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Helmholtz free energy bounds

Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature... Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature...
The quantity U represents the total energy of the system, the quantity TS is called by Helmholtz the bound or unavailable energy, so that the difference of the two, namely (U - TS), may be called the free energy (Helmholtz) and denoted by / This is the strict definition of/ We can thus wnte the above equation— df = - S dt - pdv - dw... [Pg.101]

Helmholtz s energy represents the maximum amount of work which a system can exert on its surroundings. Hence, it constitutes an upper bound for the work done by the explosion. Helmholtz s free energy is not normally found in tables. Therefore calculations based on Eq. (2.46) make use of the internal energy and entropy differences. We then have... [Pg.52]

Once solved Laplace s equation for the bounded problem, we are able to determine the potentials within the cavity, c, and in the bulk, e. The charges within the cavity induce a polarisation in the dielectric, giving rise to a reaction potential, OR(ri), which acts back on the dissolved charges. Once determined the analytical expression for R(rt), Helmholtz s free energy of this interaction is just the difference between the reversible work of assembling the charge distribution in the presence of the dielectric and under vacuum, and simply reads ... [Pg.457]


See other pages where Helmholtz free energy bounds is mentioned: [Pg.131]    [Pg.21]    [Pg.28]    [Pg.129]    [Pg.116]    [Pg.171]    [Pg.403]    [Pg.501]    [Pg.502]   
See also in sourсe #XX -- [ Pg.53 ]




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