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.

Generalized van der Waals theory. I. Basic formulation and application to uniform fluids

1980 ◽  
Vol 33 (9) ◽  
pp. 2013 ◽  
Author(s):  
S Nordholm ◽  
ADJ Haymet
Keyword(s):  
Free Energy ◽  
Equation Of State ◽  
Particle Density ◽  
Van Der Waals ◽  
Three Dimensional ◽  
Energy Functional ◽  
Coarse Grained ◽  
Uniform Structure ◽  
Free Energy Functional ◽  
Van Der Waals Theory

A generalized van der Waals theory is derived on the basis of simple physical and mathematical arguments. The derivation results in a free- energy functional wherein the independent variable is a coarse-grained particle density. It is assumed that a well defined particle density dominates the free energy and this density is to be obtained by minimizing the free energy functional. The variational theory so obtained can be applied to non-uniform fluids. In the present work the possibility of stable non-uniform structure is neglected and the theory is applied to uniform fluids. It then produces an equation of state identical in form to that proposed originally by van der Waals but the excluded volume is only about half as large in the three-dimensional case. Applications to several two- and three-dimensional systems indicate that the new equation of state is a distinct improvement over the traditional van der Waals theory when the full range of fluid densities is considered. The quantitative accuracy in the case of simple uniform fluids is sufficient to warrant further development and exploitation of the theory.

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Q. Yang ◽  
Y. R. Liu ◽  
X. Q. Feng ◽  
S. W. Yu
Keyword(s):  
Free Energy ◽  
Critical Point ◽  
Thermodynamic Equilibrium ◽  
Energy Function ◽  
Hessian Matrix ◽  
Energy Functional ◽  
Internal Variables ◽  
Free Energy Function ◽  
Free Energy Functional ◽  
Appl Math

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

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 On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals Droplet minimizers for the Gates–Lebowitz–Penrose free energy functional

Nonlinearity ◽  
2009 ◽  
Vol 22 (12) ◽  
pp. 2919-2952 ◽  
Author(s):  
E A Carlen ◽  
M C Carvalho ◽  
R Esposito ◽  
J L Lebowitz ◽  
R Marra
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Free Energy Functional

scholarly journals Droplet minimizers for the Cahn-Hilliard free energy functional

2006 ◽  
Vol 16 (2) ◽  
pp. 233-264 ◽  
Author(s):  
E. A. Carlen ◽  
M. C. Carvalho ◽  
R. Esposito ◽  
J. L. Lebowitz ◽  
R. Marra
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Free Energy Functional

scholarly journals Homogenization of composite ferromagnetic materials

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150365 ◽  
Author(s):  
François Alouges ◽  
Giovanni Di Fratta
Keyword(s):  
Free Energy ◽  
Ferromagnetic Material ◽  
Energy Functional ◽  
Ferromagnetic Materials ◽  
Rigorous Derivation ◽  
Free Energy Functional ◽  
Landau Free Energy ◽  
Ferromagnetic Body ◽  
The Media

The objective of this paper is to perform, by means of Γ - convergence and two-scale convergence , a rigorous derivation of the homogenized Gibbs–Landau free energy functional associated with a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the media. We thus describe the Γ -limit of the Gibbs–Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

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