Quantum Mechanical Ensemble Averages and Statistical Thermodynamics

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Free Energy ◽  
Thermal Energy ◽  
Helmholtz Free Energy ◽  
Chemical Potential ◽  
Quantum Statistical Mechanics ◽  
Bose Condensation ◽  
Distribution Functions ◽  
Ensemble Average ◽  
Statistical Thermodynamic ◽  
Quantum Mechanical

Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.

scholarly journals Hamiltonian Thermodynamics

2020 ◽  
Vol 16 (4) ◽  
pp. 557-580
Author(s):  
S.A. Rashkovskiy ◽  
Keyword(s):  
Free Energy ◽  
Hamiltonian Systems ◽  
Thermal Energy ◽  
Helmholtz Free Energy ◽  
Random Processes ◽  
Carnot Cycle ◽  
Thermodynamic Process ◽  
Thermodynamic Cycles ◽  
Thermodynamic State ◽  
Laws Of Thermodynamics

It is believed that thermodynamic laws are associated with random processes occurring in the system and, therefore, deterministic mechanical systems cannot be described within the framework of the thermodynamic approach. In this paper, we show that thermodynamics (or, more precisely, a thermodynamically-like description) can be constructed even for deterministic Hamiltonian systems, for example, systems with only one degree of freedom. We show that for such systems it is possible to introduce analogs of thermal energy, temperature, entropy, Helmholtz free energy, etc., which are related to each other by the usual thermodynamic relations. For the Hamiltonian systems considered, the first and second laws of thermodynamics are rigorously derived, which have the same form as in ordinary (molecular) thermodynamics. It is shown that for Hamiltonian systems it is possible to introduce the concepts of a thermodynamic state, a thermodynamic process, and thermodynamic cycles, in particular, the Carnot cycle, which are described by the same relations as their usual thermodynamic analogs.

Free energy

2018 ◽  
Author(s):  
Dennis Sherwood ◽  
Paul Dalby
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Helmholtz Free Energy ◽  
Chemical Potential ◽  
Free Energies ◽  
Central Concept ◽  
Mathematical Properties ◽  
Closed Systems ◽  
Coupled Reactions ◽  
Enthalpy And Entropy

A critical chapter, explaining how the principles of thermodynamics can be applied to real systems. The central concept is the Gibbs free energy, which is explored in depth, with many examples. Specific topics addressed are: Spontaneous changes in closed systems. Definitions and mathematical properties of Gibbs free energy and Helmholtz free energy. Enthalpy- and entropy-driven reactions. Maximum available work. Coupled reactions, and how to make non-spontaneous changes happen, with examples such as tidying a room, life, and global warming. Standard Gibbs free energies. Mixtures, partial molar quantities and the chemical potential.

A Thermodynamic-Consistent Model for the Thermo-Chemo-Mechanical Couplings in Amorphous Shape-Memory Polymers

2021 ◽  
pp. 2150022
Author(s):  
Lu Dai ◽  
Rui Xiao
Keyword(s):  
Free Energy ◽  
Energy Density ◽  
Shape Memory ◽  
Helmholtz Free Energy ◽  
Chemical Potential ◽  
Shape Memory Polymers ◽  
Free Energy Density ◽  
Entropy Inequality ◽  
Limited Attention ◽  
Solvent Molecules

Chemically-responsive amorphous shape-memory polymers (SMPs) can transit from the temporary shape to the permanent shape in responsive to solvents. This effect has been reported in various polymer-solvent systems. However, limited attention has been paid to the constitutive modeling of this behavior. In this work, we develop a fully thermo-chemo-mechanical coupled thermodynamic framework for the chemically-responsive amorphous SMPs. The framework shows that the entropy, the chemical potential and the stress can be directly obtained if the Helmholtz free energy density is defined. Based on the entropy inequality, the evolution equation for the viscous strain, the temperature and the number of solvent molecules are also derived. We also provide an explicit form of Helmholtz free energy density as an example. In addition, based on the free volume concept, the dependence of viscosity and diffusivity on the temperature and solvent concentration is defined. The theoretical framework can potentially advance the fundamental understanding of chemically-responsive shape-memory effect. Meanwhile, it can also be used to describe other important physical processes such as the diffusion of solvents in glassy polymers.

Coarse Grained Free Energy Functional for Lennard-Jones Systems

1999 ◽  
Vol 580 ◽  
Author(s):  
M. E. Gracheva ◽  
J. M. Rickman ◽  
J. D. Gunton ◽  
D. C. Coffey
Keyword(s):  
Free Energy ◽  
Chemical Potential ◽  
Canonical Ensemble ◽  
Particle Density ◽  
Energy Functional ◽  
Distribution Functions ◽  
Coarse Grained ◽  
Free Energy Function ◽  
Gas Phases ◽  
Free Energy Functional

AbstractResults are presented for the coarse grained distribution function and Ginzburg-Landau free energy function for coexistence of liquid and gas phases. These distribution functions were obtained by two different methods: 1) the compilation of particle density information from different coarse-grained cells using the canonical ensemble, and 2) the compilation of energy and density information from a single simulation cell by tuning the chemical potential using the grand-canonical ensemble. Both methods permit the calculation of a coarse-grained free energy functional which links the atomic and mesoscopic length scales.

scholarly journals The principle of ‘maximum energy dissipation’: a novel thermodynamic perspective on rapid water flow in connected soil structures

2010 ◽  
Vol 365 (1545) ◽  
pp. 1377-1386 ◽  
Author(s):  
Erwin Zehe ◽  
Theresa Blume ◽  
Günter Blöschl
Keyword(s):  
Free Energy ◽  
Energy Dissipation ◽  
Soil Water ◽  
Water Flow ◽  
Preferential Flow ◽  
Helmholtz Free Energy ◽  
Chemical Potential ◽  
Cohesive Soil ◽  
Soil Structures ◽  
Soil Water Flow

Preferential flow in biological soil structures is of key importance for infiltration and soil water flow at a range of scales. In the present study, we treat soil water flow as a dissipative process in an open non-equilibrium thermodynamic system, to better understand this key process. We define the chemical potential and Helmholtz free energy based on soil physical quantities, parametrize a physically based hydrological model based on field data and simulate the evolution of Helmholtz free energy in a cohesive soil with different populations of worm burrows for a range of rainfall scenarios. The simulations suggest that flow in connected worm burrows allows a more efficient redistribution of water within the soil, which implies a more efficient dissipation of free energy/higher production of entropy. There is additional evidence that the spatial pattern of worm burrow density at the hillslope scale is a major control of energy dissipation. The pattern typically found in the study is more efficient in dissipating energy/producing entropy than other patterns. This is because upslope run-off accumulates and infiltrates via the worm burrows into the dry soil in the lower part of the hillslope, which results in an overall more efficient dissipation of free energy.

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

scholarly journals A 4d $$ \mathcal{N} $$ = 1 Cardy Formula

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Joonho Kim ◽  
Seok Kim ◽  
Jaewon Song
Keyword(s):  
Black Holes ◽  
Free Energy ◽  
Asymptotic Behavior ◽  
High Temperature ◽  
Gauge Theory ◽  
Chemical Potential ◽  
Temperature Limit ◽  
Superconformal Index ◽  
High Temperature Limit ◽  
Cardy Formula

Abstract We study the asymptotic behavior of the (modified) superconformal index for 4d $$ \mathcal{N} $$ N = 1 gauge theory. By considering complexified chemical potential, we find that the ‘high-temperature limit’ of the index can be written in terms of the conformal anomalies 3c − 2a. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound 1/2 < a/c < 3/2 for the interacting SCFT is satisfied. We study $$ \mathcal{N} $$ N = 1 theories that are dual to AdS5 × Yp,p and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

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