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.     

Algorithmic information and simplicity in statistical physics

Applications of algorithmic information theory to statistical physics rely (a) on the fact that average conditional algorithmic information can be approximated by Shannon information and (b) on the existence ofsimple states described by short programs. More precisely, given a list ofN states with probabilities 0<p ₁ ≤ ... ≤ p _N , the average conditional algorithmic informationĪ to specify one of these states obeys the inequalityH≤ Ī<H+O(1), whereH=−Σp _j log₂ p _j andO(1) is a computer-dependent constant. We show how any universal computer can be slightly modified in such a way that (a) the inequality becomesH≤ Ī<H+1 and (b) states that are simple with respect to the original computer remain simple with respect to the modified computer, thereby eliminating the computer-dependent constant from statistical physics.     

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.     

Fractal behavior in quantum statistical physics

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.     

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.     

统计物理中常用的级数和积分

给出了级数∞∑n=11/n^2和∞∑n=11/n^4的求解方法，给出了Boltzmann统计、Fermi—Dirac统计和Bose—Einstein统计中常用的一些积分的普遍求解方法，并将公式化的结果加以系统地归纳。     

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.     

On linearized hydrodynamic modes in statistical physics

We formulate the linearized generalized Boltzmann equation as an (asymmetric) eigenvalue problem. This problem has five eigenvalues which tend to zero when the uniformity parameter tends to zero: to second order in this parameter, they correspond to damped sound (two modes), diffusing shear flow (two modes), and diffusing entropy flow (one mode). The microscopic expressions deduced from these results for the transport coefficients agree with the correlation-function formulas. Moreover, the corresponding eigenfunctions are explicitly displayed to lowest order in the uniformity parameter: they are microscopic analogs, in terms ofone-particle distribution functions, of the well-known linearized hydrodynamic modes of macroscopic physics. All results are established to all orders in the interactions.     

Quantum theoretical physics is statistical and relativistic

We present a new theoretical framework for the quantum mechanism. We base it on a strict deterministic behavior of single systems. The conventional QM equation, however, is found to describe statistical results of many classical systems. We will see, moreover, that a rigorous synthesis of our theory requires relativistic kinematics.So, QM is not only a classical statistical theory, it is, of necessity, a relativistic theory. The equation of our theory does not just duplicate QM, it indicates an inherent nonlinearity in QM which is subject to experimental verification. We show, therefore, that conventional QM is a corollary of classical deterministic principles. We suggest this concept of nature conflicts with that prevalent in modern physics.     

Problem of distribution moments in statistical physics

The statistical possibilities of studying intrasystem processes in multielement systems are determined by solving the problem of the moments of distributions specified in mathematical and physical formulations. The characteristic properties of the distributions can be revealed by analyzing the simple and product moments (integer, fractional, and functional) and the relationships between them. The proposed method of complex analysis of experimental data and theoretical calculations is illustrated using the examples of the Maxwellian velocity distribution of molecules, the Planck radiation spectral power distribution, and the size distribution of microscopic particles in a disperse-hardened alloy. State University, Zaporozhye. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 92–97, April, 1997.     

Surname distribution in population genetics and in statistical physics

Surnames tend to behave like neutral genes, and their distribution has attracted a growing attention from genetists and physicists. We review the century-long history of surname studies and discuss the most recent developments. Isonymy has been regarded as a tool for the measurement of consanguinity of individuals and populations and has been applied to the analysis of migrations. The analogy between patrilineal surname transmission and the propagation of Y chromosomes has been exploited for the genetic characterization of families, communities and control groups. Surname distribution is the result of a stochastic dynamics, which has been studied either as a Yule process or as a branching phenomenon: both approaches predict the asymptotic power-law behavior which has been observed in many empirical researches. Models of neutral evolution based on the theory of disordered systems have suggested the application of field-theoretical techniques, and in particular the Renormalization Group, to describe the dynamics leading to scale-invariant distributions and to compute the related (critical) exponents.     

统计物理历史上的宏观不可逆性与微观可逆性之争

牛顿力学的时间可逆性与宏观热力学过程不可逆性的矛盾,很长时间以来一直是物理学和哲学中争论最多的问题之一．本文通过玻尔兹曼引入的H定理所引起的不可逆性之争,回顾了历史上关于这一问题的争论,并讨论了最近几十年关于这一问题研究的历史进展．在现代遍历性理论和混沌动力学的框架下,微观可逆与宏观不可逆之间的矛盾得到了很好的回答,并得到了很好的统一和解决.     

Economic uncertainty and econophysics

The objective of this paper is to provide a methodological link between econophysics and economics. I will study a key notion of both fields: uncertainty and the ways of thinking about it developed by the two disciplines. After having presented the main economic theories of uncertainty (provided by Knight, Keynes and Hayek), I show how this notion is paradoxically excluded from the economic field. In economics, uncertainty is totally reduced by an a priori Gaussian framework—in contrast to econophysics, which does not use a priori models because it works directly on data. Uncertainty is then not shaped by a specific model, and is partially and temporally reduced as models improve. This way of thinking about uncertainty has echoes in the economic literature. By presenting econophysics as a Knightian method, and a complementary approach to a Hayekian framework, this paper shows that econophysics can be methodologically justified from an economic point of view.     

Current challenges for statistical physics in fracture and plasticity

Statistical physics has been applied in the last decades to several problems in mechanics, including fracture and plasticity. Concept drawn from percolation, fractal geometry, phase-transitions, and interface depinning have been used with varying degrees of success to understand these problems. In this colloquium, I describe recent successes and current challenging problems for statistical physics in fracture and plasticity, focusing on the roughness of cracks, fracture size effects and micron-scale plasticity.