A New Approach to Probability Theory with Reference to Statistics and Statistical Physics

A new approach to probability theory is presented with reference to statistics and statistical physics. At the outset, it is recognized that the “average man” of a population and the “average particle” of a gas are only objects of thought, and not real entities which exist in nature. The concept of average (man) is generalized as a new concept of represental (man) whose epistemological status is intermediate between those of the particular (the man) and the universal (a man). This new concept has become necessary as a result of emergence of statistics as a new branch of human knowledge at the beginning of the nineteenth century. Probability is defined with reference to the represental. The concept of probability is the same in probability theory and in physics. But whereas in statistics the probabilities are estimated using random sequences, in statistical physics they are determined either by the laws of physics alone or by making use of the laws of probability also. Thus in physics we deal with probability at a more basic level than in statistics. This approach is free from most of the controversies we face at present in interpreting probability theory and quantum mechanics.

Physical Kinetics, Relativism and Nonlocal Physics

The old classical problems of theoretical physics are revisited from the point of view of nonlocal physics. Nonlocal physics leads to very complicated mathematical apparatus. Here, we explain the main principles of nonlocal physics using transparent considerations and animations.

Restricted Boltzmann machine: Recent advances and mean-field theory

This review deals with restricted Boltzmann machine(RBM) under the light of statistical physics.The RBM is a classical family of machine learning(ML) models which played a central role in the development of deep learning.Viewing it as a spin glass model and exhibiting various links with other models of statistical physics,we gather recent results dealing with mean-field theory in this context.First the functioning of the RBM can be analyzed via the phase diagrams obtained for various statistical ensembles of RBM,leading in particular to identify a compositional phase where a small number of features or modes are combined to form complex patterns.Then we discuss recent works either able to devise mean-field based learning algorithms;either able to reproduce generic aspects of the learning process from some ensemble dynamics equations or/and from linear stability arguments.

Control of Fractal Erosion of Safe Basins in a Holmes-Duffing System via Delayed Position Feedback

A linear delayed position feedback control is applied to control the erosion of safe basins in a Holmes-Duffing system. The conditions of fractal erosion of the safe basin of the controlled system on the basis that the range of time delay leading to good control is obtained by the Melnikov method. It is found that the increasing time delay can reduce the basin erosion under a weak and positive feedback g^in. Then the evolutions of safe basins with time delay are presented in detail by the fourth Runge-Kutta and Monte-Carlo methods, which shows that the safe basin of the controlled Holmes Dulling system can be expanded, and its fractal can be reduced by the increasing time delay. These results suggest that delayed position feedbacks can be used as a good approach to control the erosion of safe basins.

Modeling random telegraph signal noise in CMOS image sensor under low light based on binomial distribution

The random telegraph signal noise in the pixel source follower MOSFET is the principle component of the noise in the CMOS image sensor under low light. In this paper, the physical and statistical model of the random telegraph signal noise in the pixel source follower based on the binomial distribution is set up. The number of electrons captured or released by the oxide traps in the unit time is described as the random variables which obey the binomial distribution. As a result,the output states and the corresponding probabilities of the first and the second samples of the correlated double sampling circuit are acquired. The standard deviation of the output states after the correlated double sampling circuit can be obtained accordingly. In the simulation section, one hundred thousand samples of the source follower MOSFET have been simulated,and the simulation results show that the proposed model has the similar statistical characteristics with the existing models under the effect of the channel length and the density of the oxide trap. Moreover, the noise histogram of the proposed model has been evaluated at different environmental temperatures.

Effect of a force-free end on the mechanical property of a biopolymer--A path integral approach

We study the effect of a force-free end on the mechanical property of a stretched biopolymer.The system can be divided into two parts.The first part consists of the segment counted from the fixed point（i.e.,the origin） to the forced point in the biopolymer,with arclength L_f.The second part consists of the segment counted from the forced point to the force-free end with arclength △L.We apply the path integral technique to find the relationship between these two parts.At finite temperature and without any constraint at the end,we show exactly that if we focus on the quantities related to the first part,then we can ignore the second part completely.Monte Carlo simulation confirms this conclusion.In contrast,the effect for the quantities related to the second part is dependent on what we want to observe.A force-free end has little effect on the relative extension,but it affects seriously the value of the end-to-end distance if △L is comparable to L_f.

Voter model on adaptive networks

Voter model is an important basic model in statistical physics.In recent years,it has been more and more used to describe the process of opinion formation in sociophysics.In real complex systems,the interactive network of individuals is dynamically adjusted,and the evolving network topology and individual behaviors affect each other.Therefore,we propose a linking dynamics to describe the coevolution of network topology and individual behaviors in this paper,and study the voter model on the adaptive network.We theoretically analyze the properties of the voter model,including consensus probability and time.The evolution of opinions on dynamic networks is further analyzed from the perspective of evolutionary game.Finally,a case study of real data is shown to verify the effectiveness of the theory.

Rapidly calculating the partition function of macroscopic systems

It has remained an open problem to accurately compute the partition function of macroscopic systems since the establishment of statistical physics. A rapid method approaching this problem was presented and was strictly tested by molecular dynamic （MD） simulations on Ar atoms in both dense gaseous and liquid states. The outcomes from the method on the internal energy and the work of isothermal expansion （and therefore the free energy） are in good agreement with the MD simulations, suggesting the method would be immediately applied in vast areas.

Understanding axon guidance: attraction, repulsion, and statistical physics

Understanding axon guidance is important for developing therapies to restore neuronal connections damaged by injury or disease. Axons migrate in response to extraceUular guidance molecules that induce or inhibit axon outgrowth activity within the axon. The direction of guidance is determined by the attractive and repulsive responses that the axon has to the guidance cues. In a deterministic model of guidance, the direction of guidance can be precisely determined if the attractive and repulsive effect that each cue has on the axon is known. But what if there are numerous attractive and repulsive responses induced by multiple guidance cues, and the direction of the attractive and repulsive events fluctuates？ If the effect that each attractive and repulsive event has on guidance becomes too complex to measure then understand- ing how each molecular cue influences the guidance decision becomes impossible.

Quantum Phase Transition of the Bosonic Atoms near the Feshbach Resonance in an Optical Lattice

The quantum phase transition from the Mott insulator to the superfluid phases of the bosonic atoms trapped in an optical lattice, in which the on-site interaction carl be tuned by a Feshbach resonance, is investigated by a variational approach within mean-field theory. We derive an extended Bos^Hubbard model to describe this ultracold atomic system. By theoretical calculation and analysis, the phase diagram is shown clearly, and we find an exciting and novel phenomenon that is the appearance of the Mort insulator-sea （MI-sea）. Meanwhile, the experimental feasibility of observing the MI-sea is discussed by analyzing the published data related to the Fashbaeh resonance at present. Finally, the potential application of the MI-sea for quantum information processing and quantum computation is also discussed in detail

Distinction of self-synchronous scrambled linear blockcodes based on multi-fractal spectrum

This study proposes a novel multi-fractal spectrumbasedapproach to distinguish linear block codes from its selfsynchronousscrambled codes. Given that the linear block codeand self-synchronous scrambled linear block code share the propertyof linear correlation, the existing linear correlation-basedidentification method is invalid for this case. This drawback can becircumvented by introducing a novel multi-fractal spectrum-basedmethod. Simulation results show that the new method has highrobustness and under the same conditions of bit error, the lowerthe code rate, the higher the recognition rate. Thus, the methodhas significant potential for future application in engineering.

Ground State of Fermions in a 1D Trap with δ Function Interaction

The ground state of fermions in a 1D trap with δ function interaction is studied mathematically with group theory ideas.

Spinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction I： General Theory

In these two papers, we solve the N body 1D harmonically trapped spinless Boson problem with repulsive δ function interaction in the limit N→∞. The general theory is given in paper I and the numerical solutions will be given in paper II.

Understanding Model Independent Genetic Mutations through Trends in Increase in Entropy

Introduction: A mutation, in general, can be defined as a change in the genetic sequence. Mutations can be changes as small as the substitution of a single DNA building block, or nucleotide base, with another nucleotide base. There can be larger mutations which can affect many genes on a chromosome. In this study we have tried to understand a normal mutation and a failed mutation from the point of view of entropy. We have shown that the entropy range of a normal mutation is less compared to the entropy range of a failed mutation. In this article we have plotted the increase of entropy of both types of mutations mentioned above. Statistical Physics of Partition Function and Entropy: In this section we have used statistical physics to define the partition function of an ensemble. Based on the partition function we have expressed how to calculate physical quantities such as average energy and entropy. Model Independent Mutation Entropy: The entropy of all processes increases. This is true even for biological systems. We have shown the difference between the entropy of a successful mutation and a failed mutation. Conclusion: In conclusion we have shown how the entropy of a successful mutation differs from that of a failed mutation. This opens up future research opportunities where we can apply this to specific biological systems.

Condensation of Self-Driven Particles in Scale-Free Networks

A new simple model of self-driven particles in scale-free networks is introduced to understand the emergence of condensation in the natural world. In the model, at each time step, particles are driven to choose their next habitats according to the particle numbers at both the present habitats and neighbors. It is found that the hub effect results in the condensation. The present numerical results as well as the theoretical analysis of condensation transition show the criterion of condensation. Both simulations and theoretical analysis display that there are three phases for different hopping probabilities δ： non-condensation （δ〈δc）, partial condensation （δc≤ δ〈1）, and complete condensation （δ≥δc＇= 1）, and the mean occupation particle number at the stationary state is also obtained. Moreover, the noise effect on self-driven particles is studied, and it makes particle numbers at nodes tend towards being identical.

Heat Transport Behaviour in One-Dimensional Lattice Models with Damping

We investigate the heat transport behaviours of two typical lattice models, the Fermi-Pasta-Ulam-beta model and the phi(4) lattice model, in the presence of damping which imitates the effect of the thermal radiation and the thermal diffusion to the surroundings through the sample boundary. It is found that the damping does not affect the thermal conductivity, but can change the heat flux dumped into the lattice chain. We also discuss possible applications under the heuristic guidance of our numerical results. In particular, we suggest a way to measure the thermal conductivity experimentally in the presence of large energy loss arisen from the radiation and the diffusion.

Progress in physical properties of Chinese stock markets

In the past two decades, statistical physics was brought into the field of finance, applying new methods and concepts to financial time series and developing a new interdiscipline ＂econophysics＂. In this review, we introduce several commonly used methods for stock time series in econophysics including distribution functions, correlation functions, detrended fluctuation analysis method, de- trended moving average method, and multifractal analysis. Then based on these methods, we review some statistical properties of Chinese stock markets including scaling behavior, long-term correla- tions, cross-correlations, leverage effects, antileverage effects, and multifractality. Last, based on an agent-based model, we develop a new option pricing model -- financial market model that shows a good agreement with the prices using real Shanghai Index data. This review is helpful for people to understand and research statistical physics of financial markets.

Adsorption of rare earth elements onto diatomite M45:Experimental investigations and modeling with statistical physics theory

The separation of rare earth elements using diatomite M45 from aqueous solutions was studied.The experimental isotherms for the adsorption of trivalent lanthanum,cerium,and neodymium cations on this adsorbent were quantified under strongly acidic conditions(pH 2)at 298-328 K.The adsorption equilibria of these earth elements were analyzed using two statistical physics models(homogeneous and heterogeneous monolayer models).The results show that the adsorption of these ions implies a multiionic mechanism,which is exothermic.Si-containing functional groups are responsible for the adsorption of these rare-earth elements on the diatomite surface.A heterogeneous statistical physics model confirms that two Si-based functional groups participate in the separation of these cations.The calculated adsorption capacities at saturation follow the order:neodymium>cerium>lanthanum.Calculated interaction energies range from 28600 to 40100 J/mol,indicating physical adsorption on diatomite M45.This study demonstrates that diatomite M45 is a promising separation medium that can be used for the recovery of REEs dissolved in aqueous solutions via adsorption.

SDE decomposition and A-type stochastic interpretation in nonequilibrium processes

An innovative theoretical framework for stochastic dynamics based on the decomposition of a stochas- tic differential equation （SDE） into a dissipative component, a detailed-balance-breaking component, and a dual-role potential landscape has been developed, which has fruitful applications in physics, engineering, chemistry, and biology. It introduces the A-type stochastic interpretation of the SDE beyond the traditional Ito or Stratonovich interpretation or even the a-type interpretation for multi- dimensional systems. The potential landscape serves as a Hmniltonian-like function in nonequilibrimn processes without detailed balance, which extends this important concept from equilibrium statistical physics to the nonequilibrium region. A question on the uniqueness of the SDE decomposition was recently raised. Our review of both the mathematical and physical aspects shows that uniqueness is guaranteed. The demonstration leads to a better understanding of the robustness of the novel frame- work. In addition, we discuss related issues including the limitations of an approach to obtaining the potential function from a steady-state distribution.

A modified Euler–Maclaurin formula in 1D and 2D with applications in statistical physics

The Euler–Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts,which includes both the Euler–Maclaurin summation formula and the Poisson summation formula as special cases.By making use of the modified formula,a possible numerical summation method is obtained and the remainder can be controlled.The modified formula is also generalized from one dimension to two dimensions.Approximate expressions of partition functions of a classical particle in square well in 1D and 2D and that of a quantum rotator are obtained with error estimation.