Minimum entropy

Thus, the aforementioned directionality arises from the combination of (A. 11) and (A. 13). All closed systems tend to the state of maximum entropy (minimum free energy), that is, the equilibrium. 

The solution of a protein crystal structure can still be a lengthy process, even when crystals are available, because of the phase problem. In contrast, small molecule (< 100 atoms) structures can be solved routinely by direct methods. In the early fifties it was shown that certain mathematical relationships exist between the phases and the amplitudes of the structure factors if it is assumed that the electron density is positive and atoms are resolved [255]. These mathematical methods have been developed [256,257] so that it is possible to solve a small molecule structure directly from the intensity data [258]. For example, the crystal structure of gramicidin S [259] (a cyclic polypeptide of 10 amino acids, 92 atoms) has been solved using the computer programme MULTAN. Traditional direct methods are not applicable to protein structures, partly because the diffraction data seldom extend to atomic resolution. Recently, a new method derived from information theory and based on the maximum entropy (minimum information) principle has been developed. In the immediate future the application will require an approximate starting phase set. However, the method has the potential for an ab initio structure determination from the measured intensities and a very small sub-set of starting phases, once the formidable problems in providing numerical methods for the solution of the fundamental equations have been solved. 

In principle, it is possible to compute a very large number of electron density maps (Eqn. 2, section 2(a)) based on all possible combinations of trial values for the phases. How can the correct solution be selected Maximum entropy (minimum information) provides a method for introducing constraints which reflect prior knowledge (e.g., the electron density must be positive) and provides a criterion for selection of the best map. 

Structural anomaly region can be bounded by using the local order parameters or by excess entropy minimum and maximum. Here we apply the definition via excess entropy. 

The accepted explanation for the minimum is that it represents the point of complete coverage of the surface by a monolayer according to Eq. XVII-37, Sconfig should go to minus infinity at this point, but in real systems an onset of multilayer adsorption occurs, and this provides a countering positive contribution. Some further discussion of the behavior of adsorption entropies in the case of heterogeneous adsorbents is given in Section XVII-14. 

Of these the last eondition, minimum Gibbs free energy at eonstant temperahire, pressure and eomposition, is probably the one of greatest praetieal importanee in ehemieal systems. (This list does not exhaust the mathematieal possibilities thus one ean also derive other apparently ununportant eonditions sueh as tliat at eonstant U, S and Uj, Fisa minimum.) However, an experimentalist will wonder how one ean hold the entropy eonstant and release a eonstraint so that some other state fiinetion seeks a minimum. 

Wlien H has reached its minimum value this is the well known Maxwell-Boltzmaim distribution for a gas in themial equilibrium with a unifomi motion u. So, argues Boltzmaim, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor (-/fg, in fact), differences in H are the same as differences in the themiodynamic entropy between initial and final equilibrium states. Boltzmaim thought that his //-tiieorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

The second law of thermodynamics may be used to show that a cyclic heat power plant (or cyclic heat engine) achieves maximum efficiency by operating on a reversible cycle called the Carnot cycle for a given (maximum) temperature of supply (T ax) and given (minimum) temperature of heat rejection (T jn). Such a Carnot power plant receives all its heat (Qq) at the maximum temperature (i.e. Tq = and rejects all its heat (Q ) at the minimum temperature (i.e. 7 = 7, in) the other processes are reversible and adiabatic and therefore isentropic (see the temperature-entropy diagram of Fig. 1.8). Its thermal efficiency is... 

For points 2-3, there is constant entropy (S) compression for a one pound of air from Pg to P3. From points 3-5 the air cools at constant pressure, and gives up heat, Q, to the intercooler. From points 5-6 the air is compressed at constant S to the final pressure Pg. Note that point Tj = point Tg for constant temperature. For minimum work Tg = T3. Then the heat, Q, equals the Work, Wl of Figure 12-36B. Figure 12-38 is convenient for estimating the moisture condensed from an airstream, as well as establishing the remaining water vapor in the gas-air. 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

Hydrogen has been suggested as the fuel of the future. One way to store it is to convert it to a compound that can be heated to release the hydrogen. One such compound is calcium hydride, CaH2. This compound has a heat of formation of —186.2 kj/mol and a standard entropy of 42.0 J/mol-K. What is the minimum temperature to which calcium hydride would have to be heated to produce hydrogen at one atmosphere pressure ... 

Theorem 4-4 can now be used to obtain a simple relationship between tire entropy of a source and the minimum average length of a set of binary code words for the source. 

The equilibrium is evidently stable when the entropy is a maximum, for then every possible change would diminish the entropy. The equilibrium will be unstable when the entropy is a minimum for a given value of the energy. This implies that if there are several conceivable neighbouring states with the same energy, that with the least entropy will correspond with a state of unstable equilibrium, whilst the others with more entropy will be essentially unstable states, except the one with the greatest amount of entropy, which will be the state of stable... 

The condition that the equilibrium shall be stable, unstable, or neutral is that the entropy shall be a maximum, a minimum or stationary respectively ... 

Equation (2.38) relates an entropy change to the flow of an infinitesimal quantity of heat in a reversible process. Earlier in this chapter, we have shown that in the reversible process, the flow of work 6 ir is a minimum for the reversible process.51 Since ir and q are related through the first law expression... 

The Hg/dimethyl formamide (DMF) interface has been studied by capacitance measurements10,120,294,301,310 in the presence of various tetraalkylammonium and alkali metal perchlorates in the range of temperatures -15 to 40°C. The specific adsorption of (C2H5)4NC104 was found to be negligible.108,109 The properties of the inner layer were analyzed on the basis of a three-state model. The temperature coefficient of the inner-layer potential drop has been found to be negative at Easo, with a minimum at -5.5 fiC cm-2. Thus the entropy of formation of the interface has a maximum at this charge. These data cannot be described... 

The second law also describes the equilibrium state of a system as one of maximum entropy and minimum free energy. For a system at constant temperature and pressure the equilibrium condition requires that the change in free energy is zero ... 

It follows from the second law of thermodynamics that the optimal free energy of a hydrocarbon-water mixture is a function of both maximal enthalpy (from hydrogen bonding) and minimum entropy (maximum degrees of freedom). Thus, nonpolar molecules tend to form droplets with minimal exposed surface area, reducing the number of water molecules affected. For the same reason, in the aqueous environment of the hving cell the hydrophobic portions of biopolymers tend to be buried inside the structure of the molecule, or within a lipid bilayer, minimizing contact with water. 

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