Thermodynamical entropy

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

The relationships between thermodynamic entropy and Shannon s information-theoretic entropy and between physics and computation have been explored and hotly debated ever since. It is now well known, for example, that computers can, in principle, provide an arbitrary amount of reliable computation per kT of dissipated energy ([benu73], [fredkin82] see also the discussion in section 6.4). Whether a dissipationless computer can be built in practice, remains an open problem. We must also remember that computers are themselves physical (and therefore, ultimately, quantum) devices, so that any exploration of the limitations of computation will be inextricably linked with the fundamental limitations imposed by the laws of physics. 

Physical information is a function of (1) structural organization, (2) thermodynamic entropy, and/or (3) the amount of useful work required to create it. 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

We have shown that the statistical and thermodynamic entropies lead to the same conclusions. We can expect their more general properties to be the same, too ... 

Doubling the number of molecules increases the number of microstates from W to W2, and so the entropy changes from k In W to k In W2, or 2k In W. Therefore, the statistical entropy, like the thermodynamic entropy, is an extensive property. 

The equations used to calculate changes in the statistical entropy and the thermodynamic entropy lead to the same result. 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

In thermodynamics, entropy enjoys the status as an infallible criterion of spontaneity. The concept of entropy could be used to determine whether or not a given process would take place spontaneously. It has been found that in a natural or spontaneous process there would be an increase in the entropy of the system. This is the most general criterion of spontaneity that thermodynamics offers however, to use this concept one must consider the entropy change in a process under the condition of constant volume and internal energy. Though infallible, entropy is thus not a very convenient criterion. There have, therefore, been attempts to find more suitable thermodynamic functions that would be of greater practical... 

Linear momentum (L) operator, time reversal symmetry and, 243-244 Linear scaling, multiparticle collision dynamics, nonideal fluids, 137 Linear thermodynamics entropy production, 20-23 formalities, 8-11... 

Maximum Dissipation Principle, linear thermodynamics, entropy production, 21—23... 

The entropies per unit time as well as the thermodynamic entropy production entering in the formula (101) can be interpreted in terms of the numbers of paths satisfying different conditions. In this regard, important connections exist between information theory and the second law of thermodynamics. 

We notice that the only difference between both dynamical entropies is the exchange of oa and oa in the transition probabilities appearing in the logarithm. According to Eq. (101), the thermodynamic entropy production of this process would be equal to... 

The number of typical paths generated by the random process increases as exp(/ifl). In this regard, the Kolmogorov-Sinai entropy per unit time is the rate of production of information by the random process. On the other hand, the time-reversed entropy per unit time is the rate of production of information by the time reversals of the typical paths. The thermodynamic entropy production is the difference between these two rates of information production. With the formula (101), we can recover a result by Landauer [50] and Bennett [51] that erasing information in the memory of a computer is an irreversible process of... 

The total heat released during cell discharge is the sum of the thermodynamic entropy contribution plus the irreversible contribution. This heat is released inside the battery at the reaction site on the surface of the electrode structures. Heat release is not a... 

In thermodynamics, entropy change is defined in a reversible process as... 

Work from Sturtevant s laboratory detailed the kinetics and thermodynamics of zinc binding to apocarbonic anhydrase (carbonate dehydratase) selected data are recorded in Table II (Henkens and Sturtevant, 1968 Henkens etal., 1969). The thermodynamic entropy term A5 at pH 7.0 is 88 e.u. (1 e.u. = 1 cal/mol-K), and this is essentially matched by the binding of zinc to the hexadentate ligand cyclohexylenediamine tetraacetate where AS = 82 e.u. At pH 7.0 the enthalpy of zinc-protein association is 9.8 kcal/mol, but this unfavorable term is overwhelmed by the favorable entropic contribution to the free energy (AG = AH - T AS), where —TAS = -26.2 kcal/mol at 298 K (25°C). Hence, the kinetics and thermodynamics of protein-zinc interaction in this example are dominated by very favorable entropy effects. 

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

Clausius recognition of the thermodynamic entropy S was a crucial watershed in thermodynamic theory, allowing the subsequent development of the theory to proceed in a far more accurate and coherent manner. 

These two additional properties of the H in (5.6), together with its mono-tonic decrease, has led to its identification with the entropy defined by the second law of thermodynamics. It must be realized, however, that H is a functional of a non-equilibrium probability distribution, whereas the thermodynamic entropy is a quantity defined for thermodynamic equilibrium states. The present entropy is therefore a generalization of the thermodynamic entropy the generalized entropy is... 

In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium. Entropy increase plays a large role in irreversible thermodynamics. If each of the reference cells were an isolated system, the right-hand side of Eq. 2.4 could only increase in a kinetic process. However, because energy, heat, and mass may flow between cells during kinetic processes, they cannot be treated as isolated systems, and application of the second law must be generalized to the system of interacting cells. 

Chapter 17 Thermodynamics Entropy, Free Energy, and Equilibrium... 

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