderings of $N$ particles. In the near unitary limit, this larger symmetry is broken, and different shapes of traps break the symmetry to different degrees. This brief note exploits these symmetries to present a useful, geometric analogy with graph theory and build an algebraic framework for calculating energy splitting in the near unitary limit.

Symmetrization of the nuclear wavefunctions defined by the quantum trajectory dynamics

In a rigorous quantum-mechanical (QM) description of indistinguishable particles, the correct symmetry is often built-in into the form of an approximate wavefunction, with the electronic wavefunctions constructed as Slater determinants of the single-particle functions being the prime example. In contrast, when evaluating QM effects for the nuclei, often described by approximate wavefunctions of full dimensionality, the wavefunction symmetry can be included directly into calculation of expectation values. The straightforward implementation, however, may be impractical for a large system due to factorial scaling of particle permutations. In this work, the leading correction due to the wavefunction symmetrization within the quantum trajectory (QT) framework is presented. The correction is based on the non-symmetrized wavefunction evolved using QT dynamics with empirical friction, yielding the lowest energy states. Use of symmetry improves the accuracy and efficiency of this dynamics approach as shown on model systems of up to four dimensions.

Graph states of prime-power dimension from generalized CNOT quantum circuit.

We construct multipartite graph states whose dimension is the power of a prime number. This is realized by the finite field, as well as the generalized controlled-NOT quantum circuit acting on two qudits. We propose the standard form of graph states up to local unitary transformations and particle permutations. The form greatly simplifies the classification of graph states as we illustrate up to five qudits. We also show that some graph states are multipartite maximally entangled states in the sense that any bipartition of the system produces a bipartite maximally entangled state. We further prove that 4-partite maximally entangled states exist when the dimension is an odd number at least three or a multiple of four.

Applications of quantum cryptographic switch: various tasks related to controlled quantum communication can be performed using Bell states and permutation of particles

Recently, several aspects of controlled quantum communication (e.g., bidirectional controlled state teleportation, controlled quantum secure direct communication, controlled quantum dialogue, etc.) have been studied using $n$-qubit ($n\geq3$) entanglement. Specially, a large number of schemes for bidirectional controlled state teleportation are proposed using $m$-qubit entanglement ($m\in\{5,6,7\}$). Here, we propose a set of protocols to illustrate that it is possible to realize all these tasks related to controlled quantum communication using only Bell states and permutation of particles (PoP). As the generation and maintenance of a Bell state is much easier than a multi-partite entanglement, the proposed strategy has a clear advantage over the existing proposals. Further, it is shown that all the schemes proposed here may be viewed as applications of the concept of quantum cryptographic switch which was recently introduced by some of us. The performances of the proposed protocols as subjected to the amplitude damping and phase damping noise on the channels are also discussed.

Efficient protocols for unidirectional and bidirectional controlled deterministic secure quantum communication: different alternative approaches

Recently, Hassanpour and Houshmand have proposed a protocol of controlled deterministic secure quantum communication (Quant. Info. Process, DOI 10.1007/s11128-014-0866-z (2014)). The authors compared the efficiency of their protocol with that of two other existing protocols and claimed that their protocol is efficient. Here, we have shown that the efficiency of Hassanpour Houshmand (HH) protocol is not high, and there exist several approaches through which more efficient protocols for the same task can be designed. To establish this point, we have proposed an efficient protocol of controlled deterministic secure quantum communication which is based on permutation of particles (PoP) technique and is considerably efficient compared to HH protocol. We have also generalized this protocol into its bidirectional counterpart. Interestingly, bipartite entanglement (Bell state) is sufficient for the realization of the proposed protocols, but HH protocol and other existing protocols require at least tripartite entanglement. Further, we have shown that it is possible to construct a large number of efficient protocols of unidirectional and bidirectional controlled deterministic secure quantum communication by using various alternative approaches and different quantum states. These alternative protocols can be realized by modifying the existing protocols of quantum secure direct communication and deterministic secure quantum communication. We have also shown that it is possible to design completely orthogonal-state-based protocols for unidirectional and bidirectional controlled deterministic secure quantum communication.

Resonance tunnelling of clusters through repulsive barriers

Quantum tunnelling of a cluster comprised of several identical particles, coupled via an oscillator-type potential, through short-range repulsive barrier potentials is studied in the s-wave approximation of the symmetrized coordinate representation. A procedure is briefly described that allows the construction of states, symmetric or asymmetric with respect to permutations of A identical particles, from the harmonic oscillator basis functions expressed via the newly introduced symmetrized coordinates. In the coupled-channel approximation of the -matrix approach, the effect of quantum transparency is analysed; it manifests itself in non-monotonic resonance dependence of the transmission coefficient upon the energy of the particles, their number A = 3, 4 and the symmetry types of their states. The total transmission coefficient is shown to demonstrate resonance behaviour with probability density growth in the vicinity of the potential energy local minima, which is a manifestation of the barrier quasi-stationary states, embedded in the continuum.

Nonlinear supersymmetry in the quantum Calogero model

It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g+1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order n(n-1)(2g-1)/2 in the momenta and squares to a polynomial in the Liouville charges. With a natural Z_2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.

Spectroscopy for a few atoms harmonically trapped in one dimension

Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3--5$, are provided.

Imaginary-time nonuniform mesh method for solving the multidimensional Schrödinger equation: Fermionization and melting of quantum Lennard-Jones crystals

An imaginary-time nonuniform mesh method for diagonalizing multidimensional quantum Hamiltonians is proposed and used to find the first 50 eigenstates and energies of up to $D=5$ strongly interacting spinless quantum Lennard-Jones particles trapped in a one-dimensional harmonic potential. We show that the use of tailored grids allows exploitation of the symmetries of the system (in our case the $D!$ degeneracy derived from all possible permutations of distinguishable particles), reducing drastically the computational effort needed to diagonalize the Hamiltonian. This leads to a favorable scaling with dimensionality, requiring for the five-dimensional system four orders of magnitude fewer grid points than the equivalent regular grid. Solutions to both bosonic and fermionic counterparts of this strongly interacting system are constructed, the bosonic case clustering as a Tonks-Girardeau crystal exhibiting the phenomenon of fermionization. The numerically exact excited states are used to describe the melting of this crystal at finite temperature.

Six-Bodies Calculations Using the Hyperspherical Harmonics Method

In this work we show results for light nuclear systems and small clusters of helium atoms using the hyperspherical harmonics basis. We use the basis without previous symmetrization or antisymmetrization of the state. After the diagonalization of the Hamiltonian matrix, the eigenvectors have well defined symmetry under particle permutation and the identification of the physical states is possible. We show results for systems composed up to six particles. As an example of a fermionic system, we consider a nucleon system interacting through the Volkov potential, used many times in the literature. For the case of bosons, we consider helium atoms interacting through a potential model which does not present a strong repulsion at short distances. We have used an attractive gaussian potential to reproduce the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the most widely used He–He interaction, the LM2M2 potential. In addition, we include a repulsive hypercentral three-body force to reproduce the trimer binding energy.