On an inverse problem in quantum statistical physics

Restoration of the full probability distribution by means of its asymptotics is considered. In the present contribution such cases are discussed where information about the asymptotics can be deduced from statistical properties of real physical systems represented, e.g., by light beams in the case of the photoproduction of electrons in the field of quantum optics or by (charged) secondaries produced by collisions observed in the field of high energy physics. Arbitrarily many modes with superposition of stochastic and coherent components are taken into account including also the case of pure coherent fields. The difference of the last case compared with the superposition alone is emphasized. To solve the corresponding inverse problems the Poisson transform is applied. Scaling properties of asymptotic probability distributions and some open problems are mentioned too.     

Tomographic representation of quantum mechanics and statistical physics

The tomographic representation for operators dependent of continuous variables, as an example of star-product quantization, and the relationship between the tomographic representations of quantum mechanics and classical statistical mechanics are discussed.     

Estimating perturbative coefficients in quantum field theory and statistical physics

We present a method for estimating perturbative coefficients in quantum field theory and statistical physics. We are able to obtain reliable error bars for each estimate. The results are in excellent agreement with known exact calculation.     

Theorems on estimating perturbative coefficients in quantum field theory and statistical physics

We present rigorous proofs for several theorems on using Padé approximants to estimate coefficients in perturbative quantum field theory and statistical physics. As a result, we find new trigonometric and other identities where the estimates based on this approach are exact. We discuss hypergeometric functions, as well as series from both perturbative quantum field theory and statistical physics.     

Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

Inference in quantum physics

In quantum physics all experimental information is discrete and stochastic. But the values of physical quantities are considered to depict definite properties of the physical world. Thus physical quantities should be identified with mathematical variables which are derived from the experimental data, but which exhibit as little randomness as possible. We look for such variables in two examples by investigating how it is possible to arrive at a value of a physical quantity from intrinsically stochastic data. With the aid of standard probability calculus and elementary information theory, we are necessarily led to the quantum theoretical phases and state vectors as the first candidates for physical quantities.     

Fractal diffusion processes in particle physics

A series of consistent measurements of kinematic variables for pion diffraction production processes by pions with an initial momentum of about 4 GeV/c were analyzed: π ⁺ + p → p + 2π ⁺ + π ^? and π ^? + p → p + 2π ^? + π ⁺. The Hurst method analysis discovered the presence of the memory effect for both data arrays. The distributions of the transition probability density appeared to seek some equilibrium shape, characteristic of the fractal Brownian motion (FBM). The process can be defined by the special diffusion Fokker-Planck equation (FPE). The obtained values of Hurst coefficient 0.5 < H < 1, which is a parameter of FPE, mean that the processes explored are realized in fractal generalized phase space with fractional dimension.     

物理学中的全量子化效应与新的探索

在一般的第一性原理计算中,原子核总是被近似成经典粒子.然而,在一些特殊的体系中,原子核的量子效应对体系的物理性质和物理过程有着至关重要的影响.在相关问题的模拟中,考虑了原子核量子效应的全量子化计算,展示了其独有的准确性.目前,路径积分分子动力学是被广泛采用的全量子化计算方法.而第一性原理的路径积分分子动力学不仅保留了第一性原理计算中电子结构和电子基态能量计算的方法,同时还应用费恩曼(Feynman)路径积分原理,得到了包含原子核量子信息的运动方程.张千帆等人应用第一性原理路径积分分子动力学,计算了BaZrO3中氢核的输运情况.结果表明,原子核的量子化对输运中两个不同的子过程有不同程度的影响,它使得有氢氧键断裂的T过程的势垒下降更多,使T过程成为快过程,从而验证了红外光谱实验的结果,同时否定了传统计算给出的T过程是慢过程的结论.     

Multiplicative stochastic processes in statistical physics

For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck equation is solved exactly by means of analytic methods. We obtain a universal eigenvalue spectrum and the corresponding set of eigenfunctions.     

GPU-computing in econophysics and statistical physics

This article focuses on the analysis of financial time series and their correlations. A method is used for quantifying pattern based correlations of a time series. With this methodology, evidence is found that typical behavioral patterns of financial market participants manifest over short time scales, i.e., that reactions to given price patterns are not entirely random, but that similar price patterns also cause similar reactions. Based on the investigation of the complex correlations in financial time series, the question arises, which properties change when switching from a positive trend to a negative trend. An empirical quantification by rescaling provides the result that new price extrema coincide with a significant increase in transaction volume and a significant decrease in the length of corresponding time intervals between transactions. These findings are independent of the time scale over 9 orders of magnitude, and they exhibit characteristics which one can also find in other complex systems in nature (and in physical systems in particular). These properties are independent of the markets analyzed. Trends that exist only for a few seconds show the same characteristics as trends on time scales of several months. Thus, it is possible to study financial bubbles and their collapses in more detail, because trend switching processes occur with higher frequency on small time scales. In addition, a Monte Carlo based simulation of financial markets is analyzed and extended in order to reproduce empirical features and to gain insight into their causes. These causes include both financial market microstructure and the risk aversion of market participants.     

The q-exponential family in statistical physics

The notion of a generalized exponential family is considered in the restricted context of non-extensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the micro-canonical ensemble belong to the q-exponential family.      

Old mathematical errors in statistical physics

In this paper, we discuss the following questions of quantum statistics: (1) the absence of Bose condensate in ideal Bose gas in the two-dimensional and one-dimensional case; (2) the concentration of the Bose condensate in an ideal Bose gas at the lowest level of the spectrum of the Schrödinger operator. In classical statistics, we discuss the discrepancy between the notion of Boltzman-Maxwell ideal gas with the notion of saturated vapor. The appearance of clusters requires the complete revision of the Clayperon equation as an equation depending on the number of degrees of freedom. For the new ideal gas and the ideal virtual liquid, we describe the phase transition of the first kind by specifying the concept of negative pressure for ideal virtual liquids.     

Conditional entropy in algebraic statistical physics

The notion of conditional entropy as entropy of conditional state on C^*-algebra with respect to its C^*-subalgebra ₁ is introduced. It is proved that for a compatible state σ on (which admits the conditional expectation of Umegaki-Takesaki) the mean conditional entropy in an a priori state σ₁ on ₁ is equal to the difference of the entropy of the state σ on and the entropy of the state σ₁ on ₁. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.     

物理学中的全量子化效应与新的探索

在一般的第一性原理计算中,原子核总是被近似成经典粒子.然而,在一些特殊的体系中,原子核的量子效应对体系的物理性质和物理过程有着至关重要的影响.在相关问题的模拟中,考虑了原子核量子效应的全量子化计算,展示了其独有的准确性.目前,路径积分分子动力学是被广泛采用的全量子化计算方法.而第一性原理的路径积分分子动力学不仅保留了第一性原理计算中电子结构和电子基态能量计算的方法,同时还应用费恩曼（Feynman）路径积分原理,得到了包含原子核量子信息的运动方程.张千帆等人应用第一性原理路径积分分子动力学,计算了BaZrO3中氢核的输运情况.结果表明,原子核的量子化对输运中两个不同的子过程有不同程度的影响,它使得有氢氧键断裂的T过程的势垒下降更多,使T过程成为快过程,从而验证了红外光谱实验的结果,同时否定了传统计算给出的T过程是慢过程的结论.     

Biological evolution and statistical physics

This review is an introduction to theoretical models and mathematical calculations for biological evolution, aimed at physicists. The methods in the field are naturally very similar to those used in statistical physics, although the majority of publications have appeared in biology journals. The review has three parts, which can be read independently. The first part deals with evolution in fitness landscapes and includes Fisher's theorem, adaptive walks, quasispecies models, effects of finite population sizes, and neutral evolution. The second part studies models of coevolution, including evolutionary game theory, kin selection, group selection, sexual selection, speciation, and coevolution of hosts and parasites. The third part discusses models for networks of interacting species and their extinction avalanches. Throughout the review, attention is paid to giving the necessary biological information, and to pointing out the assumptions underlying the models, and their limits of validity.