Entropy formulation

Singer, A., Maximum entropy formulation of the Kirkwood superposition approximation. J. Chem. Phys. 121, 3657-3666 (2004). 

However, even for convection zones close to the stellar surface, where convection becomes strongly non-adiabatic and Lconv —> 0, T (r) = T(r) > 1 is not a criterion for instability. This can be seen when the Schwarzschild criterion is written in entropy formulation as < 0. Since oc yv iy )/y (with a being the adiabatic sound speed), we have for adiabatic convection (4s- =0) As in a hydrostatic situation... 

In a recent paper Bricogne [263] has shown a complete equivalence between the maximum entropy formulation and traditional direct methods. He has developed a theory capable of unifying and simplifying the whole of direct methods while avoiding the limitations and approximations inherent in the traditional approaches. Thus, maximum entropy should become general for both small and large structures. 

Both the energy formulation and the entropy formulation serve to obtain the position of equilibria. In general, it is stated that both methods are equivalent. This is basically true. However, there is a subtle difference from the point of view. 

In the energy formulation, the entropy is faced as a thermodynamic variable, formally equivalent to the other thermodynamic variables such as volume, mol numbers. The energy is treated as a function of these variables. In contrast to this view, in the entropy formulation, the entropy as such is the function, and the energy is positioned side on side with the other thermodynamic variables, such as volume, mol numbers. 

In this paper we have presented an approach to model uncertainties in the distributions of random variables by maximum entropy formulations based on the first four stochastic moments. Since these moments can not be estimated exactly for small-sample observations we model them as imcertain parameters. Based on a given set of observations we estimated the uncertainties utilizing the bootstrap method. We observed an almost normal distribution of the mean value, the standard deviation and the skewness but a skewed distribution of the kurtosis. 

We have estimated the variation of the computed failure probabihty for a bearing failure problem of a shallow foimdation based on the obtained variation of the stochastic parameters. As an outcome we can summarize that the consideration of the imcertainties in the skewness and kurtosis, which can be represented with the maximimi entropy formulation, leads to remarkable larger variations of the failure probability as obtained with a standard distribution type, where only the mean value and the standard deviation are imcertain. 

An alternative formulation of the alxtve expressions can be obtained by treating the entropy in terms of the internal energy, volume, and composition, that is, 5( /,K A )- This leads to the entropy formulation of Gibbs fundamental equation and, while not conunonly used in thermodynamics, provides certain advantages in statistic thermodynamics. The following expressions are then obtained ... 

The pressure p of a gas is exerted by random molecular impacts, which seek to enlarge the volume available to the gas and consequently to increase the number of configuration possibilities, which means an increase in entropy. Exactly in the same way, the stress random thermal motions of the individual links, tends to shorten the length of the chain, because the shorter length has greater possibilities of realization. Hence, it is more probable and corresponds to a higher entropy. Formulated, these points become... 

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

The constant of equilibrium of the whole reaction may be formulated as product of the constants of elementary steps, because the same heat and entropy of formation is expected for every single step. 

If the system is not isolated, its entropy may either increase or decrease. Thus, if a mass of gas is compressed in a cylinder impervious to heat, its entropy increases, but if heat is allowed to pass out into a medium, the entropy of the gas may decrease. By including the"gas and medium in a larger isolated system, we can apply (10) of 45, and hence show Jhat the medium gains more entropy than the gas loses. An extended assimilation of this kind shows that, if every body affected in a change is taken into account, the entropy of the whole must increase by reason of irreversible changes occurring in it. This is evidently what Clausius (1854) had in mind in the formulation of his famous aphorism The entropy of the universe strives towards a maximum. The word universe is to be understood in the sense of an ultimately isolated system. 

The significance of this relationship is that although qx and q2 are not state functions, q/T is a state function, since the sum of the q/T terms in the cycle add to zero. This important observation led to the formulation of the entropy function. 

For convenience and in accordance with a familiar formulation of the third law of thermodynamics, let us take our starting point for entropy measurements such that the entropy of the crystal is zero at the extremely low temperature involved. Starting with the crystal let us then form by reversible evaporation one mole of vapor at the vapor pressure. The entropy of the gas thus formed will evidently be... 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

This formulation is of advantage only when the constant ho (cq) is given a physical meaning (118, 119) or a supposed general linear relation between ho and /3—the so called hypercompensation effect (6)—is looked for (26, 102) or when it can be shown that ho is equal to zero (30,45, 172). Usually, or at least in kinetics, ho and Co are simply seen as intercepts without any special meaning and without a general relationship to Of course, eq. (11) can be written with interchanged variables, and in this case the intercept So can be interpreted as the so called model entropy (6). 

To reach W = 1 and S = 0, we must remove as much of this vibrational motion as possible. Recall that temperature is a measure of the amount of thermal energy in a sample, which for a solid is the energy of the atoms or molecules vibrating in their cages. Thermal energy reaches a minimum when T = 0 K. At this temperature, there is only one way to describe the system, so — 1 and — 0. This is formulated as the third law of thermodynamics, which states that a pure, perfect crystal at 0 K has zero entropy. We can state the third law as an equation, Equation perfect crystal T=0 K) 0... 

In the formulation of the nonequilibrium second law, Eq. (2), dynamic structure was said to be equivalent to a rate or flux. This may be seen more clearly from the present definition of the second entropy, since the coarse velocity can be defined as... 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

This section provides a short introductory survey of an area of science which is not only mathematically exacting, but also of fundamental importance for certain aspects of biogenesis. Thermodynamics, a sub-discipline of physics, deals not only with heat and dynamics , but formulated more generally, thermodynamics is concerned with energy and entropy and deals with theorems which are valid across almost all areas of physics. 

Once the total entropy of a composite system has been formulated as a function of the various extensive parameters of the subsystems, the extrema of this total entropy function may in principle be located by direct differentiation and classified as either maxima, minima or inflection points from the sign of the second derivative. Of these extrema, only the maxima represent stable equilibria. 

In order to formulate entropy production, local entropy is assumed to depend on local extensive parameters Xk by the same functional relationship that exists at equilibrium. Thus... 

This definition suggests a reasonable formulation of entropy current density as... 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

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