Dealing with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed matter physics and statistical mechanics, this book provides an introduction to computer simulations in physics. The 5th edition contains extensive new material describing numerous powerful algorithms and methods that represent recent developments in the field. New topics such as active matter and machine learning are also introduced. Throughout, there are many applications, examples, recipes, case studies, and exercises to help the reader fully comprehend the material. This book is ideal for graduate students and researchers, both in academia and industry, who want to learn techniques that have become a third tool of physical science, complementing experiment and analytical theory.
Significance Understanding how statistical ensembles arise from the out-of-equilibrium dynamics of isolated pure systems has been a fascinating question since the early days of quantum mechanics. Recently, it has been proposed that the thermodynamic entropy of the long-time statistical ensemble is the stationary entanglement of a large subsystem in an infinite system. Here, we combine this concept with the quasiparticle picture of the entanglement evolution and integrability-based knowledge of the steady state to obtain exact analytical predictions for the time evolution of the entanglement in arbitrary 1D integrable models. These results explicitly show the transformation between the entanglement and thermodynamic entropy during the time evolution. Thus, entanglement is the natural witness for the generalized microcanonical principle underlying relaxation in integrable models. Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics. Recently, the study of quantum quenches revealed that these concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure state maintains the system at zero entropy, local properties at long times are captured by a statistical ensemble with nonzero thermodynamic entropy, which is the entanglement accumulated during the dynamics. Therefore, understanding the entanglement evolution unveils how thermodynamics emerges in isolated systems. Alas, an exact computation of the entanglement dynamics was available so far only for noninteracting systems, whereas it was deemed unfeasible for interacting ones. Here, we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the steady state and its excitations, leads to a complete understanding of the entanglement dynamics in the space–time scaling limit. We thoroughly check our result for the paradigmatic Heisenberg chain.
This paper reviews a paper from 1906 by J. Henri Poincaré on statistical mechanics with a background in his earlier work and notable connections to J. Willard Gibbs. Poincaré's paper presents important ideas that are still relevant for understanding the need for probability in statistical mechanics. Poincaré understands the foundations of statistical mechanics as a many-body problem in analytical mechanics (reflecting his 1890 monograph on The Three-Body Problem and the Equations of Dynamics) and possibly influenced by Gibbs independent development published in chapters in his 1902 book, Elementary Principles in Statistical Mechanics. This dynamical systems approach of Poincaré and Gibbs provides great flexibility including applications to many systems besides gasses. This foundation benefits from close connections to Poincaré's earlier work. Notably, Poincaré had shown (e.g. in his study of non-linear oscillators) that Hamiltonian dynamical systems display sensitivity to initial conditions separating stable and unstable trajectories. In the first context it precludes proving the stability of orbits in the solar system, here it compels the use of ensembles of systems for which the probability is ontic and frequentist and does not have an a priori value. Poincaré's key concepts relating to uncertain initial conditions, and fine- and coarse-grained entropy are presented for the readers' consideration. Poincaré and Gibbs clearly both wanted to say something about irreversibility, but came up short.
The present article deals with the problem of characterizing a widely large class of associative and possibly non-commutative rings. So, we define and explore the class of rings R for which each element in R is a sum of a tripotent element from R and an element from the subring ∆(R) of R which commute with each other, calling them strongly ∆-tripotent rings, or shortly just SDT rings. Succeeding in obtaining a complete description of these rings R modulo their Jacobson radical J(R) as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Ko¸san-Yildirim-Zhou in Can. Math. Bull. (2019). Specifically, it is proved that if a ring R is SDT, then the factor ring R/J(R) is always reduced and 6 lies in J(R). Even something more, as already noticed before, it is shown that the quotient R/J(R) is a tripotent ring, which means that each of its elements satisfies the cubic equation x3 = x. Furthermore, examining triangular matrix rings Tn(R), we succeeded to classify its structure rather completely in the case where R is a local ring and n≥ 3 by establishing a satisfactory necessary and sufficient condition in terms of the ring R and its sections, resp., divisions.
When light meets sound, a new dimension of analysis unfolds. This work explores black hole observations through the lens of signal theory and acoustic wave mechanics, revealing a resonant bridge between electromagnetic and mechanical waves. Using Event Horizon Telescope EHT data, black hole imagery is treated as a three-dimensional digital signal, where the analytic Hilbert envelope and normalized Discrete Fourier Transform DFT expose hidden structures.
The gravitational shadow is interpreted not as silence, but as a measurable energy dip—an imprint of absorption rather than absence. Euler’s identity is employed to map signal phase and symmetry into polar and complex domains, providing an intuitive mathematical pathway toward the event horizon.
By applying foundational acoustic concepts such as resonance, interference, and entropy, the field surrounding the black hole is reinterpreted as a complex communication signal. This interdisciplinary framework unifies digital signal processing, electromagnetic theory, and acoustics into a novel methodology for astronomical analysis. Notably, when a full noise assessment is conducted, EHT images exhibit a significant enhancement in resolution and information transmission
This paper presents a novel approach to developing a work roll prediction model that takes into account both the mechanism and condition influences on work roll wear. This was accomplished by conducting an analytic calculation of work roll mechanism influence, constructing a work roll wear model, and combining the wear mechanism with actual wear data. The resulting model is applicable to both symmetric and asymmetric wear of the work roll, and experimental results showed that the relative error between measured and predicted values was less than 5%, with a maximum error of below 15%. This level of accuracy is sufficient for predicting roll wear and lays the foundation for improved strip shape control and roll design. Furthermore, this approach has the potential to generate significant economic benefits and has wide-ranging applications.
Rough set theory is a worth noticing approach for inexact and uncertain system modelling. When rough set theory accompanies with fuzzy set theory, which both are a complementary generalization of set theory, they will be attended by potency in theoretical discussions. In this paper a definition for fuzzy Cayley subsets is put forward as well as fuzzy Cayley graphs of fuzzy subsets on groups inspired from the definition of Cayley graphs. We introduce rough approximation of a Cayley graph with respect to a fuzzy normal subgroup. We introduce the approximation rough fuzzy Cayley graphs and fuzzy rough fuzzy Cayley graphs. The last approximation is the mixture of the other approximations. Some theorems and properties are investigated and proved.
In this paper, we consider a linear Benney–Luke type partial integro-differential equation of higher order with degenerate kernel and two redefinition functions given at the endpoint of the segment and two parameters. To find these redefinition functions we use two intermediate data. Dirichlet boundary value conditions are used with respect to spatial variable. The Fourier series method of variables separation is applied. The countable system of functional-integral equations is obtained. Theorem on a unique solvability of countable system for functional-integral equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The triple of solutions of the inverse problem is obtained in the form of Fourier series. Absolutely and uniformly convergences of Fourier series are proved.
In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.
The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α -cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α -weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α -cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information.
This article deals with two-dimensional Navier-Stokes system of equations with rapidly oscillating terms in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier-Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.
In this paper, we obtain some closed forms of hypergeometric summation theorems for Appell’s function of first kind F1having the arguments -1, 1/2 with suitable convergence conditions, by adjustment of parameters and arguments in generalized form of first, second and third summation theorems of Ku¨mmer and others.
This work establishes necessary and sufficient conditions for the boundedness of one variable differential operator acting from a weighted Sobolev space Wlp,v to a weighted Lebesgue space on the positive real half line. The coefficients of differential operators are often assumed to be pointwise multipliers of function spaces. The author introduces pointwise multipliers in weighted Sobolev spaces; obtains the description of the space of multipliers M(W1→ W2) for a pair of weighted Sobolev spaces (W1,W2) with weights of general type.
Matthew Grasinger, Kaushik Dayal, Gal deBotton
et al.
Constitutive modeling of dielectric elastomers has been of long standing interest in mechanics. Over the last two decades rigorous constitutive models have been developed that couple the electrical response of these polymers with large deformations characteristic of soft solids. A drawback of these models is that unlike classic models of rubber elasticity they do not consider the coupled electromechanical response of single polymer chains which must be treated using statistical mechanics. The objective of this paper is to compute the stretch and polarization of single polymer chains subject to a fixed force and fixed electric field using statistical mechanics. We assume that the dipoles induced by the applied electric field at each link do not interact with each other and compute the partition function using standard techniques. We then calculate the stretch and polarization by taking appropriate derivatives of the partition function and obtain analytical results in various limits. We also perform Markov chain Monte Carlo simulations using the Metropolis and umbrella sampling methods, as well as develop a new sampling method which improves convergence by exploiting a symmetry inherent in dielectric polymer chains. The analytical expressions are shown to agree with the Monte Carlo results over a range of forces and electric fields. Our results complement recent work on the statistical mechanics of electro-responsive chains which obtains analytical expressions in a different ensemble.
Mark A. Miller, Subrahmanyam Duvvuri, Marcus Hultmark
The variety of configurations for vertical-axis wind turbines (VAWTs) make the development of universal scaling relationships for even basic performance parameters difficult. Rotor geometry changes can be characterized using the concept of solidity, defined as the ratio of solid rotor area to the swept area. However, few studies have explored the effect of this parameter at full-scale conditions due to the challenge of matching both the non-dimensional rotational rate (or tip speed ratio) and scale (or Reynolds number) in conventional wind tunnels. In this study, experiments were conducted on a VAWT model using a specialized compressed-air wind tunnel where the density can be increased to over 200 times atmospheric air. The number of blades on the model was altered to explore how solidity affects performance while keeping other geometric parameters, such as the ratio of blade chord to rotor radius, the same. These data were collected at conditions relevant to the field-scale VAWT but in the controlled environment of the lab. For the three highest solidity rotors (using the most blades), performance was found to depend similarly on the Reynolds number, despite changes in rotational effects. This result has direct implications for the modelling and design of high-solidity field-scale VAWTs.
A.A. Aniyarov, S.A. Jumabayev, D.B. Nurakhmetov
et al.
The inverse problem of determining the weight of three intermediate masses on a uniform beam from the known three natural frequencies has been solved. The performed numerical analysis allows restoring the value of only the second mass in a unique way. The inverse problem of determining the weight of three intermediate masses is solved uniquely except in the case when the first and the third masses are located geometrically symmetric relative to the middle of the beam. The hybrid algorithm for the unique solving inverse problem of determining the weight of three intermediate masses has been developed. The first three natural frequencies of the beam are calculated numerically by using the Maple computer package. Analytical relations between the masses are found.
Research in the theory of functions of an h-complex variable is of interest in connection with existing applications in non-Euclidean geometry, theoretical mechanics, etc. This article is devoted to the study of the properties of h-differentiable functions. Criteria for h-differentiability and h-holomorphy are found, formulated and proved a theorem on finite increments for an h-holomorphic function. Sufficient conditions for h-analyticity are given, formulated and proved a uniqueness theorem for h-analytic functions.