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Statistical Mechanics
Published in Marc J. Assael, Geoffrey C. Maitland, Thomas Maskow, Urs von Stockar, William A. Wakeham, Stefan Will, Commonly Asked Questions in Thermodynamics, 2022
Marc J. Assael, Geoffrey C. Maitland, Thomas Maskow, Urs von Stockar, William A. Wakeham, Stefan Will
Chapter 1 dealt with the definitions of many of the quantities required for the macroscopic description of the thermodynamic behavior of systems viewed as continua, including the definition of a system. However, we are familiar with the notion that all matter is made up of atomic or molecular entities and it is the purpose of statistical mechanics to provide a microscopic description of the behavior of a thermodynamic system in terms of the properties, interactions and motions of the atoms or molecules that make up the system. Because macroscopic thermodynamic systems contain very large numbers of molecules, the task of statistical mechanics is not to describe exactly what happens to every single molecule, but rather to derive results that pertain to the complete assembly of molecules that comprise the system in a probabilistic manner. The atoms and molecules that comprise the system are best described using quantum mechanics rather than classical mechanics so that is the basis for the development of the theory of statistical mechanics.
Drude-Lorentz Free Electron Model
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
Collisions and thermal equilibrium: Collisions of electrons with ions, referred to as scattering of electrons, act as a means by which the electrons attain thermal equilibrium with their surroundings. After a collision, the electron loses memory of its pre-collision velocity and the new velocity is acquired by thermal equilibrium at the collision location. In statistical mechanics, the law of equipartition of energy relates the temperature of a system to its average energy. It states that in thermal equilibrium at absolute temperature T, each quadratic term in the equation for energy of a system of particles will contribute 1/2 kBT to the average energy; here kB is the Boltzmann constant. Degree of freedom of a system is the minimum number of independent parameters such as coordinates needed to specify its configuration. The motion of an electron has three degrees of freedom pertaining to the x, y, and z components of its momentum. Owing to the quadratic appearance of these momenta in the equation for kinetic energy of the electron, each electron is ascribed a kinetic energy of (3/2) kBT. Hence, the post-collision velocity v of the electron of mass m is determined by the equality 12mv2=32kBT
Linear Viscoelasticity
Published in Timothy P. Lodge, Paul C. Hiemenz, Polymer Chemistry, 2020
Timothy P. Lodge, Paul C. Hiemenz
We have inserted a front factor, Gp, which gives the amplitude of each mode; Gp must have the units of a modulus. We now invoke the equipartition theorem of statistical mechanics: each degree of freedom, or normal mode, acquires kT of thermal energy. Furthermore, we can assume that the modulus will increase linearly with the number of chains per unit volume because each chain can store the same amount of elastic energy under deformation. The number of chains per unit volume is cNav/M, where c is the concentration in g/mL. Therefore we can equate Gp with (cNav/M) kT = cRT/M:G(t)=cRTM∑p=1Nexp(−t/τp)
The paradigm of complex probability and Ludwig Boltzmann's entropy
Published in Systems Science & Control Engineering, 2018
Firstly, in this introductory section an overview of statistical mechanics and entropy will be done. Statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system whose exact state is uncertain. Statistical mechanics is commonly used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics. Microscopic mechanical laws do not contain concepts such as temperature, heat, or entropy; however, statistical mechanics shows how these concepts arise from the natural uncertainty about the state of a system when that system is prepared in practice. The benefit of using statistical mechanics is that it provides exact methods to connect thermodynamic quantities (such as heat capacity) to microscopic behaviour, whereas, in classical thermodynamics, the only available option would be to just measure and tabulate such quantities for various materials. Statistical mechanics also makes it possible to extend the laws of thermodynamics to cases which are not considered in classical thermodynamics, such as microscopic systems and other mechanical systems with few degrees of freedom. (Wikipedia, the free encyclopedia, Statistical Mechanics; Wikipedia, the free encyclopedia, Entropy; Wikipedia, the free encyclopedia, Entropy (statistical thermodynamics); Wikipedia, the free encyclopedia, Thermodynamics; Gibbs, 1902; Tolman, 1938; Balescu, 1975; Gibbs, 1993).
Why an urban population continues to grow under intensifying water scarcity: an answer from generalized water resources
Published in Urban Water Journal, 2018
Entropy is an extensive property of a thermodynamic system in statistical mechanics, and closely related to the number Ω of microscopic configurations that are consistent with the macroscopic quantities that characterize the system. In this paper, generalized water entropy is an indicator that refers to the degree of how virtual water contributes to an economy.