The significant digit law in statistical physics

The occurrence of the nonzero leftmost digit, i.e., 1,2,…,9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic distribution, named Benford’s law. We investigate three kinds of widely used physical statistics, i.e., the Boltzmann-Gibbs (BG) distribution, the Fermi-Dirac (FD) distribution, and the Bose-Einstein (BE) distribution, and find that the BG and FD distributions both fluctuate slightly in a periodic manner around Benford’s distribution with respect to the temperature of the system, while the BE distribution conforms to it exactly whatever the temperature is. Thus Benford’s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature. Furthermore, various elegant properties of Benford’s law, especially the mantissa distribution of data sets, are discussed.     

Convergence and gauge dependence properties of the resummed one-loop quark-quark scattering amplitude in perturbative QCD

The one-loop QCD effective charge α _s ^eff for quark-quark scattering is derived by diagrammatic resummation of the one-loop amplitude using an arbitrary covariant gauge. Except for the particular choice of gauge parameter ξ = −3, α _s ^eff is found to increase with increasing physical scale, Q, as lnQ or ln² Q. For ξ = −3, α _s ^eff decreases with increasing Q and satisfies a renormalization group equation. Also, except for the case ξ = 19/9, convergence radii of geometric series are found to impose upper limits on Q. The text was submitted by the author in English.     

非广延统计力学与完全开放系统的统计分布

简介了非广延统计力学的Tsallis统计,用其计算了理想气体;推导出了以含有非广延熵常数的Shannon熵为基础和以Tsallis熵为基础的非广延统计力学的完全开放系统的统计分布及计算热力学量的公式;讨论表明:Tsallis熵对应的统计分布及计算热力学量的公式在非广延参量q→1时,完全过渡到了Shannon熵对应的形式.     

The Tsallis statistical distribution in a completely open system

We derive the statistical distributions, partition functions and thermodynamic formulas for a completely open system on the basis of Tsallis entropy. These results are derived for two types of constraints, using the method of maximum entropy.     

Parton distribution functions and nuclear EMC effect in a statistical model

A new and simple statistical approach is performed to calculate the parton distribution functions (PDFs) of the nucleon in terms of light-front kinematic variables.Analytic expressions of x-dependent PDFs are obtained in the whole x region.And thereafter,we treat the temperature T as a parameter of the atomic number A to explain the nuclear EMC effect in the region x ∈ [0.2,0.7].We give the predictions of PDF ratios,and they are very different from those by other models,thus experiments aiming at measuring PDF ratios are suggested to provide a discrimination of different models.     

Parton distribution functions and nuclear EMC effect in a statistical model

A new and simple statistical approach is performed to calculate the parton distribution functions （PDFs） of the nucleon in terms of light-front kinematic variables.Analytic expressions of x-dependent PDFs are obtained in the whole x region.And thereafter,we treat the temperature T as a parameter of the atomic number A to explain the nuclear EMC effect in the region x ∈ [0.2,0.7].We give the predictions of PDF ratios,and they are very different from those by other models,thus experiments aiming at measuring PDF ratios are suggested to provide a discrimination of different models.     

诱惑态量子密钥分配系统中统计涨落的研究

针对实用的量子密钥分配(QKD)系统是基于强衰减的弱激光脉冲作为单光子源, 光子数分束攻击极大限制了通信双方在非理想条件下QKD的传输距离和密钥生成率,采用大数定律对诱惑态协议中单光子的计数率、单光子增益和误码率分别进行统计涨落分析, 利用双诱惑态比较了1310 nm和1550 nm条件下,编码脉冲的长度为(N = 10⁶-N = 10¹²)实际QKD协议中密钥的生成率与安全传输距离之间的关系、安全传输距离随编码长度的变化的关系, 得出脉冲编码长度增大到N = 10¹²时,密钥的最大安全传输距离为135 km.     

Renormalization group evaluation of exponents in family name distributions

According to many phenomenological and theoretical studies the distribution of family name frequencies in a population can be asymptotically described by a power law. We show that the Galton-Watson process corresponding to the dynamics of a growing population can be represented in Hilbert space, and its time evolution may be analyzed by renormalization group techniques, thus explaining the origin of the power law and establishing the connection between its exponent and the ratio between the population growth and the name production rates.     

Asymmetric twin-field quantum key distribution with both statistical and intensity fluctuations

Twin-field quantum key distribution(TF-QKD) is a disruptive innovation which is able to overcome the rate-distance limit of QKD without trusted relays. Since the proposal of the first TF-QKD protocol, theoretical and experimental breakthroughs have been made to enhance its ability. However, there still exist some practical issues waiting for settlement. In this paper, we examine the performances of asymmetric TF-QKD protocol with unstable light sources and limited data sizes. The statistical fluctuations of the parameters are estimated employing Azuma's inequality. Through numerical simulations, we compare the secret key rates of the asymmetric TF-QKD protocol with different data sizes and variant intensity fluctuation magnitudes. Our results demonstrate that both statistical and intensity fluctuations have significant impacts on the performance of asymmetric TF-QKD.     

A generalized formulation for population distribution in a molecular laser

A generalized theoretical model for calculating the population distribution in a molecular laser is proposed. This molecule population is found to be quantized and depending upon temperature. In addition, we show, as a particular case, that population distribution in a diatomic gasdynamic laser arises from our model.     

Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations

Reference-frame-independent quantum key distribution(RFI QKD) can generate secret keys without the alignment of reference frames, which is very robust in real-life implementations of QKD systems. However, the performance of decoystate RFI QKD with both source errors and statistical fluctuations is still missing until now. In this paper, we investigate the performance of decoy-state RFI QKD in practical scenarios with two kinds of light sources, the heralded single photon source(HSPS) and the weak coherent source(WCS), and also give clear comparison results of decoy-state RFI QKD with WCS and HSPS. Simulation results show that the secret key rates of decoy-state RFI QKD with WCS are higher than those with HSPS in short distance range, but the secret key rates of RFI QKD with HSPS outperform those with WCS in long distance range.     

物理学中常用的高斯与类高斯型积分

本文展示了基于大学数学基础对不同积分限的高斯与类高斯型积分的求解方法,并列举了高斯与类高斯型积分在“热力学与统计物理”、“光学”、“量子力学”、“量子光学”等物理学科中的若干应用实例.     

Continuity of the temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics

For a classical system of interacting particles we prove, in the microcanonical ensemble formalism of statistical mechanics, that the thermodynamic-limit entropy density is a differentiable function of the energy density and that its derivative, the thermodynamic-limit inverse temperature, is a continuous function of the energy density. We also prove that the inverse temperature of a finite system approaches the thermodynamic-limit inverse temperature as the volume of the system increases indefinitely. Finally, we show that the probability distribution for a system of fixed size in thermal contact with a large system approaches the Gibbs canonical distribution as the size of the large system increases indefinitely, if the composite system is distributed microcanonically.Supported by The British Council and the Universidad Nacional Autónoma de México.     

Inherent correlations between thermodynamics and statistical physics in extensive and nonextensive systems

With the help of a general expression of the entropies in extensive and nonextensive systems, some important relations between thermodynamics and statistical mechanics are revealed through the views of thermodynamics and statistical physics. These relations are proved through the MaxEnt approach once again. It is found that for a reversible isothermal process, the information contained in the first and second laws of thermodynamics and the MaxEnt approach is equivalent. Moreover, these relations are used to derive the probability distribution functions in nonextensive and extensive statistics and calculate the generalized forces of some interesting systems. The results obtained are of universal significance.     

On the Maxwell-Boltzmann statistical distribution function of an ideal gaseous assembly in mass motion

The invariance of the Maxwell-Boltzmann statistical distribution function has been established using Hsu’s space-light transformations. The transformations of temperature and heat turn out to be the same as given by Ott, contradicting Planck-Einstein’s views regarding their transformation. Incidently the invariance of entropy is obtained.     

中国人口分布规律及演化机理研究

近20年来,中国的总人口近似地按指数规律增加,但增加的速度有减缓的趋势.省、市、县三级的人口分布具有相似的规律：人口数k小的各级人口分布随k缓慢地变化;而k较大时,人口分布随k急剧变化,满足齐普夫定律.研究结果表明中国人口系统的各级分布具有自相似性.采用无标度网络上的聚集体迁移模型较好地解释了中国人口分布的演化规律. 关键词： 人口分布 齐普夫定律 迁移 动力学行为     

Localization length exponent,critical conductance distribution and multifractality in hierarchical network models for the quantum Hall effect

We study hierarchical network models which have recently been introduced to approximate the Chalker-Coddington model for the integer quantum Hall effect (A.G. Galstyan and M.E. Raikh, PRB 5 (1997) 1422; Arovas et al., PRB 56 (1997) 4751). The hierarchical structure is due to a recursive method starting from a finite elementary cell. The localization-delocalization transition occurring in these models is displayed in the flow of the conductance distribution under increasing system size. We numerically determine this flow, calculate the critical conductance distribution, the critical exponent of the localization length, and the multifractal exponents of critical eigenstates.     

Effect of relaxation rate on population distribution, EIT and refraction in Λ and V systems

In the three-level Λ-type and V-type systems, we find that the same pump-effect parameter R\left(=\dfrac{2|V_{c}|^{2}γ}{(ω_{ac}-ω_{c})^{2}+γ^{2}}\right) determines the population distributions, while the probe field leads to population fluctuations near the absorption peaks. With the increasing of the relaxation rate γ, the population distributions are shifted and their fluctuation amplitudes decrease. At the same time the electromagnetically induced transparency window becomes a broadened absorption peak, and the dispersion at the zero probe detuning changes from positive to negative. Large population fluctuation and non-fluctuation observed in different systems are attributed to the different relaxation rates.     

Probability theory and polymer physics

This study applies the theory of stochastic processes to the equilibrium statistical physics of polymers in solution. The topics treated include random copolymers and randomly branching polymers, with self-consistent mean field effects. A new and more natural way of dealing with Boltzmann weighting is discussed, which makes it possible from the beginning of a calculation to consider only the physical polymer conformations. We also show that in general the random copolymer problem can be reduced to the ordinary polymer problem, and treat the self-consistent field problem for a general branching polymer.     

Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology

We investigate the effect of varying boundary conditions on the renormalization group flow in a recently developed noncommutative geometry model of particle physics and cosmology. We first show that there is a sensitive dependence on the initial conditions at unification, so that, varying a parameter even slightly can be shown to have drastic effects on the running of the model parameters. We compare the running in the case of the default and the maximal mixing conditions at unification. We then exhibit explicitly a particular choice of initial conditions at the unification scale, in the form of modified maximal mixing conditions, which have the property that they satisfy all the geometric constraints imposed by the noncommutative geometry of the model at unification, and at the same time, after running them down to lower energies with the renormalization group flow, they still agree in order of magnitude with the predictions at the electroweak scale.