Constructing Thermal Equilibrium (1866–1871)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Statistical Mechanics ◽  
Thermal Equilibrium ◽  
Entropy Function ◽  
Fertile Period ◽  
Ergodic Hypothesis ◽  
Least Action ◽  
Kinetic Molecular Theory ◽  
Principle Of Least Action ◽  
Basic Concepts ◽  
Boltzmann Law

This chapter is the first subset of a set of critical summaries Boltzmann’s writings on kinetic-molecular theory. It covers a first period in which he tried to construct the laws of thermal equilibrium, including the existence of the entropy function and the Maxwell–Boltzmann law, by various means including the principle of least action, Maxwell’s collision formula, the ergodic hypothesis, and a procedure of adiabatic variation. This is an immensely fertile period in which Boltzmann introduced several of the basic concepts, problems, and difficulties of modern statistical mechanics.

A PHASE SPACE FORMULATION OF QUANTUM STATE FUNCTIONS

1993 ◽  
Vol 07 (18) ◽  
pp. 3255-3272 ◽  
Author(s):  
Y. SOBOUTI ◽  
S. NASIRI
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Canonical Quantization ◽  
Quantum Statistical Mechanics ◽  
Least Action ◽  
Virtual Paths ◽  
Principle Of Least Action ◽  
Space Formulation ◽  
Special Case ◽  
State Functions

Allowing for virtual paths in phase space permits an extension of Hamilton’s principle of least action, of lagrangians and of hamiltonians to phase space. A subsequent canonical quantization, then, provides a framework for quantum statistical mechanics. The classical statistical mechanics and the conventional quantum mechanics emerge as special case of this formalism. Von Neumann’s density matrix may be inferred from it. Wigner’s functions and their evolution equation may also be obtained by a unitary transformation.

On the Shoulders of Giants

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Eighteenth Century ◽  
Robert Hooke ◽  
New Approach ◽  
Isaac Newton ◽  
Least Action ◽  
New Science ◽  
Principle Of Least Action ◽  
Parabolic Trajectory ◽  
Leonhard Euler ◽  
New Generation

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley. The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes. Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics. Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

The Analogical Turn (1884–1887)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Boltzmann Equation ◽  
Statistical Mechanics ◽  
Mechanical System ◽  
Thermal Equilibrium ◽  
Special Kind ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Equipartition Theorem ◽  
Royal Road ◽  
H Theorem

This chapter recounts how Boltzmann reacted to Hermann Helmholtz’s analogy between thermodynamic systems and a special kind of mechanical system (the “monocyclic systems”) by grouping all attempts to relate thermodynamics to mechanics, including the kinetic-molecular analogy, into a family of partial analogies all derivable from what we would now call a microcanonical ensemble. At that time, Boltzmann regarded ensemble-based statistical mechanics as the royal road to the laws of thermal equilibrium (as we now do). In the same period, he returned to the Boltzmann equation and the H theorem in reply to Peter Guthrie Tait’s attack on the equipartition theorem. He also made a non-technical survey of the second law of thermodynamics seen as a law of probability increase.

scholarly journals Nonlinear vibrations and time delay control of an extensible slowly rotating beam

2020 ◽  
Author(s):  
Jerzy Warminski ◽  
Lukasz Kloda ◽  
Jaroslaw Latalski ◽  
Andrzej Mitura ◽  
Marcin Kowalczuk
Keyword(s):  
Time Delay ◽  
Control Strategies ◽  
Vibration Reduction ◽  
Least Action ◽  
Active Elements ◽  
Principle Of Least Action ◽  
Second Mode ◽  
Delay Control ◽  
Speed Range ◽  
System With Time Delay

AbstractNonlinear dynamics of a rotating flexible slender beam with embedded active elements is studied in the paper. Mathematical model of the structure considers possible moderate oscillations thus the motion is governed by the extended Euler–Bernoulli model that incorporates a nonlinear curvature and coupled transversal–longitudinal deformations. The Hamilton’s principle of least action is applied to derive a system of nonlinear coupled partial differential equations (PDEs) of motion. The embedded active elements are used to control or reduce beam oscillations for various dynamical conditions and rotational speed range. The control inputs generated by active elements are represented in boundary conditions as non-homogenous terms. Classical linear proportional (P) control and nonlinear cubic (C) control as well as mixed ($$P-C$$ P - C ) control strategies with time delay are analyzed for vibration reduction. Dynamics of the complete system with time delay is determined analytically solving directly the PDEs by the multiple timescale method. Natural and forced vibrations around the first and the second mode resonances demonstrating hardening and softening phenomena are studied. An impact of time delay linear and nonlinear control methods on vibration reduction for different angular speeds is presented.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

Application of principle of least action to beam problems

2000 ◽  
Vol 142 (1-4) ◽  
pp. 235-243 ◽  
Author(s):  
B. Tabarrok ◽  
W. L. Cleghorn
Keyword(s):  
Least Action ◽  
Principle Of Least Action

scholarly journals Evaluation of Functioning Quality of Local Electrical Systems by the Criterion Method Based on Markov Processes

2019 ◽  
pp. 169-172 ◽  
Author(s):  
Petro Lezhniuk ◽  
Vyacheslav Komar ◽  
Natalya Sobchuk ◽  
Olena Sikorska
Keyword(s):  
Markov Processes ◽  
Renewable Energy Sources ◽  
Network Development ◽  
Least Action ◽  
Electrical Systems ◽  
Principle Of Least Action ◽  
Intensive Development ◽  
Criterion Method ◽  
The Ideal

The article proposes to use of a combination of the criterion method and Markov processes to evaluate the quality of functioning of renewable energy sources (RES) in the form of integrated readiness characteristic of the electricity network with RES or a local electrical system (LES). This is possible throughthe analysis of the problems of ensuring the quality of electricity supply in the conditions of intensive development of RES and defined by the qualimetric characteristics of the electricity networks, which are important for the provision of quality electricity. This contribute the development of generalizedsolutions and network development strategies, especially when it comes to building RES. The components of the integral index are determined as the probability of matching the actual regime to the "ideal". The "ideal" mode is determined on the basis of the principle of least action and corresponds to the circuit diagram of the network formed by the r-scheme. The basis thus determined in this way reduces the subjectivity of both evaluations and decisions taken on the basis of it.

scholarly journals The Principle of Maximal Simplicity for Modular Inorganic Crystal Structures

Crystals ◽  
2021 ◽  
Vol 11 (12) ◽  
pp. 1472
Author(s):  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structures ◽  
Structure Prediction ◽  
Structural Information ◽  
Structural Complexity ◽  
Configurational Entropy ◽  
Sensu Stricto ◽  
Least Action ◽  
Inorganic Crystal ◽  
Principle Of Least Action ◽  
Structure Description

Modularity is an important construction principle of many inorganic crystal structures that has been used for the analysis of structural relations, classification, structure description and structure prediction. The principle of maximal simplicity for modular inorganic crystal structures can be formulated as follows: in a modular series of inorganic crystal structures, the most common and abundant in nature and experiments are those arrangements that possess maximal simplicity and minimal structural information. The latter can be quantitatively estimated using information-based structural complexity parameters. The principle is applied for the modular series based upon 0D (lovozerite family), 1D (biopyriboles) and 2D (spinelloids and kurchatovite family) modules. This principle is empirical and is valid for those cases only, where there are no factors that may lead to the destabilization of simplest structural arrangements. The physical basis of the principle is in the relations between structural complexity and configurational entropy sensu stricto (which should be distinguished from the entropy of mixing). It can also be seen as an analogy of the principle of least action in physics.

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