Entropy conservation

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 have called this average the Boltzmann-Gibbs entropy, where Sbg is the entropy per molecule and n is the molecular number density. For s = 1, we recover Boltzmann s definition 

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. 

to obtain an entropy conservation equation, we can work with Eq. (5.9) modified to include the time dependence in a itself, or it is somewhat easier to work directly with the reduced Liouville equation, Eq. (3.20) or (3.24), for pairwise additive systems we choose the latter representation. 

For j =/ i, the first term on the right-hand side of Eq. (5.82) vanishes by the virtue of Gauss theorem and the properties  

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To improve understanding, we will contrast an entropy conserving process with one that generates entropy in two simple experiments. In order for an undesired exchange of entropy with the surroundings not to falsify the results, the samples must be well insulated, or the experiments must be carried out very quickly. Let us begin with the entropy conserving process, the expansion of rubber (Experiment 3.9). 

Fig. 64. Left panel Electronic specific heat ACjT vs. temperature (Vollmer et al., 2003). Solid line is an entropy conserving construction leading to SC transitions at T =1.85 K and T 2 = 1-75. Total specific heat jump ACsc/E Tc- i- Right panel Corresponding jumps in the volume thermal expansion at Tj-i 2 (Oeschler et al., 2003,...

Expanding the left-hand side and eliminating the equation of continuity, we get the equivalent form of the entropy conservation as... 

For the energy and entropy conservation equations, we can write similar and equally important expressions, viz.. 

John Gamble Kirkwood Collected Works, I. Oppenheim, (ed.), Gordon Breach, New York, 1967. [Except for entropy conservation, most of the material of the chapter is based on Ref. 1. Kirkwood contributed heavily to numerous aspects of the statistical mechanics of equilibrium and nonequilibrimn systems, including pioneering work in polymer kinetic theory, time correlations, and brownian motion. Some of the most significant papers of Kirkwood and colleagues are reproduced in this book.]... 

Finally, we turn to the equation of entropy conservation and look at the specific expressions for the entropy flux and entropy generation terms. With entropy defined in the s = 1 space, the entropy flux from Eq. (5.86) in dimensionless terms is ... 

When all operations are unitary, a stricter bound than imposed by entropy conservation was derived by Sprensen [7]. In practice, due to experimental limitations, such as the efEciency of the algorithm, relaxation times, and off-resonance effects, the obtained cooling is significantly below the bound given in eq 2. 

Two subsystems a. and p, in each of which the potentials T,p, and all the p-s are unifonn, are pennitted to interact and come to equilibrium. At equilibrium all infinitesimal processes are reversible, so for the overall system (a + P), which may be regarded as isolated, the quantities conserved include not only energy, volume and numbers of moles, but also entropy, i.e. there is no entropy creation in a system at equilibrium. One now... 

Side chain generation is often a source of error. It will be most reliable if certain rules of thumb are obeyed. Start with structurally conserved side chains and hold them fixed. Then look at the energy and entropy of rotamers for the remaining side chains. Conventional conformation search techniques are often used to place each side chain. 

Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Thus, in adiabatic processes the entropy of a system must always increase or remain constant. In words, the second law of thermodynamics states that the entropy of a system that undergoes an adiabatic process can never decrease. Notice that for the system plus the surroundings, that is, the universe, all processes are adiabatic since there are no surroundings, hence in the universe the entropy can never decrease. Thus, the first law deals with the conservation of energy in any type of process, while the sec-... 

In our opening remarks in this section, we mentioned that an analogy with dynamical Ising models can only be carried so far since there is no known conserved energy for the Life rule. However, Schulman and Seiden were able to discover a possible constant of the motion , namely a normalized entropy. 

C14-0029. Describe the thermod3Tiamic criteria for spontaneity. Describe in your own words what entropy is and why it is not a conserved quantity. 

To see this quantitatively the first entropy is required. Let the energies of the respective reservoirs be ET . Imagine a fixed region of the subsystem adjacent to each boundary and denote the energy of these regions by Es . Now impose the energy conservation laws... 

Using the definition of temperature, T 1 = dS/dE, the conservation laws, and a Taylor expansion, the reservoirs entropy may be written... 

The first energy moment of the isolated system is not conserved and it fluctuates about zero. According to the general analysis of Section IIB, the entropy of the isolated system may be written as a quadratic form,... 

A system can in principle undergo an indefinite number of processes under the constraint that energy is conserved. While the first law of thermodynamics identifies the allowed changes, a new state function, the entropy A, is needed to identify the spontaneous changes among the allowed changes. The second law of thermodynamics may be expressed as... 

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