Advances in statistical mechanics of rock masses and its engineering applications

To efficiently link the continuum mechanics for rocks with the structural statistics of rock masses,a theoretical and methodological system called the statistical mechanics of rock masses(SMRM)was developed in the past three decades.In SMRM,equivalent continuum models of stressestrain relationship,strength and failure probability for jointed rock masses were established,which were based on the geometric probability models characterising the rock mass structure.This follows the statistical physics,the continuum mechanics,the fracture mechanics and the weakest link hypothesis.A general constitutive model and complete stressestrain models under compressive and shear conditions were also developed as the derivatives of the SMRM theory.An SMRM calculation system was then developed to provide fast and precise solutions for parameter estimations of rock masses,such as full-direction rock quality designation(RQD),elastic modulus,Coulomb compressive strength,rock mass quality rating,and Poisson’s ratio and shear strength.The constitutive equations involved in SMRM were integrated into a FLAC3D based numerical module to apply for engineering rock masses.It is also capable of analysing the complete deformation of rock masses and active reinforcement of engineering rock masses.Examples of engineering applications of SMRM were presented,including a rock mass at QBT hydropower station in northwestern China,a dam slope of Zongo II hydropower station in D.R.Congo,an open-pit mine in Dexing,China,an underground powerhouse of Jinping I hydropower station in southwestern China,and a typical circular tunnel in Lanzhou-Chongqing railway,China.These applications verified the reliability of the SMRM and demonstrated its applicability to broad engineering issues associated with jointed rock masses.

Damage statistical mechanics model of top coal in steep top caving coal

Damage statistical mechanics model of horizontal section height in the top caving was constructed in the paper. The influence factors including supporting pressure, dip angle and characteristic of coal on horizontal section height were analyzed as well. By terms of the practice project analysis, the horizontal section height increases with the increase of dip angle β and thickness of coal seam M. Dip angle of coal seam β has tremendous impact on horizontal section height, while thickness of coal seam M has slight impact. When thickness of coal seam is below 10m, horizontal section height increases sharply. While thickness exceeds 15m, it is not major factor influencing on horizontal section height any long.

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.

Statistical Mechanics Model of Liquid Binary Ailoy

From the view of chemical short range order and uncomplete random mixing existing in liquid binary al-loy, absorbing the rational part of past statistical mechanics model. a statistical mechanics model of liquidbinary alloy is proposed in this paper. According to the model, the expressions of component activity are obtained.

A statistical mechanics model of carbon nanotube macro-films

Carbon nanotube macro-films are two-dimensional films with micrometer thickness and centimeter by centimeter in-plane dimension.These carbon nanotube macroscopic assemblies have attracted significant attention from the material and mechanics communities recently because they can be easily handled and tailored to meet specific engineering needs.This paper reports the experimental methods on the preparation and characterization of single-walled carbon nanotube macro-films,and a statistical mechanics model on the deformation behavior of this material.This model provides a capability to optimize the synthesis process by comparing with the experiments.

Statistical Mechanics for Weak Measurements and Quantum Inseparability

In weak measurement thought experiment, an ensemble consists of M quantum particles and N states. We observe that separability of the particles is lost, and hence we have fuzzy occupation numbers for the particles in the ensemble. Without sharply measuring each particle state, quantum interferences add extra possible configurations of the ensemble, this explains the Quantum Pigeonhole Principle. This principle adds more entropy to the system;hence the particles seem to have a new kind of correlations emergent from particles not having a single, well-defined state. We formulated the Quantum Pigeonhole Principle in the language of abstract Hilbert spaces, then generalized it to systems consisting of mixed states. This insight into the fundamentals of quantum statistical mechanics could help us understand the interpretation of quantum mechanics more deeply, and possibly have implication on quantum computing and information theory.

Statistical properties of ideal photons in a two-dimensional dye-filled spherical cap cavity

Within the framework of quantum statistical mechanics,we have proposed an exact analytical solution to the problemof Bose-Einstein condensation(BEC)of harmonically trapped two-dimensional(2D)ideal photons.We utilize this analyticalsolution to investigate the statistical properties of ideal photons in a 2D dye-filled spherical cap cavity.The resultsof numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC ofharmonically trapped 2D ideal photons.The analytical expressions of the critical temperature and the condensate fractionare derived in the thermodynamic limit.It is found that the 2D critical photon number is larger than the one-dimensional(1D)critical photon number by two orders of magnitude.The spectral radiance of a 2D spherical cap cavity has a sharppeak at the frequency of the cavity cutoff when the photon number exceeds the critical value determined by a temperature.

Recent Advances in the Use of Statistical Mechanics to Establish Molecular Thermodynamic Models for Electrolyte Solutions

Based on statistical mechanics, a review of recent theoretical studies of real electrolyte solutions is presented from three aspects, namely, molecular simulation, mean spherical approximation (MSA), and perturbation theory. Recent advances in studies of three kinds of electrostatic potentials of mean force, three kinds of internal energies (ion-ion, ion-dipole, and dipole-dipole interactions), and three kinds of electro-lyte models (primitive, non-primitive, and solvent primitive models) are introduced. The advantages and dis-advantages between primitive and non-primitive models, and between MSA and perturbation theory are dis-cussed. Some new equations of state (EOSs) based on MSA and perturbation theory for real electrolyte so-lutions are introduced. The one-Yukawa EOS and the two-Yukawa EOS for charged colloid systems are presented.

Statistical mechanics of a nonequilibrium steady-state classical particle system driven by a constant external force

A classical particle system coupled with a thermostat driven by an external constant force reaches its steady state when the ensemble-averaged drift velocity does not vary with time.In this work,the statistical mechanics of such a system is derived solely based on the equiprobability and ergodicity principles,free from any conclusions drawn on equilibrium statistical mechanics or local equilibrium hypothesis.The momentum space distribution is determined by a random walk argument,and the position space distribution is determined by employing the equiprobability and ergodicity principles.The expressions for energy,entropy,free energy,and pressures are then deduced,and the relation among external force,drift velocity,and temperature is also established.Moreover,the relaxation towards its equilibrium is found to be an exponentially decaying process obeying the minimum entropy production theorem.

Variational quantum simulation of thermal statistical states on a superconducting quantum processer

Quantum computers promise to solve finite-temperature properties of quantum many-body systems,which is generally challenging for classical computers due to high computational complexities.Here,we report experimental preparations of Gibbs states and excited states of Heisenberg X X and X X Z models by using a 5-qubit programmable superconducting processor.In the experiments,we apply a hybrid quantum–classical algorithm to generate finite temperature states with classical probability models and variational quantum circuits.We reveal that the Hamiltonians can be fully diagonalized with optimized quantum circuits,which enable us to prepare excited states at arbitrary energy density.We demonstrate that the approach has a self-verifying feature and can estimate fundamental thermal observables with a small statistical error.Based on numerical results,we further show that the time complexity of our approach scales polynomially in the number of qubits,revealing its potential in solving large-scale problems.

CLOSED TRANS-SCALE STATISTICAL MICRODAMAGE MECHANICS

Damage and failure due to distributed microcracks or microvoids are on the challenging frontiers of solid mechanics. This appeals strongly to tools not yet fully developed in continuum damage mechanics, in particular to irreversible statistical thermodynamics and a unified macroscopic equations of mechanics and kinetic equations of microstructural transformations. This review provides the state of the art in statistical microdamage mechanics. (1) It clarifies on what level of approximation continuum damage mechanics works. Particularly,D-level approximation with dynamic function of damage appears to be a proper closed trans-scale formulation of the problem. (2) It provides physical foundation of evolution law in damage mechanics. Essentially, the damage-dependent feature of the macroscopic evolution law is due to the movement of microdamage front, resulting from microdamage growth. (3) It is found that intrinsic Deborah numberD *, a ratio of nucleation rate over growth rate of microdamage, is a proper indication of critical damage in damage mechanics, based on the idea of damage localization. (4) It clearly distinguishes the non-equilibrium damage evolution from equilibrium phase transition, like percolation.

Theoretical study of kinetic isotope effects for vacancy diffusion of impurity in solids

Theoretical studies of the diffusionalisotope effect in solids are still stuck in the 1960s and 1970s.With the development of high spatial resolution mass spectrometers,isotopic data of mineral grains are rapidly accumulated.To dig up information from these data,molecularlevel theoretical models are urgently needed.Based on the microscopic definition of the diffusion coe fficient(D),a new theoretical framework for calculating the diffusional isotope effect(DIE(v))(intermsofD*/D)forvacancy-mediated impurity diffusion in solids is provided based on statistical mechanics formalism.The newly derived equation shows that theDIE(v)can be easily calculated as long as the vibration frequencies of isotope-substituted solids are obtained.The calculatedDIE(v)values of^(199)Au/^(195)Au and^(60)Co/^(57)Co during diffusion in Cu and Au metals are all within 1%of errors compared to the experimental data,which shows that this theoretical model is reasonable and precise.

Calculation of Vapour Pressure of Metals by Statistical-Mechanical Method With the Debye Model

Statistical expression of vapour pressure equations of metals is derived from the Debye model.The statistical distribution of T_(-p) ensemble is presented in an in-elab- orate mode and the partition function is defined.The vapour pressure of eleven metals have been calculated with the Debye equation and compared with those given by the E- instein equation and empirical equation.Comparison of results of calculation from dif- ferent methods show their evident accordance within the same orders of magnitude.

Statistical-mechanical analysis of multiuser channel capacity with imperfect channel state information

In this paper, the effect of imperfect channel state information at the receiver, which is caused by noise and other interference, on the multi-access channel capacity is analysed through a statistical-mechanical approach. Replica analyses focus on analytically studying how the minimum mean square error （MMSE） channel estimation error appears in a multiuser channel capacity formula. And the relevant mathematical expressions are derived. At the same time, numerical simulation results are demonstrated to validate the Replica analyses. The simulation results show how the system parameters, such as channel estimation error, system load and signal-to-noise ratio, affect the channel capacity.

Statistical analysis of aggregation in freeway traffic

We restudy the master-equation approach to aggregation in freeway traffic based on the theory of birth-death process, in which the clustering behaviour in one-lane freeway traffic model is investigated. The transition probabilities for the jump processes are reconstructed by using Greenshields＇ model, and the equation of the mean size of the cluster at any time t is derived from the birth^death equation. Numerical experiments show the clustering behaviours varying with time very well.

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.

Long-range interacting Stark many-body probes with super-Heisenberg precision

In contrast to interferometry-based quantum sensing,where interparticle interaction is detrimental,quantum many-body probes exploit such interactions to achieve quantum-enhanced sensitivity.In most of the studied quantum many-body probes,the interaction is considered to be short-ranged.Here,we investigate the impact of long-range interaction at various filling factors on the performance of Stark quantum probes for measuring a small gradient field.These probes harness the ground state Stark localization phase transition which happens at an infinitesimal gradient field as the system size increases.Our results show that while super-Heisenberg precision is always achievable in all ranges of interaction,the long-range interacting Stark probe reveals two distinct behaviors.First,by algebraically increasing the range of interaction,the localization power is enhanced and thus the sensitivity of the probe decreases.Second,as the interaction range becomes close to a fully connected graph its effective localization power disappears and thus the sensitivity of the probe starts to enhance again.The super-Heisenberg precision is achievable throughout the extended phase until the transition point and remains valid even when the state preparation time is incorporated in the resource analysis.As the probe enters the localized phase,the sensitivity decreases and its performance becomes size-independent,following a universal behavior.In addition,our analysis shows that lower filling factors lead to better precision for measuring weak gradient fields.

Equilibrium Energy and Entropy of Vortex Filaments in the Context of Tornadogenesis and Tornadic Flows

In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.

Surface and transport properties of Cu-Sn-Ti liquid alloys

The lack of experimental data and / or limited experimental information concerning both surface and transport properties of liquid alloys often require the prediction of these quantities. An attempt has been made to link the thermophysical properties of a ternary Cu-Sn-Ti system and its binary Cu-Sn, Cu-Ti and SnoTi subsystems with the bulk through the study of the concentration dependence of various thermodynamic, structural, surface and dynamic properties in the frame of the statistical mechanical theory in conjunction with the quasi-lattice theory （QLT）. This formalism provides valuable qualitative insight into mixing processes that occur in molten alloys.

Finding Global Minima with a New Dynamical Evolutionary Algorithm

A new dynamical evolutionary algorithm (DEA) based on the theory of statistical mechanics is presented. This algorithm is very different from the traditional evolutionary algorithm and the two novel features are the unique of selecting strategy and the determination of individuals that are selected to crossover and mutate. We use DEA to solve a lot of global optimization problems that are nonlinear, multimodal and multidimensional and obtain satisfactory results.