scholarly journals EXTRAPOLATION OF POWER SERIES BY SELF-SIMILAR FACTOR AND ROOT APPROXIMANTS

2004 ◽  
Vol 18 (22) ◽  
pp. 3027-3046 ◽  
Author(s):  
V. I. YUKALOV ◽  
S. GLUZMAN
Keyword(s):  
Quantum Theory ◽  
Power Series ◽  
Statistical Mechanics ◽  
Statistical Physics ◽  
Self Similar ◽  
Similar Factor

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on self-similar factor and root approximants, suggested earlier by the authors. It is shown that these approximants and their combinations can effectively extrapolate power series to the region of large variables, even up to infinity. Several examples from quantum and statistical mechanics are analyzed, illustrating the approach.

scholarly journals Summation of power series by self-similar factor approximants

2003 ◽  
Vol 328 (3-4) ◽  
pp. 409-438 ◽  
Author(s):  
V.I. Yukalov ◽  
S. Gluzman ◽  
D. Sornette
Keyword(s):  
Power Series ◽  
Self Similar ◽  
Similar Factor

Chapter 22. Connections to Statistical Physics

2021 ◽  
Author(s):  
Fabrizio Altarelli ◽  
Rémi Monasson ◽  
Guilhem Semerjian ◽  
Francesco Zamponi
Keyword(s):  
Phase Transitions ◽  
Low Temperature ◽  
Statistical Mechanics ◽  
Cost Function ◽  
Computer Science ◽  
Statistical Physics ◽  
Energy Landscape ◽  
Research Activity ◽  
Detailed Picture ◽  
Intense Research

This chapter surveys a part of the intense research activity that has been devoted by theoretical physicists to the study of randomly generated k-SAT instances. It can be at first sight surprising that there is a connection between physics and computer science. However low-temperature statistical mechanics concerns precisely the behaviour of the low-lying configurations of an energy landscape, in other words the optimization of a cost function. Moreover the ensemble of random k-SAT instances exhibit phase transitions, a phenomenon mostly studied in physics (think for instance at the transition between liquid and gaseous water). Besides the introduction of general concepts of statistical mechanics and their translations in computer science language, the chapter presents results on the location of the satisfiability transition, the detailed picture of the satisfiable regime and the various phase transitions it undergoes, and algorithmic issues for random k-SAT instances.

scholarly journals Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

2020 ◽  
Vol 34 (21) ◽  
pp. 2050208
Author(s):  
V. I. Yukalov ◽  
E. P. Yukalova
Keyword(s):  
Asymptotic Series ◽  
Nonlinear Problems ◽  
Numerical Convergence ◽  
Physical Problems ◽  
Variable Limit ◽  
Small Parameters ◽  
Self Similar ◽  
Similar Approximation ◽  
Similar Factor ◽  
Class Of Functions

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

scholarly journals Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces

2013 ◽  
Vol 117 (3) ◽  
pp. 517-537 ◽  
Author(s):  
T. Gulden ◽  
M. Janas ◽  
P. Koroteev ◽  
A. Kamenev
Keyword(s):  
Quantum Theory ◽  
Statistical Mechanics ◽  
Riemann Surfaces ◽  
Coulomb Gases

scholarly journals Landau's statistical mechanics for quasi-particle models

2014 ◽  
Vol 29 (10) ◽  
pp. 1450056 ◽  
Author(s):  
Vishnu M. Bannur
Keyword(s):  
Energy Density ◽  
Statistical Mechanics ◽  
Quark Gluon Plasma ◽  
Statistical Physics ◽  
Particle System ◽  
Gluon Plasma ◽  
Particle Model ◽  
Quasi Particle ◽  
Thermodynamics And Statistical Mechanics

Landau's formalism of statistical mechanics [following L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, 1980)] is applied to the quasi-particle model of quark–gluon plasma. Here, one starts from the expression for pressure and develop all thermodynamics. It is a general formalism and consistent with our earlier studies [V. M. Bannur, Phys. Lett. B647, 271 (2007)] based on Pathria's formalism [following R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford, 1977)]. In Pathria's formalism, one starts from the expression for energy density and develop thermodynamics. Both the formalisms are consistent with thermodynamics and statistical mechanics. Under certain conditions, which are wrongly called thermodynamic consistent relation, we recover other formalism of quasi-particle system, like in M. I. Gorenstein and S. N. Yang, Phys. Rev. D52, 5206 (1995), widely studied in quark–gluon plasma.

The Emergence of Statistical Mechanics

2017 ◽  
Author(s):  
Olivier Darrigol ◽  
Jürgen Renn
Keyword(s):  
Kinetic Theory ◽  
Statistical Mechanics ◽  
Statistical Physics ◽  
Second Law Of Thermodynamics ◽  
Boltzmann Distribution ◽  
Probabilistic Interpretation ◽  
Specific Heats ◽  
Mechanical Models ◽  
History Of ◽  
Thermal Phenomena

This article traces the history of statistical mechanics, beginning with a discussion of mechanical models of thermal phenomena. In particular, it considers how several circumstances, including the establishment of thermodynamics in the mid-nineteenth century, led to a focus on the model of heat as a motion of particles. It then describes the concept of heat as fluid and the kinetic theory before turning to gas theory and how it served as a bridge between mechanics and thermodynamics. It also explores gases as particles in motion, the Maxwell–Boltzmann distribution, the problem of specific heats, challenges to the second law of thermodynamics, and the probabilistic interpretation of entropy. Finally, it examines how the results of the kinetic theory assumed a new meaning as cornerstones of a more broadly conceived statistical physics, along with Josiah Willard Gibbs and Albert Einstein’s development of statistical mechanics as a synthetic framework.

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