Entropy theorem

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... 

Because of the Nemst heat theorem and the third law, standard themrodynamic tables usually do not report entropies of fomiation of compounds instead they report the molar entropy 50 7 for each element and... 

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

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. 

Pauii W Jr 1928 Uber das H-Theorem vom Anwachsen der Entropie vom Standpunkt der neuen Quantenmechanik Probleme der modernen Physik ed P Debye (Leipzig Hirzei) pp 30-45... 

Theorem 4-2. Given an arbitrary discrete memoryless source, V, of entropy H U), and given any e > 0 and 8 > 0, it is possible to find an N large enough so that all but a set of probability less than e of the N length sequences can be coded into unique binary sequences of length at most [H(U) + h]N. 

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. 

Theorem 4-5. Let Pr(ux), , Pr(u7) be the probabilities in decreasing order of the set of sequences of length N from a discrete memoryless source of entropy H(U). Then every binary prefix code for this source has an average length Nb satisfying... 

Theorem, 4-9. Let RT be the entropy per unit time of a discrete memoryless source of alphabet size M, and let CT be the capacity per unit time of a discrete memoryless channel. Let Tg and Tc be the intersymbol times for the source and channel, and let a sequence of N source letters be transmitted by at most... 

Equation (4-66) yields an implicit lower bound to Pe that is greater than 0 when BT > CT. Observe that the bound is independent of N and depends only on the source entropy, the channel capacity per source digit (CTTt), and the source alphabet size. It would be satisfying if the dependence of Eq. (4-66) on the source alphabet size could be removed. Unfortunately the dependence of Pe on M as well as (RT — Ct)Ts is necessary, as the next theorem shows. 

Suppose now that we had some other source of entropy R < R nats per channel symbol. We know from Theorem 4-2 that this source can be coded into binary digits as efficiently as desired. These binary digits can then be coded into letters of the alphabet %, , uM. Thus, aside from the equal probability assumption for the M letter source, our results are applicable to any source. 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

Then from (4), (6), (11), and (12) we find Theorem II. Ij an isopiestic p + dp is drawn to cut the three curves of transition (or their prolongations) meeting at a triple point, the central point of section corresponds u-ith the transition involving the greatest change of entropy. This theorem is due to Roozeboom (1901). 

With the entropy, however, the case is quite otherwise, and we shall now go on to show that as soon as we are in possession of a method of determining the absolute value of the entropy of a system, all the lacunae of the classical thermodynamics can be completed. The required information is furnished by a hypothesis put forward in 1906 by W. Nernst, and usually called by German writers das Nernstsche Wdrmetheorem. We can refer to it without ambiguity as Nernsfs Theorem. ... 

The theorem of Nernst applies only to chemically homogeneous condensed phases the entropy of a condensed solution phase has at absolute zero a finite value, owing to the mutual presence of the different components. 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

The conclusion that can be reached from the Nernst heat theorem is that the total entropy of the products and the reactants in a chemical reaction must be the same at 0 Kelvin. But nothing in the statement requires that the entropy of the individual substances in the chemical reaction be zero, although a value of zero for all reactants and products is an easy way to achieve the result of equation (4.17). 

Nernst heat theorem 164-5 nickel chloride, heat capacity 180 nitric acid, heat capacity 224-5 nitric oxide, entropy of 173 nitrogen... 

The end effects have been neglected here, including in the expression for change in reservoir entropy, Eq. (178). This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow. In essence this is the thermodynamic gradient version of the fluctuation theorem that was first derived by Bochkov and Kuzovlev [60] and subsequently by Evans et al. [56, 57]. It should be stressed that these versions relied on an adiabatic trajectory, macrovariables, and mechanical work. The present derivation explicitly accounts for interactions with the reservoir during the thermodynamic (here) or mechanical (later) work,... 

Closely related to the fluctuation theorem is the work theorem. Consider the average of the exponential of the negative of the entropy change,... 

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

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. 

Crooks, G. E., Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 1999,60, 2721-2726... 

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