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Exact Methods for PDEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
The Legendre transformation is used in thermodynamics when transforming the fundamental equation from internal energy (canonical variables are specific volume and specific entropy) to the Gibbs function (canonical variables are pressure and temperature), or to enthalpy (canonical variables are pressure and specific entropy), or to the Helmholtz function (canonical variables are specific volume and temperature). For more details of this application, see Kestin [683].
Finite element formulation to study thermal stresses in nanoencapsulated phase change materials for energy storage
Published in Journal of Thermal Stresses, 2020
Josep Forner-Escrig, Roberto Palma, Rosa Mondragón
The material constitution for the solid phase is calculated from the Helmholtz energy potential which is obtained by combining the first and second law of thermodynamics, by assuming that only reversible processes are considered, by applying a Legendre transformation to exchange the entropy S by T, and by assuming a natural state for which the body is undeformed and at a reference temperature Tref: where denotes small strain tensor, the displacement vector with Cartesian components the symmetric part of the displacement gradient, and hot is the abbreviation for high-order terms.
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
In [23] (p. 25), it is stated that the physical meaning of the Hamiltonian (of a classical dissipative port-Hamiltonian system) is ‘free energy’, rather than energy. In equilibrium thermodynamics, the Helmholtz free energy is a thermodynamic potential obtained from the internal energy via a Legendre transformation with respect to entropy. A potential contains all thermodynamic information about the behaviour of a material at equilibrium, see e.g [36] (p. 10). The thermodynamic potential named after Helmholtz is called a free energy because the maximum entropy principle applied to an isothermal and isochoric nonequilibrium system implies the minimization of its free energy. The difference between its free energy in the initial state and its free energy in the equilibrium state corresponds to the maximum (reversible) work production which can occur as the system passes from the initial to the equilibrium state while interacting with the isothermal reservoir at the same temperature, see e.g [68] (ch. 6). The statement that the Hamiltonian is a ‘free energy’ can thus be explained as follows: An (irrelevant) additive constant in the Hamiltonian can be identified with the combined Helmholtz free energy corresponding to all (neglected) internal energy storage of the overall system. Hence, the term ‘free energy’, as used in [23], additively combines electro-mechanical energy components and constant Helmholtz free energy components corresponding to internal energy storage in the isothermal system and environment. The electro-mechanical energy components have no entropy content since all related degrees of freedom are resolved by the model. Therefore, they are not Legendre-transformed quantities. In this (perhaps not obvious) sense, the Hamiltonian is a (Helmholtz) free energy. The GENERIC literature also mentions that Helmholtz free energy can be used as a single generator for isothermal systems [36] (p. 136).
Dynamics of particles in cold electrons plasma: fractional actionlike variational approach versus fractal spaces approach
Published in Waves in Random and Complex Media, 2021
Rami Ahmad El-Nabulsi, Alireza Khalili Golmankhaneh
For the fractal smooth Lagrangian where is a the fractal analogous tangent bundle of /( middle-η Cantor set), and assumed to be a -function with all its arguments, the fractal action is defined by where β is the fractal dimension of the fractal time set [18]. For the simplicity, let . Using the fractal calculus of variations for finding extremum of Equation (32) we obtain fractal Euler–Lagrange equation as where . Then corresponding Hamilton's equations by using Legendre transformation, and supposing the is a convex function one can get where where is might called fractal Hamiltonian. Also, and are called generalized coordinate and conjugate momentum [70]. If the density of gas molecules in the phase space has a function where and are position and momentum, respectively, then the fractal Liouville's theorem of incompressible phase space flow is as follows: Using Equation (34), we obtain In view of equation of we have which might be called fractal Boltzmann/Vlasov equation. One can rewrite Equation (39) as follows: where, and the fractal Lorentz force is defined by It is easy to verify using the previous equations that The fractal time Maxwell's equations are suggested as follows: