Boltzmann’s entropy

According to Boltzmann s entropy formula (S = kB In 2), the entropy of a polymer chain approaches zero when the chain is highly stretched by an external stretching force. That is, the entropies of the final state are the same (S 0) in the two conditions. However, it is known that the conformation of PNIPAM in water at 35°C is much more compact than that in water at RT, i.e., the initial entropy of PNIPAM at 35°C is much less than that at RT. Thus, the energy cost of a compact conformation would be less than that of a coil conformation. This factor may roughly correspond to the energy difference between water at 35°C and organic solvents. 

Before leaving this chapter, we briefly look at an important quantity known as Boltzmann s entropy, and we will examine reduced forms of the Liouville equation in generalized coordinates. 

Interlude 3.2 Poincare Recurrence Times We have seen that Boltzmann s entropy theorem leads not only to an expression for the equilibrium distribution function, but also to a specific direction of change with time or irreversibility for a system of particles or molecules. The entropy theorem states that the entropy of a closed system can never decrease so, whatever entropy state the system is in, it will always change to a higher entropy state. At that time, Boltzmann s entropy theorem was viewed to be contradictory to a well-known theorem in dynamics due to Poincare. This theorem states that... 

In previous chapters, we have introduced two seemingly different definitions of entropy, i.e., Boltzmann s entropy, Eq. (3.41), celebrated in nonequilibrium studies... 

We note that for systems at equilibrium, is independent of the locator vector r and Eq. (5.80) reduces to Gibbs entropy for equilibrium systems, Eq. (4.39).l Also, note that the introduction of Planck s constant in the logarithm term of Boltzmann s entropy, Eq. (5.79), is necessary on account of dimensional arguments, albeit it is often incorrectly left out. 

For s = 1, Eq. (5.87) becomes Boltzmann s entropy generation term, discussed in Chap. 3,... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

In relation to the introductory thermodynamic equations of the Stefan-Boltzmann law we note that Max Planck had been a professor of physics since 1889 specializing in thermodynamics. There is a very interesting history of Planck s discovery on the Internet at http //www.daviddarling. info/encyclopedia/Q/quantum theory origins.html. In fact Planck s interest was initially related to an equation he had tried to find relating Boltzmann s entropy to Wein s law. Wein s law was simply that the color of a hot object shifts with temperature and Planck developed the quantized equation to explain Wein s law. Wein s law is just an empirical observation that Planck tried to put on a firm foundation, although Planck approached the problem from a thermodynamic approach. Wien s Law relates the wavelength of the spectral maximum to temperature as [7]... 

The basis of the calculation for the entropy of the mixture is Boltzmann s entropy expression ... 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

To calculate the entropy of a substance, we use Boltzmann s formula, but the calculations are sometimes very difficult and require a lot of manipulation of Eq. 6. To measure the entropy of a substance, we use the thermodynamic definition, Eq. 1, in combination with the third law of thermodynamics. Because the third law tells us that S(0) = 0 and Eq. 2 can be used to calculate the change in entropy as a substance is heated to the temperature of interest, we can write... 

Boltzmann formula (for the entropy) The formula S = k In W, where k is Boltzmann s constant and W is the number of atomic arrangements that correspond to the same energy. 

Other than an effect on backbone solvation, side chains could potentially modulate PPII helix-forming propensities in a number of ways. These include contributions due to side chain conformational entropy and, as discussed previously, side chain-to-backbone hydrogen bonds. Given the extended nature of the PPII conformation, one might expect the side chains to possess significant conformational entropy compared to more compact conformations. The side chain conformational entropy, Y.S ppn (T = 298°K), available to each of the residues simulated in the Ac-Ala-Xaa-Ala-NMe peptides above was estimated using methods outlined in Creamer (2000). In essence, conformational entropy Scan be derived from the distribution of side chain conformations using Boltzmann s equation... 

The connection between the multiplicative insensitivity of 12 and thermodynamics is actually rather intuitive classically, we are normally only concerned with entropy differences, not absolute entropy values. Along these lines, if we examine Boltzmann s equation, S = kB In 12, where kB is the Boltzmann constant, we see that a multiplicative uncertainty in the density of states translates to an additive uncertainty in the entropy. From a simulation perspective, this implies that we need not converge to an absolute density of states. Typically, however, one implements a heuristic rule which defines the minimum value of the working density of states to be one. 

As suggested previously, the density of states has a direct connection to the entropy, and, hence, to thermodynamics, via Boltzmann s equation. Alternately, we can consider the free energy analogue, using the Laplace transform of the density of states - the canonical partition function ... 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Boltzmann s W-function is not monotonic after we perform a velocity inversion of every particle—that is, if we perform time inversion. In contrast, our -function is always monotonic as long as the system is isolated. When a velocity inversion is performed, the 7f-function jumps discontinuously due to the flow of entropy from outside. After this, the 7f-function continues its monotonic decrease [10]. Our -function breaks time symmetry, because At itself breaks time symmetry. 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

S+ =entropy of activation Q = heat of expln k = Boltzmann s constant, aod f=function... 

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