Gibbs inequality

To develop the partial average method from Eq. (4.38), we use the Gibbs inequality ... 

The effect of this envelope can be approximated by first averaging exp[ — Sy/K] over the high-order Fourier coefficients and then using the Gibbs inequality to establish a new expression for the density matrix. The... 

The partial average of the action can be effectively evaluated by employing the Gibbs inequality [33], which tells us that the exponentiated action integral, averaged over any probability distribution function, obeys the relation... 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

The Gibbs-Bogoliubov inequalities set bounds on A A of (AU)0 and (AU) which are easier a priori to estimate. These bounds are of considerable conceptual interest, but are rarely sufficiently tight to be helpful in practice. Equation (2.17) helps to explain why this is so. For distributions that are nearly Gaussian, the bounds are tight only if a is small enough. 

The Helmholtz and Gibbs energies are useful also in that they define the maximum work and the maximum non-expansion work a system can do, respectively. The combination of the Clausius inequality 7dS > dq and the first law of thermodynamics dU = dq + dw gives... 

Free energy bounds can be established via the Gibbs-Bogoliubov inequality [72], which follows from Eq. (2.6) by considering the convexity of the exponential function... 

Typically, we formally remove the term (V)o from this relation by incorporating it into H(, the Gibbs-Bogoliubov inequality [Eq. (75)] is then simply A Ag. Also, in this way the final system-bath coupling is defined to affect the system only through fluctuations, that is, at second order. 

In more general constitutive models the Gibbs equations (local equilibrium) are not valid and therefore explicit calculations of entropy are impossible. This seems to correspond to the nonuniqueness of entropy or to irreversibility of processes between nonequilibrium states [see below (1.37) and Rem. 20]. Such are some constitutive models in Sects. 2.1-2.3, but in particular models with long range memory [17, 23, 48]. Even the usefulness of entropy in situations far from equilibrium [11, 101, 114-120] seems questionable, the entropy inequality deduced and used in... 

This postulate is in agreement with traditional and reasonable expectation of achieving finite extremal values of entropy in a process occurring in an isolated system (an increase of entropy only follows from inequality (3.108)). Similarly, finite extremal values of other potentials at corresponding conditions, like the minimum of (say Gibbs) energies, etc., may be expected cf. [1, 37, 92, 110, 111]. 

Before going further, we note that analogously as in Sect. 3.8 (i.e. all velocities are considered to be zero), we can define the Gibbs stability (under isolation) of the equilibrium state (usually for a non-reacting mixture) if for every state with (4.341), (4.346), (4.347), the inequality (4.345) is valid. From this definition, the stability conditions (4.357), (4.358), (4.359) may be deduced (similarly as shown above). Reversely, these conditions express the Gibbs stability. 

The fourth inequality (8.3.9) does not provide any new constraint, but merely gives the analog of (8.3.13) for component 2. In other words, because the labeling of components is arbitrary, an expression like (8.3.13) must be obeyed by each component in the mixture. This can also be deduced in a different way for a binary, if component 1 obeys (8.3.13), then the Gibbs-Duhem equation demands that component 2 obey the analogous constraint. 

The inequality in eq. is an equivalent statement of the second law. In its original formulation, the second law specifies the equilibrium state of an isolated system (i.e., one of constant U ><, V< , and ni) as the state that maximizes entropy. If the system is by its temperature, pressure and moles, the equilibrium state corresponds to minimum Gibbs free energy. Other equivalent criteria can be developed depending on the variables that are used to fix the overall system. These statements are summarized below by the... 

The inequality goes back to Gibbs. See Isihara for a good discussion of it. 

Dyson, D.C., Contact line stability at edges comments on Gibbs s inequalities, Phys. Fluids, 31, 229, 1988. 

At the third stage the equilibrium thermodynamics was created by Clausius, Helmholtz, Boltzmann and Gibbs. Since that time the equilibrium principles started to develop as applied to macroscopic systems of any physical nature. The main, second law of thermodynamics was discovered by Clausius (Clausius, 2008). He found out the existence of the state function, entropy (S), that can change in the isolated systems exclusively towards increase. The inequality that shows such monotonicity of change... 

What has already been said at the end of Sect. 2.1.3 is also valid here To obtain accurate values of the Gibbs excess mass, it is essential to know system s parameters (g, V, ) and the sorptive gas density (p ) fairly accurately which means inequalities (Og / g) < 10 , (av / < lO , and... 

The essentials of the constitutive theory are that the experimental results are reproduced after satisfying the requirements of thermodynamics such as the Clausius-Duhem inequality and the Gibbs-Duhem relation. The constitutive theory that includes chemical processes is complex because all the conservation laws for mass, linear momentum/moment of momentum and energy are involved, and experiments to determine the parameters are extremely difficult to conduct. 

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