15 Jan 1996-Physical Review B (Phys Rev B Condens Matter)-Vol. 53, Iss: 3, pp 1006-1009

About: This article is published in Physical Review B.The article was published on 1996-01-15 and is currently open access. It has received 4 citations till now. The article focuses on the topics: Microcanonical ensemble & Canonical ensemble.

More detailed quantitative calculations have been given by Cheung et al. [2] [3] [4].

The reason is that the canonical-ensemble occupation probability is complicated and sensitive to the details of the energy levels.

The authors results show that replacing the canonical-ensemble occupation probability with the Fermi-Dirac distribution cannot lead to correct results for the persistent current.

The authors results send warnings to those using the above approximation.

III. PERSISTENT CURRENT IN THE RING

It is possible to calculate the average magnetic moment from the occupation probability of the levels.

As expected, the dominant term in the average magnetic moment is linear in magnetic field.

The authors also noted that the amplitude of the harmonics is roughly exp(Ϫ2l 2 /⌬) in the grand-canonical ensemble.

For systems with many rings, the electron number in each ring may not be the same.

Analysis starting from Eq. ͑9͒ shows that this crossover of period is true over the whole range of temperatures.

TL;DR: In this article, the typical persistent current (the component that varies randomly from ring to ring) was measured with high sensitivity, over two orders of magnitude smaller than that observed in previous studies.

Abstract: In this work, I present measurements of persistent currents in normal metal rings performed with cantilever torsional magnetometry. With this technique, the typical persistent current (the component that varies randomly from ring to ring) was measured with high sensitivity. I report measured magnitudes as low as 1 pA, over two orders of magnitude smaller than that observed in previous studies. These measurements extend the range of temperature and magnetic field over which the typical current has been observed. The wide magnetic field range allowed us to study the effect of magnetic field penetrating the ring. It also enabled the recording of many independent measurements of the current magnitude in a single sample. These independent measurements are necessary to characterize the persistent current magnitude because it is a random quantity. From these measurements of the persistent current, I also characterize the parametric dependence of the typical current on sample orientation and number of rings. In addition to presenting the experimental results, I thoroughly review the theory of the typical persistent current in the diffusive regime. I begin with the simplest model and build up to the case appropriate for the samples studied in our experiments. I also present in detail the experimental apparatus used to measure the persistent currents.

9 citations

Cites background from "Persistent current of one-dimension..."

...We note that some authors have taken issue with the use of the grand canonical ensemble [48], while others have argued that the calculation in the grand canonical ensemble can be related to the one in the canonical ensemble by making the chemical potential flux dependent so that as the energy levels change with φ the number of occupied levels remains constant [40, 49, 50]....

TL;DR: In this article, the authors developed and analyzed a model for quantum rings in which persistent currents are induced by Aharonov-Bohm (AB) or other similar effects, based on a centric and annual potential profile.

Abstract: This paper reports the work on the development and analysis of a model for quantum rings in which persistent currents are induced by Aharonov–Bohm (AB) or other similar effects. The model is based on a centric and annual potential profile. The time-independent Schrodinger equation including an external magnetic field and an AB flux is analytically solved. The outputs, namely energy dispersion and wavefunctions, are analyzed in detail. It is shown that the rotation quantum number m is limited to small numbers, especially in weak confinement, and a conceptual proposal is put forward for acquiring the flux and eventually estimating the persistent currents in a Zeeman spectroscopy. The wavefunctions and electron distributions are numerically studied and compared to one-dimensional (1D) quantum well. It is predicated that the model and its solutions, eigen energy structure and analytic wavefunctions, would be a powerful tool for studying various electric and optical properties of quantum rings.

TL;DR: In this article, temperature dependent diamagnetic-paramagnetic transitions in thin quantum rings were studied and it was shown that the magnetic susceptibility of metal/semiconductor rings can exhibit multiple sign flips at intermediate and high temperatures depending on the number of conduction electrons in the ring and whether or not spin effects are included.

Abstract: We present theoretical studies of temperature dependent diamagnetic-paramagnetic transitions in thin quantum rings. Our studies show that the magnetic susceptibility of metal/semiconductor rings can exhibit multiple sign flips at intermediate and high temperatures depending on the number of conduction electrons in the ring ($N$) and whether or not spin effects are included. When the temperature is increased from absolute zero, the susceptibility begins to flip sign above a characteristic temperature that scales inversely with the number of electrons according to ${N}^{\ensuremath{-}1}$ or ${N}^{\ensuremath{-}1/2}$, depending on the presence of spin effects and the value of $N\phantom{\rule{0.16em}{0ex}}\mathrm{mod}\phantom{\rule{0.16em}{0ex}}4$. Analytical results are derived for the susceptibility in the low and high temperature limits, explicitly showing the spin effects on the ring Curie constant.

Cites background from "Persistent current of one-dimension..."

...As has been wellestablished by now [22, 37–40], it is most appropriate to use the canonical ensemble when the number of particles on the ring is fixed, but it is often a good approximation to treat the ring using a grand canonical ensemble and Fermi-Dirac statistics....

TL;DR: In this article, a brain model based on canonical ensemble theory was proposed for the detection of Alzheimer's disease, which showed the feasibility and rationality of brain thermodynamic modelling method. But the model was not applied to describe the state of neural system.

Abstract: Objective. Modelling is an important way to study the working mechanism of brain. While the characterization and understanding of brain are still inadequate. This study tried to build a model of brain from the perspective of thermodynamics at system level, which brought a new thinking to brain modelling.
Approach. Regarding brain regions as systems, voxels as particles, and intensity of signals as energy of particles, the thermodynamic model of brain was built based on canonical ensemble theory. Two pairs of activated regions and two pairs of inactivated brain regions were selected for comparison in this study, and the analysis on thermodynamic properties based on the model proposed were performed. In addition, the thermodynamic properties were also extracted as input features for the detection of Alzheimer's disease.
Main results. The experiment results verified the assumption that the brain also follows the thermodynamic laws. It demonstrated the feasibility and rationality of brain thermodynamic modelling method proposed, indicating that thermodynamic parameters could be applied to describe the state of neural system. Meanwhile, the brain thermodynamic model achieved much better accuracy in detection of Alzheimer's disease, suggesting the potential application of thermodynamic model in auxiliary diagnosis.
Significance. (1) Instead of applying some thermodynamic parameters to analyze neural system, a brain model at system level was proposed from perspective of thermodynamics for the first time in this study. (2) The study discovered that the neural system also follows the laws of thermodynamics, which leads to increased internal energy, increased free energy and decreased entropy when system is activated. (3) The detection of neural disease was demonstrated to be benefit from thermodynamic model, implying the immense potential of thermodynamics in auxiliary diagnosis.

TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.

Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

TL;DR: The fifth edition of the book as discussed by the authors has been published in 2002 and it has been supplemented with more recent results and references where appropriate, but it still preserves the style and content of the original, but with some modifications.

Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

Q1. What have the authors contributed in "Persistent current of one-dimensional perfect rings under the canonical ensemble" ?

The variation of the persistent current in oneand higherdimensional perfect and disorder rings as a function of temperature has been studied by Cheung et al. All these studies adopted the grand-canonical ensemble, so the chemical potential is fixed and the electron occupation probability follows the Fermi-Dirac distribution. In this paper the authors present exact analysis on the persistent current on 1D perfect rings under the canonical ensemble. Their calculation logic is to allow the chemical potential to vary with magnetic flux such that the number of electrons remains fixed. At high temperatures, if one approximates the canonicalensemble occupation probability by the Fermi-Dirac distribution with a suitably chosen chemical potential, one might expect to get a reasonable answer for the persistent current.