Temperature and entropy in molecular system

2022 ◽  
pp. 1-11
Author(s):  
Leyu Wang ◽  
James D. Lee
Keyword(s):  
Kinetic Energy ◽  
Gaussian Distribution ◽  
Equilibrium Position ◽  
Velocity Vector ◽  
Stable Equilibrium ◽  
Molecular System ◽  
Solid Material ◽  
Thermal Velocity ◽  
Global Motion ◽  
Stable Equilibrium Position

The irreversibility, temperature, and entropy are identified for an atomic system of solid material. Thermodynamics second law is automatically satisfied in the time evolution of molecular dynamics (MD). The irreversibility caused by an atom spontaneously moves from a non-stable equilibrium position to a stable equilibrium position. The process is dynamic in nature associated with the conversion of potential energy to kinetic energy and the dissipation of kinetic energy to the entire system. The forward process is less sensitive to small variation of boundary condition than reverse process, causing the time symmetry to break. Different methods to define temperature in molecular system are revisited with paradox examples. It is seen that the temperature can only be rigorously defined on an atom knowing its time history of velocity vector. The velocity vector of an atom is the summation of the mechanical part and the thermal part, the mechanical velocity is related to the global motion (translation, rotation, acceleration, vibration, etc.), the thermal velocity is related to temperature and is assumed to follow the identical random Gaussian distribution for all of its [Formula: see text], [Formula: see text] and [Formula: see text] component. The [Formula: see text]-velocity (same for [Formula: see text] or [Formula: see text]) versus time obtained from MD simulation is treated as a signal (mechanical motion) corrupted with random Gaussian distribution noise (thermal motion). The noise is separated from signal with wavelet filter and used as the randomness measurement. The temperature is thus defined as the variance of the thermal velocity multiply the atom mass and divided by Boltzmann constant. The new definition is equivalent to the Nose–Hover thermostat for a stationary system. For system with macroscopic acceleration, rotation, vibration, etc., the new definition can predict the same temperature as the stationary system, while Nose–Hover thermostat predicts a much higher temperature. It is seen that the new definition of temperature is not influenced by the global motion, i.e., translation, rotation, acceleration, vibration, etc., of the system. The Gibbs entropy is calculated for each atom by knowing normal distribution as the probability density function. The relationship between entropy and temperature is established for solid material.

On the equivalent systems for concurrent springs and dampers – Part 1: Small in-plane oscillations

2014 ◽  
Vol 228 (14) ◽  
pp. 2520-2531 ◽  
Author(s):  
Dragi Radomirovic ◽  
Ivana Kovacic
Keyword(s):  
Equilibrium Position ◽  
Stable Equilibrium ◽  
Damping Coefficient ◽  
Viscous Dampers ◽  
Stiffness Coefficients ◽  
Stable Equilibrium Position ◽  
Equivalent Systems ◽  
Stiffness And Damping ◽  
Plane Oscillations

In this work, concurrent linear springs placed in the system that performs small in-plane oscillations around the stable equilibrium position are considered. New theorems defining how they can be replaced by two mutually orthogonal springs are provided. The same concept is applied to find two mutually orthogonal linear viscous dampers that can replace a system of concurrent linear viscous dampers. The directions of such springs and dampers correspond to the principal stiffness and damping axes, respectively. So far unknown invariants related to the sum of stiffness coefficients and damping coefficient of the original and equivalent systems are presented. A few examples are given to illustrate the use and benefits of this approach. In addition, it is shown how the concept of two mutually orthogonal springs can be beneficially used for analysing problems concerned with oscillations of a particle on elastic frames.

scholarly journals EDWIN B. WILSON, MORE THAN A CATALYTIC INFLUENCE FOR PAUL SAMUELSON’S FOUNDATIONS OF ECONOMIC ANALYSIS

2019 ◽  
Vol 41 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Juan Carvajalino
Keyword(s):  
Economic Analysis ◽  
Equilibrium Position ◽  
Differential Calculus ◽  
Stable Equilibrium ◽  
Neoclassical Economics ◽  
Discrete Inequalities ◽  
Stable Equilibrium Position ◽  
Definition Of

This paper is an exploration of the genesis of Paul Samuelson’s Foundations of Economic Analysis (1947) from the perspective of his commitment to Edwin B. Wilson’s mathematics. The paper sheds new light on Samuelson’s Foundations at two levels. First, Wilson’s foundational ideas, embodied in maxims that abound in Samuelson’s book, such as “Mathematics is a Language” or “operationally meaningful theorems,” unified the chapters of Foundations and gave a sense of unity to Samuelson’s economics. Second, Wilson influenced certain theoretical concerns of Samuelson’s economics. Particularly, Samuelson adopted Wilson’s definition of a stable equilibrium position of a system in terms of discrete inequalities. Following Wilson, Samuelson developed correspondences between the continuous and the discrete in order to translate the mathematics of the continuous of neoclassical economics into formulas of discrete magnitudes. In Foundations, the local and the discrete provided the best way of operationalizing marginal and differential calculus. The discrete resonated intuitively with data; the continuous did not.

Elastic Force on a Point Defect in or Near a Surface Layer

1996 ◽  
Vol 63 (4) ◽  
pp. 1042-1045 ◽  
Author(s):  
H. Yua ◽  
S. C. Sanday ◽  
D. J. Bacon
Keyword(s):  
Surface Layer ◽  
Free Surface ◽  
Point Defect ◽  
Layer Thickness ◽  
Elastic Constants ◽  
Equilibrium Position ◽  
Stable Equilibrium ◽  
Elastic Force ◽  
Image Method ◽  
Stable Equilibrium Position

The elastic force on a point defect within or near a surface layer is determined by the image method. There is no stable equilibrium position for the point defect in the surface layer, it is attracted either to the free surface or to the interface. When the point defect is in the substrate it is attracted to the interface when the surface layer is softer than the substrate and to an equilibrium position in the substrate when the surface layer is stiffer than the substrate, the equilibrium position being a function of the elastic constants and the layer thickness.

An Edge Dislocation in a Three-Phase Composite Cylinder Model

1991 ◽  
Vol 58 (1) ◽  
pp. 75-86 ◽  
Author(s):  
H. A. Luo ◽  
Y. Chen
Keyword(s):  
Edge Dislocation ◽  
Stable Equilibrium ◽  
Phase Model ◽  
Composite Cylinder ◽  
Two Phase ◽  
Trapping Mechanism ◽  
Three Phase ◽  
Cylinder Model ◽  
Stable Equilibrium Position ◽  
The Stability

An exact solution is given for the stress field due to an edge dislocation embedded in a three-phase composite cylinder. The force on the dislocation is then derived, from which a set of simple approximate formulae is also suggested. It is shown that, in comparison with the two-phase model adopted by Dundurs and Mura (1964), the three-phase model allows the dislocation to have a stable equilibrium position under much less stringent combinations of the material constants. As a result, the so-called trapping mechanism of dislocations is more likely to take place in the three-phase model. Also, the analysis and calculation show that in the three-phase model the orientation of Burgers vector has only limited influence on the stability of dislocation. This behavior is pronouncedly different from that predicted by the two-phase model.

The Stress Field Due to an Edge Dislocation Interacting With Two Circular Inclusions

2016 ◽  
Vol 33 (2) ◽  
pp. 161-172 ◽  
Author(s):  
C.-K. Chao ◽  
F.-M. Chen ◽  
T.-H. Lin
Keyword(s):  
Stable Equilibrium ◽  
Complex Stress ◽  
Explicit Dependence ◽  
Closed Form Solutions ◽  
General Series ◽  
Stable Equilibrium Position ◽  
Arbitrary Plane ◽  
Plane Elastostatics ◽  
Combination Of Material ◽  
Extended Matrix

AbstractA general series solution to the problem of interacting circular inclusions in plane elastostatics is presented in this paper. The analysis is based on the use of the complex stress potentials of Muskhelishvili and the theorem of analytical continuation. The general forms of the complex potentials are derived explicitly for the circular inhomogeneities under arbitrary plane loading. Using the alternation technique, these general expressions were subsequently employed to treat the problem of an infinitely extended matrix containing two arbitrarily located inhomogeneities. The major contribution of the present proposed method is shown to be capable of yielding approximate closed-form solutions for multiple inclusions, thus providing the explicit dependence of the solution on the pertinent parameters. The result shows that the dislocation has a stable equilibrium position at a certain combination of material constants. The case of an inhomogeneity interacting with a circular hole under a remote uniform load is also investigated.

Measurement of Microwave Induced Forces

1992 ◽  
Vol 269 ◽  
Author(s):  
John L. Watkins ◽  
H. W. Jackson ◽  
M. Barmatz
Keyword(s):  
Electric Field ◽  
Stable Equilibrium ◽  
Resonant Cavity ◽  
Single Mode ◽  
Microgravity Environment ◽  
Force Measurements ◽  
Stable Equilibrium Position ◽  
Infinite Set ◽  
Frequency Radiation ◽  
Made In

ABSTRACTElectrically polarizable materials in an inhomogeneous electric field experience a dielectrophoretic force which is proportional to the gradient of the square of the electric field. For high frequency radiation, the time averaged electromagnetic field provides a dc force. The force exerted on a sapphire and a metal sphere by the microwave field in a single mode resonant cavity has been measured. The force measurements have been made at equally spaced points along three orthogonal axes centered in a cylindrical cavity operating in the TE111 mode using 20 watts of microwave power at about 4.84 GHz. These data are compared to the simple theory of dielectrophoresis on a small sphere and verify that, for this mode and in the absence of other forces, there is a unique and stable equilibrium position at the center of the cavity. Measurements of the quality factor of the cavity provide a means of making an absolute check to the theory. The TE111 mode is one of an infinite set of modes predicted to have stable equilibrium positions that exist away from the cavity walls. It is proposed that dielectrophoresis in a microwave resonant cavity can be used to position samples for material processing applications in a microgravity environment. Knowledge of these forces may also be important for interpreting thermogravimetric measurements made in a microwave oven.

Reliability-Based Optimal Design of a Bistable Compliant Mechanism

1993 ◽  
Author(s):  
L. L. Howell ◽  
S. S. Rao ◽  
A. Midha
Keyword(s):  
Stable Equilibrium ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Probabilistic Constraints ◽  
Probabilistic Design ◽  
Design Studies ◽  
Stable Equilibrium Position ◽  
Crank Mechanism ◽  
Insight Into ◽  
Input Torque

Abstract Compliant mechanisms obtain at least some of their motion from the deflection of their flexible members. Advantages of such mechanisms include the reduction of manufacturing and assembly cost and time. Bistable mechanisms are particularly useful in applications where two stable equilibrium positions are required, such as switches, gates, and closures. Fatigue is a major concern in many compliant mechanisms due to the cyclic stresses induced on the flexible members. In this paper, a method for the probabilistic design of a bistable compliant slider-crank mechanism is proposed. Link lengths, material properties, and cross-section dimensions are taken as random variables. Probabilistic constraints on the maximum and minimum required input torque, location of stable equilibrium position, and overall size are included. The objective function is the maximization of the mechanism reliability in fatigue. Several design studies are performed to gain further insight into the nature of the problem.

Reliability-Based Optimal Design of a Bistable Compliant Mechanism

1994 ◽  
Vol 116 (4) ◽  
pp. 1115-1121 ◽  
Author(s):  
L. L. Howell ◽  
S. S. Rao ◽  
A. Midha
Keyword(s):  
Stable Equilibrium ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Probabilistic Constraints ◽  
Probabilistic Design ◽  
Design Studies ◽  
Stable Equilibrium Position ◽  
Crank Mechanism ◽  
Insight Into ◽  
Input Torque

Compliant mechanisms obtain at least some of their motion from the deflection of their flexible members. Advantages of such mechanisms include the reduction of manufacturing and assembly cost and time. Bistable mechanisms are particularly useful in applications where two stable equilibrium positions are required, such as switches, gates, and closures. Fatigue is a major concern in many compliant mechanisms due to the cyclic stresses induced on the flexible members. In this paper, a method for the probabilistic design of a bistable compliant slider-crank mechanism is proposed. Link lengths, material properties, and cross-section dimensions are taken as random variables. Probabilistic constraints on the maximum and minimum required input torque, location of stable equilibrium position, and overall size are included. The objective function is the maximization of the mechanism reliability in fatigue. Several design studies are performed to gain further insight into the nature of the problem.

Bistable Compliant Extension Aid for a Polycentric Prosthetic Knee

2009 ◽  
Author(s):  
Adam D. Roetter ◽  
Craig P. Lusk ◽  
Rajiv Dubey
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Stable Equilibrium ◽  
Sitting Position ◽  
Element Analysis ◽  
Sitting Posture ◽  
Prosthetic Knee ◽  
Stable Equilibrium Position ◽  
Design Requirements ◽  
Prosthetic Knees

A compliant add-on mechanism for polycentric prosthetic knees was designed to provide two additional features beyond those already provided by the knee, the additional features being a stable equilibrium position when the knee is bent for the sitting posture and a moment-rotation profile that helps prevent excessive heel rise. The mechanism, dubbed the Bistable Compliant Extension Aid (BCEA), was developed and analyzed using finite-element-analysis (FEA) software. The BCEA was shown to satisfy the design requirements, prevention of excessive heel rise and providing a stable sitting position, based on its reaction moments for knee flexions ranging between 0 and 90 degrees.

On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations

2014 ◽  
Vol 228 (14) ◽  
pp. 2532-2544 ◽  
Author(s):  
Dragi Radomirovic ◽  
Ivana Kovacic
Keyword(s):  
Potential Energy ◽  
Stable Equilibrium ◽  
Dissipative Function ◽  
Equivalent System ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Out Of Plane ◽  
Stable Equilibrium Position ◽  
Equivalent Systems ◽  
Plane Oscillations

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.

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