Nonequilibrium Statistical Physics Subject to the Anomalous Langevin Equation in Liouville Space

Considering that thermodynarmic irreversibility and hydrodynamic equations can not be derived rigorously and unifiedly from the Liouville equations, the anomalous Langevin equation in the Liouville space is proposed as a fundamental equation of statistical physics. This equation reflects that the law of motion of particles obeying reversible, deterministic laws in dynamics becomes irreversible and stochastic in thermodynamics. From this the fundamental equations of nonequilibrium thermodynamics, the principle of entropy increase and the theorem of minimum entropy production have been derived. The hydrodynamic equations, such as the generalized Navier-Stokes equation and the mass drift-diffusion equation etc. have been derived rigorously from the kinetic kinetic equation which is reduced from the anomalous Langevin equation in Liouville space. All these are unified and self consistent. But it is difficult to prove that entropy production density σ can never be negative everywhere for all the isolated inhomogeneous systems far from equilibrium.

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.

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.

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.

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.

A Framework for Non-Equilibrium Statistical Ensemble Theory

Since Gibbs synthesized a general equilibrium statistical ensemble theory, many theorists have attempted to generalized the Gibbsian theory to non-equilibrium phenomena domain, however the status of the theory of non-equilibrium phenomena can not be said as firm as well established as the Gibbsian ensemble theory. In this work, we present a framework for the non-equilibrium statistical ensemble formalism based on a subdynamic kinetic equation (SKE) rooted from the Brussels-Austin school and followed by some up-to-date works. The constructed key is to use a similarity transformation between Gibbsian ensembles formalism based on Liouville equation and the subdynamic ensemble formalism based on the SKE. Using this formalism, we study the spin-Boson system, as cases of weak coupling or strongly coupling, and obtain the reduced density operators for the Canonical ensembles easily.

Statistical Interaction Term of One-Dimensional Anyon Models

A form of statistical interaction term of one-dimensional anyons is introduced, based on which one-dimensional anyon models are theoretically realized, and the statistical transmutation between bosons （or fermions） and anyons is established in quantum mechanics formalism. Two kinds of anyon models which are being studied are recovered and reexplained naturally in our formalism.

Statistical Mechanical Entropy of a （4 ＋ n）-Dimensional Static Spherically Symmetric Black Hole

Considering corrections to all orders in the Planck length on the quantum state density from the generalized uncertainty principle and using the quantum state density to all degrees of freedom including extra dimensions, we calculate the statistical entropy of the scalar field in the higher-dimensional static spherically symmetric black hole spacetime without any artificial cutoff. Calculation shows that the entropy is proportional to the horizon area. The coefficient of proportionality is 1/4 when the minimal length parameter is selected appropriately.

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.

Evolution of a Thermo Vacuum State in a Single-Mode Amplitude Dissipative Channel

We investigate how an initial thermo vacuum state, in the context of thermo field dynamics, evolves in a single-mode amplitude dissipative channel, and find that in this process the thermo squeezing effect decreases while the fictitious-mode vacuum becomes chaotic.

Interest-Driven Model for Human Dynamics

Empirical observations indicate that the interevent time distribution of human actions exhibits heavy-tailed features. The queuing model based on task priorities is to some extent successful in explaining the origin of such heavy tails, however, it cannot explain all the temporal statistics of human behavior especially for the daily entertainments. We propose an interest-driven model, which can reproduce the power-law distribution of interevent time. The exponent can be analytically obtained and is in good accordance with the simulations. This model well explains the observed relationship between activities and power-law exponents, as reported recently for web-based behavior and the instant message communications.

Tracking Desired Trajectory in a Vibro-Impact System Using Backstepping Design

Based on the backstepping design of smooth systems, we developed a new control law to achieve chaos control for a vibro-impact system. In our control strategy, a novel and effective controller is designed such that the output of the vibro-impact system can track any desired trajectory in its domain. The single-degree-of-freedom vibro-impact system is taken as an example to show this control procedure. Numerical simulations are provided to verify the effectiveness of the proposed method.

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.

Robustness of Complex Networks under Attack and Repair

To study the robustness of complex networks under attack and repair, we introduce a repair model of complex networks. Based on the model, we introduce two new quantities, i.e. attack fraction fa and the maximum degree of the nodes that have never been attacked ~Ka, to study analytically the critical attack fraction and the relative size of the giant component of complex networks under attack and repair, using the method of generating function. We show analytically and numerically that the repair strategy significantly enhances the robustness of the scale-free network and the effect of robustness improvement is better for the scale-free networks with a smaller degree exponent. We discuss the application of our theory in relation to theunderstanding of robustness of complex networks with reparability.

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.

Cluster Model for Wave-Like Motions of a 2D Vertically Vibrated Granular System

The fact that trapezoid clusters exist in 2D vertically vibrated granular systems leads us to construct a cluster model, in which wave-like motions are explained as the result of cluster-plate and cluster-cluster collisions. By analyzing the collision of one cluster with the plate in detail, we deduce a basic equation from velocity relationship, which could be separated into two correlative equations： one relates wave-like motion with exciting acceleration, and we call it the excitation condition; the other relates wavelength with exciting frequency, viz., the dispersion relation. The theoretical results are in agreement with the experimental ones, which supports the idea of the cluster model. Moreover, from the cluster model, we also predict a possibility of abnormal dispersion relation of a 213 granular system.

Effects of Slip on Unsteady Mixed Convective Flow and Heat Transfer Past a Stretching Surface

Unsteady mixed convective boundary layer flow and heat transfer over a stretching vertical surface in the presence of slip axe investigated. It is noted that fluid velocity decreases due to the increasing velocity slip parameter resulting in an increase in the temperature field. The rate of heat transfer decreases with the velocity slip parameter while it increases with unsteadiness parameter. The same feature is also noticed for thermal slip parameter.

Analysis of High Codimensional Bifurcation and Chaos for the Quad Bundle Conductor＇s Galloping

A two-degree-of-freedom model of iced, electrical quad bundle conductor is developed to comprehensively describe the different galloping behaviors observed. By applying centre manifold and invertible linear transformation, the co-dimension-2 bifurcation is analyzed. The relationships of parameters between this system and the original system are obtained to analyze and to control the galloping of the quad iced bundle conductor. The space trajectory, Lyapunov exponent and Lyapunov dimension are investigated via numerical simulation to present a rigorous proof of existence of chaos.

Stochastic Resonance in a Time-Delayed Mono-Stable System with Multiplicative and Additive Noise

The stochastic resonance （SR） in a time-delayed mono-stable system driven by multiplicative white noise, additive white noise, additive dichotomous noise as well as a periodic square-wave signal is considered from the view of the signal-to-noise ratio （SNR）. It is found that the SNR increases monotonically with the increase of the delay time. The SNR exhibits the SR behavior when it is plotted as a function of intensities of the noises, displaying the asymmetry of the dichotomous noise. The SNR varies non-monotonically with the increase of the system parameter and the amplitude of the input square-wave signal.

Entropic Stochastic Resonance Driven by Colored Noise

The phenomenon of entropic stochastic resonance （ESR） in a two-dimensional confined system driven by a transverse periodic force is investigated when the colored fluctuation is included in the system. Applying the method of unified colored noise approximation, the approximate Fokker-Planck equation can be derived in the absence of the periodic force. Through the escaping rate of the Brownian particle from one well to the other, the power spectral amplification can be obtained. It is found that increasing the values of the noise correlation time and the signal frequency can suppress the ESR of the system.