有限体系统计力学及其温度涨落研究
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摘要
随着纳米科技的兴起和发展,已经发现许多纳米体系具有非热力学极限效应,例如负热容,高熔点等。类似的非热力学极限效应也在核物理、玻色-爱因斯坦凝聚、分子物理等领域广泛出现。现在,有限体系(或曰小体系,或曰少粒子体系)统计力学正在建立之中。本文首先综述了非热力学极限的物理现象,然后综述了处理和解释这些现象的理论工具。最“原始”的理论工具就是跟踪集体中每个个体,然后了解集体的行为。这一方法没有概念上的突破,但是需要计算能力。越来越多的文献利用微正则系综加上一些假设来处理有限体系,这些假设中影响最大的就是改变了相变的定义已使之能应用于小体系。在这个方面,我们认为,一种正在发展中的有限体系的相变的负热容判据可能最终将证明是对统计物理的贡献。
在考察了非热力学极限的物理现象和处理的理论工具之后,发现没有一个彻底的从等概率原理出发的处理离散体系的统计理论。作为本论文的主体之一,我们从等概率原理出发建立了多项式分布的统计力学。在多项式分布中,粒子总数为N,占据r个状态,其中第i (i = 1,2,...r )个状态的能量为εi,(ε1 <ε2 <ε3... <εr)。没有理由认为这N个粒子中的任何一个有特别的力学性质,也就是任何粒子占据其中任何一个状态的概率是一样的。这样就可以利用等概率原理和标准的概率论建立一个适用于有限粒子数的多项式分布统计力学。在这个统计力学中的热力学量在热力学极限下都将给出通常的结果。
本论文的主体之二是利用多项式分布统计力学解决了一个理论困难,并分析这个理论困难给我们的启示。在统计物理中,所有顺磁性固体,Debye固体(声子气体)等体系,由于热容量趋于零的速度比温度的平方快,所以在温度趋于零时温度涨落发散。在物理学中,任何发散都是理论困难。我们研究了最简单的顺磁体系——磁场中的自旋1/2体系——中的温度涨落问题。首先发展一个有限粒子正则系综,先引入构形量,热力学量是构形量的系综平均。熵和能量等热力学量和通常统计力学的结果符合得很好,在我们的理论中,所有的热力学量都有涨落。然后我们区分了温度涨落本身和作为强度量的温度涨落这两个物理量。在热力学极限下,我们的理论能回复到所有合理的结果,包括作为强度量的温度涨落都能给出通常的结果。唯一不同的是,温度涨落本身的发散这个本来就不可接受的结果没有了。作为强度量的温度涨落具有统计力学基础,显示了少粒子体系中温度的定义与粒子数相关。由于少粒子体系不满足可加性,不同统计系综具有不等价性,所以我们还讨论了如何从微正则系综出发讨论问题得到相同的结果。这说明了我们的结果具有普适性。这个普适性的含义不仅仅说对于这个磁场中的自旋1/2体系的结论是系综无关的,还说明对于一切通常的理论认为有温度涨落发散的体系,其实都是不发散的。
本论文的主体之三就是定量研究了在纳米体系中,在温度低到什么程度需要多少粒子通常的温度概念有效。考虑到物理学的唯象基础、温度涨落和热力学第三定律,我们发现存在一个关于低温下可以定义温度的普遍适用的判据。这个判据认为,随温度趋于零,体系的粒子数会急剧增加,也就是当温度足够低的时候,10 23个粒子数也是不足够定义温度的。本文将这个判据应用到一些具体问题例如理想气体,声子气体,理想玻色和费米气体,得到了文献中在特殊情况下通过数值计算得到的结果。本文还将这个判据应用到了Ising模型,也得到了合理的结果。
本论文的主体已经在Annals of Physics (N.Y.), American Journal of Physics等刊物上发表。特别是,关于低温下温度定义的普适判据发表后,已经被美国物理研究所(AIP)美国物理学会(APS)的联合创办的收集纳米科技重要文献的数据库“Virtual Journal of Nanoscale Science & Technology”收录。
With the technological advances allowing us to make measurement on small samples, e.g., the measurement of thermodynamic quantities like temperature with a spatial resolution on the nanometer scale, there is a growing demand for better understanding of thermal properties of nanoscale devices, individual nanostructures, and nanostructured materials. All these reveal rich non-thermodynamical limit effects. Besides nano-physics, the same effects present in nuclear physics, Bose-Einstein condensation, molecular physics etc. Mentioned explicitly or implicitly, a so-called statistical mechanics for small system or finite number of particles is in progress, in which the requirement of thermodynamical limit is not met. This dissertation first reviews the physical phenomena and the theories dealing with them.
After carefully examining the progresses in exploring the non-thermodynamical limit effects, no full statistical mechanics has been found that can deal with discrete energy systems. The fist part of the main body of this dissertation is dedicated to develop a formalism of statistical mechanics for multi-polynomial distribution. In this distribution, there are N particles distributed among r states and each state possesses a discrete energy. Since no particle has a peculiar mechanics property, i.e., every particle occupies any state with equal probability. We develop a statistical mechanics for this multi-polynomial distribution based on the equal a priori probabilities and the probability theory. All results given by the formalism turns out to be the usual ones in thermodynamic limit.
The second part of the main body of the dissertation is using the above established statistical mechanics for multi-polynomial distribution to resolve a theoretical difficulty. The usual statistical mechanics for all paramagnetic materials, and phonon gas, etc. gives divergent temperature fluctuations when the temperature itself approaches to the zero Kelven because their heat capacities goes to zero faster than the temperature square. We explicitly calculate the simplest paramagnetic material: finite N independent spin-1/2 paramagnets in a constant magnetic field. In our generalized canonical ensemble for finite number of particles, all fundamental thermodynamic quantities fluctuate, including the entropy and temperature, and all fluctuations are gradually vanishing as the temperature itself approaches zero. Our approach presents a remedy of infinite temperature fluctuations and reproduces other reasonable results that are given by standard theory, which includes an intensive temperature fluctuation that is introduced in this dissertation. Since the finite sized system does not satisfy the additivity, and different statistical ensembles are not equivalent to each other, the microcanonical ensemble treatment yields the identical result. So, our conclusion has universality. By universality we mean that the conclusion the temperature fluctuation for this system is not divergent is ensemble independent, and moreover this conclusion holds for other systems that in usual treatment temperature fluctuations appear divergent as temperature approaches zero.
The last part of the main body of the dissertation presents a universal criterion for the existence of an equilibrium state at low temperatures, which is established on the base of the requirement that the temperature fluctuations be small and the third law of thermodynamics. The criterion implies that at sufficiently low temperatures the minimum number of particles increases as the temperature decreases. The application of the criterion to the phonon gas, ideal Bose gas, and the ideal Fermi gas gives quantitative results that are compatible with recent results for nanoscale systems which have been given in literature with numerical methods. In addition, the Ising models are also treated and reasonable results are obtained.
The main body of this dissertation has been published in Annals of Physics (N. Y.) and American Journal of Physics, etc. Especially, the paper on the universal criterion for the existence of an equilibrium state at low temperatures has been selected into the“Virtual Journal of Nanoscale Science & Technology”established by both AIP and APS.
在考察了非热力学极限的物理现象和处理的理论工具之后,发现没有一个彻底的从等概率原理出发的处理离散体系的统计理论。作为本论文的主体之一,我们从等概率原理出发建立了多项式分布的统计力学。在多项式分布中,粒子总数为N,占据r个状态,其中第i (i = 1,2,...r )个状态的能量为εi,(ε1 <ε2 <ε3... <εr)。没有理由认为这N个粒子中的任何一个有特别的力学性质,也就是任何粒子占据其中任何一个状态的概率是一样的。这样就可以利用等概率原理和标准的概率论建立一个适用于有限粒子数的多项式分布统计力学。在这个统计力学中的热力学量在热力学极限下都将给出通常的结果。
本论文的主体之二是利用多项式分布统计力学解决了一个理论困难,并分析这个理论困难给我们的启示。在统计物理中,所有顺磁性固体,Debye固体(声子气体)等体系,由于热容量趋于零的速度比温度的平方快,所以在温度趋于零时温度涨落发散。在物理学中,任何发散都是理论困难。我们研究了最简单的顺磁体系——磁场中的自旋1/2体系——中的温度涨落问题。首先发展一个有限粒子正则系综,先引入构形量,热力学量是构形量的系综平均。熵和能量等热力学量和通常统计力学的结果符合得很好,在我们的理论中,所有的热力学量都有涨落。然后我们区分了温度涨落本身和作为强度量的温度涨落这两个物理量。在热力学极限下,我们的理论能回复到所有合理的结果,包括作为强度量的温度涨落都能给出通常的结果。唯一不同的是,温度涨落本身的发散这个本来就不可接受的结果没有了。作为强度量的温度涨落具有统计力学基础,显示了少粒子体系中温度的定义与粒子数相关。由于少粒子体系不满足可加性,不同统计系综具有不等价性,所以我们还讨论了如何从微正则系综出发讨论问题得到相同的结果。这说明了我们的结果具有普适性。这个普适性的含义不仅仅说对于这个磁场中的自旋1/2体系的结论是系综无关的,还说明对于一切通常的理论认为有温度涨落发散的体系,其实都是不发散的。
本论文的主体之三就是定量研究了在纳米体系中,在温度低到什么程度需要多少粒子通常的温度概念有效。考虑到物理学的唯象基础、温度涨落和热力学第三定律,我们发现存在一个关于低温下可以定义温度的普遍适用的判据。这个判据认为,随温度趋于零,体系的粒子数会急剧增加,也就是当温度足够低的时候,10 23个粒子数也是不足够定义温度的。本文将这个判据应用到一些具体问题例如理想气体,声子气体,理想玻色和费米气体,得到了文献中在特殊情况下通过数值计算得到的结果。本文还将这个判据应用到了Ising模型,也得到了合理的结果。
本论文的主体已经在Annals of Physics (N.Y.), American Journal of Physics等刊物上发表。特别是,关于低温下温度定义的普适判据发表后,已经被美国物理研究所(AIP)美国物理学会(APS)的联合创办的收集纳米科技重要文献的数据库“Virtual Journal of Nanoscale Science & Technology”收录。
With the technological advances allowing us to make measurement on small samples, e.g., the measurement of thermodynamic quantities like temperature with a spatial resolution on the nanometer scale, there is a growing demand for better understanding of thermal properties of nanoscale devices, individual nanostructures, and nanostructured materials. All these reveal rich non-thermodynamical limit effects. Besides nano-physics, the same effects present in nuclear physics, Bose-Einstein condensation, molecular physics etc. Mentioned explicitly or implicitly, a so-called statistical mechanics for small system or finite number of particles is in progress, in which the requirement of thermodynamical limit is not met. This dissertation first reviews the physical phenomena and the theories dealing with them.
After carefully examining the progresses in exploring the non-thermodynamical limit effects, no full statistical mechanics has been found that can deal with discrete energy systems. The fist part of the main body of this dissertation is dedicated to develop a formalism of statistical mechanics for multi-polynomial distribution. In this distribution, there are N particles distributed among r states and each state possesses a discrete energy. Since no particle has a peculiar mechanics property, i.e., every particle occupies any state with equal probability. We develop a statistical mechanics for this multi-polynomial distribution based on the equal a priori probabilities and the probability theory. All results given by the formalism turns out to be the usual ones in thermodynamic limit.
The second part of the main body of the dissertation is using the above established statistical mechanics for multi-polynomial distribution to resolve a theoretical difficulty. The usual statistical mechanics for all paramagnetic materials, and phonon gas, etc. gives divergent temperature fluctuations when the temperature itself approaches to the zero Kelven because their heat capacities goes to zero faster than the temperature square. We explicitly calculate the simplest paramagnetic material: finite N independent spin-1/2 paramagnets in a constant magnetic field. In our generalized canonical ensemble for finite number of particles, all fundamental thermodynamic quantities fluctuate, including the entropy and temperature, and all fluctuations are gradually vanishing as the temperature itself approaches zero. Our approach presents a remedy of infinite temperature fluctuations and reproduces other reasonable results that are given by standard theory, which includes an intensive temperature fluctuation that is introduced in this dissertation. Since the finite sized system does not satisfy the additivity, and different statistical ensembles are not equivalent to each other, the microcanonical ensemble treatment yields the identical result. So, our conclusion has universality. By universality we mean that the conclusion the temperature fluctuation for this system is not divergent is ensemble independent, and moreover this conclusion holds for other systems that in usual treatment temperature fluctuations appear divergent as temperature approaches zero.
The last part of the main body of the dissertation presents a universal criterion for the existence of an equilibrium state at low temperatures, which is established on the base of the requirement that the temperature fluctuations be small and the third law of thermodynamics. The criterion implies that at sufficiently low temperatures the minimum number of particles increases as the temperature decreases. The application of the criterion to the phonon gas, ideal Bose gas, and the ideal Fermi gas gives quantitative results that are compatible with recent results for nanoscale systems which have been given in literature with numerical methods. In addition, the Ising models are also treated and reasonable results are obtained.
The main body of this dissertation has been published in Annals of Physics (N. Y.) and American Journal of Physics, etc. Especially, the paper on the universal criterion for the existence of an equilibrium state at low temperatures has been selected into the“Virtual Journal of Nanoscale Science & Technology”established by both AIP and APS.
引文
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