Equivalence of Gibbs ensembles for quantum systems obeying Maxwell-Boltzmann statistics

A generating functional of the canonical ensemble is introduced for quantum systems of particles obeying Maxwell-Boltzmann statistics. Power estimates for the quantum analogs of the particle distribution functions of the canonical and grand canonical ensembles are obtained for systems of particles with hard cores. It is shown on the basis of the power estimates that limiting generating functionals exist for the systems under consideration and satisfy the same Bogolyubov equation. The ensembles are equivalent in this sense.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1984.The authors thank É. A. Arinshtein for useful discussion.     

Derivation and classical limit of the mean-field equation for a quantum Coulomb system: Maxwell-Boltzmann statistics

The mean-field density matrix of a changed plasma of quantum particles with Maxwell-Boltzmann statistics in a confining external potential is obtained as a limit of theN-body canonical states for suitably scaled charges. Also, it is shown that the density profile of the quantum mean-field theory converges to the solution of the classical mean-field equation when the Planck's constant tends to zero.     

Quantum particles from classical statistics

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being “classical” or “quantum” ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross‐fertilization between classical statistics and quantum physics.     

Settling statistics of hard sphere particles

Direct imaging of settling, non-Brownian, hard sphere, particles allows measurement of particle occupancy statistics as a function of time and sampling volume dimension. Initially random relative particle number fluctuations, (2)>/ = 1, become suppressed, anisotropic, and dependent. Fitting to a simple Gaussian pair correlation model suggests a minute long ranged correlation leads to strong if not complete suppression of number fluctuations. Calflisch and Luke predict a divergence in velocity fluctuations with increasing sample volume size based on random (Poisson) statistics. Our results suggest this is not a valid assumption for settling particles.     

Quantum liquids of particles with generalized statistics

We propose a phenomenological approach to quantum liquids of particles obeying generalized statistics of a fermionic type, in the spirit of the Landau Fermi liquid theory. The approach is developed for fractional exclusion statistics. We discuss both equilibrium (specific heat, compressibility, and Pauli spin susceptibility) and nonequilibrium (current and thermal conductivities, thermopower) properties. Low-temperature quantities have the same temperature dependences as for the Fermi liquid, with the coefficients depending on the statistics parameter. The novel quantum liquids provide an explicit realization of systems with a non-Fermi liquid Lorentz ratio in two and more dimensions. Consistency of the theory is verified by deriving the compressibility and f-sum rules.     

Interactions of particles having small violations of statistics

I give the first formalism which obeys the requirements of non-relativistic quantum theory and allows interactions of particles with small violations of Fermi or Bose statistics. I conclude with a summary of the seminal experiment of Ramberg and Snow which verifies the Pauli exclusion principle to high precision.     

Acceleration statistics of inertial particles in turbulent flow

Turbulent transport of material inclusions plays an important role in many natural and industrial situations. Being able to accurately model and predict the dynamics of dispersed particles transported by a turbulent carrier flow, remains a challenge. One critical and difficult point is to develop models which correctly describe the dynamics of particles over a wide range of sizes and densities. Our measurements show that acceleration statistics of particles dispersed in a turbulent flow do exhibit specific, and so far unpredicted, size and density effects and that they preserve an extremely robust turbulent signature with lognormal fluctuations, regardless of particles size and density. This has important consequences in terms of modeling for the turbulent transport of dispersed inclusions.     

Indistinguishability of orthogonal time-separated bell states

This paper proves that it is impossible to identify orthogonally time-separated Bell states. If two qubits of a Bell state interact with the measurement apparatus at different time, any attempt to identify this state will disturb it.     

On the classical approximation in the quantum statistics of equivalent particles

It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This IQC (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the N! paradox is thereby resolved. The method of correct Boltzmann counting and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation.This work was supported in part by the Air Force Office of Scientific Research, through Grants No. AF-AFSOR 557-64 and 557-67.     

麦克斯韦-玻耳兹曼统计分布律的相对论修正

为了避免光速极限带来的复杂性,首先给出了理想气体分子动量的分布律公式．在此基础上,进一步讨论了麦克斯韦一玻耳兹曼统计分布律的相对论修正．解析和数值计算均表明,这一修正几乎没有可观测的物理效应．这说明,大学物理教学中介绍的麦克斯韦一玻耳兹曼速度分布律公式对实际应用已经足够精确．当然,我们决不可据此近似公式断言气体分子有一定的概率可以超光速运动．     

Re-examination of constraints on the Maxwell-Boltzmann distribution by Helioseismology

Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution, the well-known Maxwell-Boltzmann Distribution (MBD). Strong constraints on such deviations were made previously with the detailed helioseismic information of the solar structure. With a small deviation parameterized with a factor exp[-δ (E/kT)²], it was shown δ restricted between -0.005 and +0.002. These constraints have been carefully re-examined in the present work. We find that a normalization factor was missed in the previous modified MBD. In this work, the normalization factor c is calculated as a function of δ. It shows the factor c is almost unity within the range 0< δ ≤0.002, which supports the previous conclusion. However, it demonstrates that δ cannot take a negative value from the normalization point of view. As a result, a stronger constraint on δ is defined as 0≤ δ ≤0.002. The astrophysical implication on the solar neutrino fluxes is simply discussed based on a positive δ value of 0.003. The reduction of the ⁷Be and ⁸B neutrino fluxes expected from the modified MBD can possibly shed alternative light on the solar neutrino problem. In addition, the resonant reaction rates for the ¹⁴N(p, γ)¹⁵O reaction are calculated with a standard MBD and a modified MBD, respectively. It shows that the rates are quite sensitive even to a very small δ. This work demonstrates the importance and necessity of experimental verification or testing of the well-known MBD at high temperatures.     

Local distinguishability of multipartite unitary operations

We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguished by local operations and classical communication when a finite number of runs is allowed. Intuitively, this result indicates that the lost identity of a nonlocal unitary operation can be recovered locally. No entanglement between distant parties is required.     

Fractional exclusion statistics in general systems of interacting particles

I show that fractional exclusion statistics is manifested in general interacting systems and I calculate the exclusion statistics parameters. Most importantly, I prove that the mutual exclusion statistics parameters are proportional to the dimension of the Hilbert space on which they act [D.V. Anghel, J. Phys. A: Math. Theor. 40 (2007) F1013].     

Local distinguishability of multipartite orthogonal quantum states

We consider one copy of a quantum system prepared in one of two orthogonal pure states, entangled or otherwise, and distributed between any number of parties. We demonstrate that it is possible to identify which of these two states the system is in by means of local operations and classical communication alone. The protocol we outline is both completely reliable and completely general; it will correctly distinguish any two orthogonal states 100% of the time.     

Relativistic constraints on the distinguishability of orthogonal quantum states

The constraints imposed by special relativity on the distinguishability of quantum states are discussed. An explicit expression relating the probability of an error in distinguishing two orthogonal single-photon states to their structure, the time t at which a measurement starts, and the interval of time T elapsed from the start of the measurement until the time at which the outcome is obtained by an observer is given as an example.     

Photon time-interval statistics of laser light scattered by brownian particles

Measurements are reported and some theoretical expressions are given for half-open photon time-interval statistics of laser light scattered by particles in brownian motion. Expressions in terms of cumulants are shown not to be valid in general.