**Information Thermodynamics and Halting Problem**

### Bohdan Hejna

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61900

#### **Abstract**

The formulations of the *undecidability* of the Halting Problem assume that the com‐ puting process, being observed, the description of which is given on the input of the 'observing' Turing Machine, is, at any given moment, the exact copy of the com‐ puting process running in the observing machine itself (the Cantor diagonal argu‐ ment). In this way an *infinite cycle* is created shielding what is to be possibly discovered - the possible infinite cycle in the observed computing process. By this type of our consideration and in the thermodynamic sense the equilibrium status of a certain thermodynamic system is described or, even created. This is a thermody‐ namic image of the Cantor diagonal method used for seeking a possible infinite cy‐ cle and which, as such, has the property of the Perpetuum Mobile - the structure of which is recognizable and therefore we can avoid it. Thus we can show that it is possible to recognize the infinite cycle as a certain *original* equilibrium, but with a *'step-aside'* or a time delay in evaluating the trace of the observed computing proc‐ ess.

The trace is a record of the se*quence of configurations* of the observed Turing ma‐ chine. These configurations can be simplified to their *common configuration types*, creating now a word of a *regular language*. Furthermore, the control unit of any Tu‐ ring Machine is a *finite automaton*. Both these facts enable the *Pumping Lemma* in the observing Turing Machine to be usable. In compliance with the Pumping Lemma, we know (the observing Turing Machine knows) that certain common configura‐ tion types must be *periodically repeated* in the case of the *infinite length* of their regu‐ lar language. This fact enables (in a finite time) us (the observing Turing Machine) to decide that the observed computing process has entered into an infinite cycle.

Considerations of the real sense of the Gibbs Paradox are used to illustrate the idea of the term 'step-aside' which is our main methodological tool for looking for the infinite cycle in a Turing computing process and which enables us to avoid the com‐ monly used attempts to solve the Halting Problem.

**Keywords:** Heat and Information Entropy, Observation, Carnot Cycle, Information Channel, Turing Machine, Infinite Cycle

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **ŗǯȱ**

ȱ ȱ ȱ ȱ *¢*ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱȱǰȱȱȱȱ ȱȱȱȱȱȱȱȱȂȂȱ ǰȱȱȱ¡ȱ¢ȱȱȱȱȱȱȱȱȱȱ ǻȱȱȱȱȱȱȂȱȱǽŗŞǾǼǯȱ¢ȱȱ ¢ȱȱ*Ȭ*ȱ ȱ*ȱ¢*ȱȱȱȱȱȱ*Ȭ*ȱȱȱȱȱȱ ȱȱ*¢ȱǻǼȱ*ȱȱ¢ȱȱȱǰȱȱ ȱ ȱȱ ȱ¢ȱȱ Ȭȱ ȱȱ¢ȱȱ ȱ ȱ ȱȱ Ȭȱ ȱ ȱ ȱǯȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱ ȱȱȱȱ ȱǰȱȱȱȱ¢ǰȱȱȱǽŗŞǾǯȱȱȱǰȱǰȱ ȱȱȱȱȱȱȱ¢ǯȱȱȱȱǰȱȱȱ¢ ȱȱ ǰȱȱȱȱ ȱȱȱȱȱȱȱȬ ¢ȱ ȱǰȱ ȱȱȱǻ ǰȱǯǯǰȱȱȱ*¡* ƽ *¡* Ƹ ŗȱ ȱȱǼȱȱ¢ȱ ¡ȱ¢ȱȱǯ

ȱȱȱ£ȱȱȱȱȱȱȱȱȱȱȱȱȱ£ ¢ȱȱȱ*ȂȬȂ*ǯȱȱǰȱ ȱȱȱ¢ȱȱǽśǰȱŜǰȱşǰȱŗŗǾȱȱ ȱ¢ȱ¢ȱȱȱ¢ȱȱȱ¢ȱȱȱȱȬ ȱȱȱ¢ȱȱȱȱȱȱ ȱȱȱȱ ȱ¢ȱ ȱȱ ȱȱȱȱȱȱȱȱǯȱȬ ȱȱ ȱ¡ȱǽŝǰȱŞǰȱŗŗǾȱȱȱȱȱȱȱȱȱȱȱȂȬȂ ȱȱȱȱȱȱȱȱȱȱȱ¢ȱȱȱȱ ȱȱ ȱȱȱȱȱȱȱȱȱȱȱ ȱǯ ȱȱȱȱȱȱȱȱȱȱȱ¢ȱȱȱȱȱ ȱȱ¢ȱ ȱȱȱȱȱȂȬȂȱȱȱ¢ǯȱȱȱȱȱ ȱȱ¢ȱȱȱȱȱǰȱȱǰȱǯȱȱ ȱȱ*ȱ¢*ȱȱȱȂȬ Ȃȱǽȱȱ¢ȱǻȱǼǾȱȱȱ ȱ ȱȂȂȱȱȱȂȱȱȱȂȱ ȱ ȱȱ¢ȱȱȱȂȂȱȱǯȱȱȱȱȱȂȬȂȱȱȱ ȱȱȱȱȱȱȱ ȱ¡ȱȱȱȱȱǽŝǰȱŞǰȱŗŗǰȱŗŘǰ ŗŚǾǰȱȱȱȱȱȱȱȱ ȱȱ*ǯ ȱȱ¢*ȱȱȱ ǯ

ǰȱ ȱ ȱȱȱȱȱȱ£ȱȱȱ¢ǰȱȱ ȱȱ*ȱ¢*ȱȱ ȱǻȱȱȱȱ¢ȱȱȱȱǰȱȱȱȱȱȱȱȱ ȱǼȱȱȱȱȱȱȱȱȱǯȱȱȱȱȱ ȱȱǰȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱǻȱǰȱȱ*Ȭ*ȱȱȱ*¢ȱ*ȱȱȱȱȱǼǯ ȱȱȱȱȱȱȱǻ ȱǼȱȱȱȱDZȱȈȱȱ ȱ¢ǵȈȱ ȱȱȱȱȱȱȱȱ ǯȱȱȱǰȱ ȱȱȱ*ȱȱȱ*ȱȱȱȱȱǯȱȱ ȱ ȱ ȱ ȱ ȱ *ȱ ȱ ¢*ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱǽŗŘǰȱŗŚǾǯȱǰȱȱȱȱȱ¢ȱȱȱȱȱ*ȱ*ǯ ȱȱȱȱȱ*ȱ*ȱȱȱȱȱȱȱȱǯȱ ȱ ȱȱȱǰȱ ȱ ȱǻȱȱȱȱ Ǽȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ *¢ȱ* ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱǯȱȱȱȱȱǻȱȱȱǼȱȱ ȱȱȱȱȱȱȱȱȱȱ¢ǯȱȱȱȱ ȱȱȱȱȱǰȱ¢ȱȱ ¢ǰȱ£ȱȱȱȱǰȱǯȱȱȱ ȱȱȱȱȱȱ*ȱȱ*ǯȱ¢ȱȱȱȂȂȱȱȱȱ ȱȱǰȱ ȱȱȱȱȱȱȱǯȱȱȱȱ ȱȱȱȱȱ ȱȱȱǯȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱȱȱ*ȱ*ȱȱǯ

ȱȱȱȱȱȱȱȱȱǯȱ¢ȱȱȱȱ¢ȱ ¢ǰȱ¢ȱ¢ȱȱ¢ȱȱȱǻǼȱ¢ȱȱȱȬ ȱȱȱ*ǯ ȱȱ¢*ȱȱȱ¢ȱȱȱȱ ȱȱȱ¢ǰȱȱ¡ȱȱȱȱȱȱ¢ȱǯ

#### **ŘǯȱȱȱȬ**

**ŗǯȱ**

128 7KHUPRG\QDPLFV

ȱȱȱȱ

ǯ

¡ȱ¢ȱȱǯ

ȱ ȱ ȱ ȱ *¢*ȱ ȱ ȱ

ǰȱȱǰȱȱȱȱ ȱȱȱȱȱȱȱȱȂȂȱ ǰȱȱȱ¡ȱ¢ȱȱȱȱȱȱȱȱȱȱ ǻȱȱȱȱȱȱȂȱȱǽŗŞǾǼǯȱ¢ȱȱ ¢ȱȱ*Ȭ*ȱ ȱ*ȱ¢*ȱȱȱȱȱȱ*Ȭ*ȱȱȱȱȱȱ ȱȱ*¢ȱǻǼȱ*ȱȱ¢ȱȱȱǰȱȱ ȱ ȱȱ ȱ¢ȱȱ Ȭȱ ȱȱ¢ȱȱ ȱ ȱ ȱȱ Ȭȱ ȱ ȱ ȱǯȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱ

ȱȱȱȱȱȱȱ¢ǯȱȱȱȱǰȱȱȱ¢ ȱȱ ǰȱȱȱȱ ȱȱȱȱȱȱȱȬ ¢ȱ ȱǰȱ ȱȱȱǻ ǰȱǯǯǰȱȱȱ*¡* ƽ *¡* Ƹ ŗȱ ȱȱǼȱȱ¢ȱ

ȱȱȱ£ȱȱȱȱȱȱȱȱȱȱȱȱȱ£ ¢ȱȱȱ*ȂȬȂ*ǯȱȱǰȱ ȱȱȱ¢ȱȱǽśǰȱŜǰȱşǰȱŗŗǾȱȱ ȱ¢ȱ¢ȱȱȱ¢ȱȱȱ¢ȱȱȱȱȬ ȱȱȱ¢ȱȱȱȱȱȱ ȱȱȱȱ ȱ¢ȱ ȱȱ ȱȱȱȱȱȱȱȱǯȱȬ ȱȱ ȱ¡ȱǽŝǰȱŞǰȱŗŗǾȱȱȱȱȱȱȱȱȱȱȱȂȬȂ ȱȱȱȱȱȱȱȱȱȱȱ¢ȱȱȱȱ

ȱȱȱȱȱȱȱȱȱȱȱ¢ȱȱȱȱȱ ȱȱ¢ȱ ȱȱȱȱȱȂȬȂȱȱȱ¢ǯȱȱȱȱȱ ȱȱ¢ȱȱȱȱȱǰȱȱǰȱǯȱȱ ȱȱ*ȱ¢*ȱȱȱȂȬ Ȃȱǽȱȱ¢ȱǻȱǼǾȱȱȱ ȱ ȱȂȂȱȱȱȂȱȱȱȂȱ ȱ ȱȱ¢ȱȱȱȂȂȱȱǯȱȱȱȱȱȂȬȂȱȱȱ ȱȱȱȱȱȱȱ ȱ¡ȱȱȱȱȱǽŝǰȱŞǰȱŗŗǰȱŗŘǰ ŗŚǾǰȱȱȱȱȱȱȱȱ ȱȱ*ǯ ȱȱ¢*ȱȱȱ

ǰȱ ȱ ȱȱȱȱȱȱ£ȱȱȱ¢ǰȱȱ ȱȱ*ȱ¢*ȱȱ ȱǻȱȱȱȱ¢ȱȱȱȱǰȱȱȱȱȱȱȱȱ ȱǼȱȱȱȱȱȱȱȱȱǯȱȱȱȱȱ ȱȱǰȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱǻȱǰȱȱ*Ȭ*ȱȱȱ*¢ȱ*ȱȱȱȱȱǼǯ ȱȱȱȱȱȱȱǻ ȱǼȱȱȱȱDZȱȈȱȱ ȱ¢ǵȈȱ ȱȱȱȱȱȱȱȱ ǯȱȱȱǰȱ ȱȱȱ*ȱȱȱ*ȱȱȱȱȱǯȱȱ ȱ ȱ ȱ ȱ ȱ *ȱ ȱ ¢*ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱǽŗŘǰȱŗŚǾǯȱǰȱȱȱȱȱ¢ȱȱȱȱȱ*ȱ*ǯ ȱȱȱȱȱ*ȱ*ȱȱȱȱȱȱȱȱǯȱ

ȱȱ ȱȱȱȱȱȱȱȱȱȱȱ

ȱǰȱȱȱȱ¢ǰȱȱȱǽŗŞǾǯȱȱȱǰȱǰȱ

ȱ ȱ ȱ ȱ ȱ

ȱǯ

*¡ȱ*ȱǻ¡ǰȱȱ¡ǰȱǰȱǼȱȱ ȱȱȬȱ ȱȱǰȱȱȱȱȱȱȱȱȱȱȱȱȱǯ

¢ȱȱȱ¢ȱȱȱȱ*ȱǻǼȱ*ȱȬȱ ȱȱȱȱȱ ȱ ȱ ȱȱ ȱ ȱ*ȱ ǻ¢Ǽȱ*ȱ Ȭȱ ȱȱ ȱ ȱȱ ȱȱȱȂȂȱǯȱȱ¢ȱȱ¢ȱȱȱȱ*ȬȱȬ £ȱȱȱȱ*ȱȬȱ¢ȱȱȱȱȱȱȱ*ȂȬȂȱȱȱȱ ȱ*ǯȱȱȱ¢ȱȱ¢ȱȱȱȱ*ȱ*ȱȱȱǯȱȱ ȱȱȱ¢ȱȱȱȱȱ ȱȱ*¢ȱ*ǯȱȱȱȱ ȱȱȱȱȱ*Ȭ*ȱȱ ȱǰȱȱ¢ȱǰȱ ¢ȱȱ ȱȱȱȱ*ǯȱȱȱ¢*ȱȱȱȱȱǽŞȬŗŚǾǯ

#### ȱȱȱȱȱ**Ȃȱ**ȱȱȱ¡ŗ DZ**ȱȱȱ ȱȱ ȱȱȱ ȱȱȱȱȱ ȱ** *ȱ***ȱȱȱȱ ¡ȱȱȱȱȱȱȱȱ¡ȱȱ***¢ȱȱ¡ȱȱ***ȱ**ǯ

ȱȱ ȱ ȱȱ ǰȱ ȱȱ ǻǼȱȱ ȱ ȱ ȱ ǰȱ ȱǻ¢Ǽȱǯȱȱ ȱȱȱ¡ȱȱȱ ȱȱ ȱȱ ȱȱȱ¡ȱȱǰȱ ȱȱȱȱȱȱ*Ȭ ȱ ȱ*ȱȱ ȱȱȱȱȱȱȱǯ

ȱȱȱ ȱȱȱȱȱȱȱȱȱȱȱȱ ȱǰȱȱȱȱ¢ȱȱ¢¢ȱȱȱ ȱȱȱ ȱ ¢ȱ¡ȱȱȱȱǯȱȱ ȱȱȱȱȱ ȱǻȱȂȬ ȂǼȱȱȱȱȱȱȱǯȱȱȱȱ*ȱȱ*ȱȱȱ ȱȱȱȱȂȂȱȱȱȱ ȱȱȱȱȱ

ŗȱǯȱǰȱǯȱǰȱ**ȱ**ǰȱŗşŗŖǰȱŗşŗŘǰȱŗşŗřȱȱŗşŘŝǯ

ǰȱȱ¢ȱ ǰȱ ȱȱ ȱ ȱȱǰȱȱ ȱ ȱ¢ȱ ǻ Ǽ ǯȱǻȱȱ¢ȱȱȱȱȱȱȱǯǼȱȱȱȱȱ ȱ ȱȱȱȱǻǼȱȱȱȱ ǯȱȱ¢ȱȱ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻ¡ǰȱ ¡ǰȱǰȱǼȱȱ¡ǯ

#### **ŘǯŗǯȱȬȱȱȱǰȱȬ**

ȱ¢ȱ*ȱȱ* #ȱȱ*ȱ*

$$H(X) - H(X \mid Y) = H(Y) - H(Y \mid X) \tag{1}$$

ȱ ȱ ȱ ǽŘřǾǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ *ȱ ǽǻǼȱȱǾ*ȱȱȱȱ#ǯ

ȱȱ ǻ Ǽǰȱ ǻ Ǽǰȱ ǻ ȩ Ǽȱȱ ǻ ȩ Ǽȱȱȱǰȱȱǰȱȱȱȱ ȱ¢ǯ

ȱȱ ǻ Ǽƺ ǻ ȩ Ǽȱȱȱȱ ǻ Ǽƺ ǻ ȩ Ǽȱȱȱ ǻ Dz Ǽȱȱȱȱ ǻ Dz Ǽȱ¢ǰ

$$H(X) - H(X \mid Y) \triangleq T(X;Y) \ = T(Y;X) \triangleq H(Y) - H(Y \mid X) \tag{2}$$

ȱȱȱȱȱȱȱǻ¢Ǽȱǻ Ǽǰȱȱ ȱȱ

ȱȱȱȱ¢ȱ ǻ ȩ Ǽǰȱ ǻ Ǽ ƽǻ ȩ Ǽǰȱǰȱ¢ǰȱȱȱ

$$T\left(X;Y\right) = 0 \quad \left[= H\left(Y\right) - H\left(Y\mid X\right)\right] \tag{3}$$


ȱȱȱ ȱȱȱȱ#ȱȱȱȱ*ȱǻ ȱȱȱǼ*ȱ ȱȱ ǻ Ǽȱȱ ȱ¢ȱȱȱȱȱ ǻ ǼȱǰȱǰȱȱȂȱȱȱ ȱȱ#ǯȱȱȱȱ¡ȱ¢ȱȱȱȱȱ¢ȱ ǻ ȩ Ǽǯȱȱǰ ȱ¢ǰȱȱǻ ȩ ǼƽŖǯ

ȱȱǻŗǼȬǻřǼȱ ȱȱȱ**ȱ**#**ȱȂ**ȱȱǻ ȱȱȱȱȱȱȱ Ǽȱȱȱȱ ȱȱȱȱȱǰȱǰȱȱȂȱ**ȱȬ** ȱǻ¢ǰȱǼȱǯȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱǰȱȱ ǯ

*¢ȱȱ*#*ȱȂȱȱȱ ȱȱȱȱȱȱȱǻ ȱȱȱ Ǽǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ Ȭȱ ȱ ȱ ȱ ȱ ¢ǯȱ ȱ ȱ ȱ ȂȬȂȱ ȱ ȱ ȱ £ȱ ȱ ǰ ǻ Ǽ ƽ ǻ Ǽ ƺ ǻ ȩ ǼǁŖǰȱȱǯ*

#### **ŘǯŘǯȱȬȱȱ¢ȱ¢**

ǰȱȱ¢ȱ ǰȱ ȱȱ ȱ ȱȱǰȱȱ ȱ ȱ¢ȱ ǻ Ǽ ǯȱǻȱȱ¢ȱȱȱȱȱȱȱǯǼȱȱȱȱȱ ȱ ȱȱȱȱǻǼȱȱȱȱ ǯȱȱ¢ȱȱ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻ¡ǰȱ

¡ǰȱǰȱǼȱȱ¡ǯ

ȱ¢ȱ*ȱȱ* #ȱȱ*ȱ*

*ǽǻǼȱȱǾ*ȱȱȱȱ#ǯ

ǻ Ǽǰȱ

ǻ Dz Ǽȱȱȱȱ ǻ Dz Ǽȱ¢ǰ

ȱȱȱȱȱȱȱǻ¢Ǽȱ

ǻ ȩ Ǽǰȱ

ǻ Ǽǰȱ

ǻ Ǽƺ

ǻ ȩ ǼƽŖǰȱ ȱȱ ǻ Dz Ǽ ƽ

ǻ ȩ ǼƽŖǯ

ǻ ȩ ǼƾŖȱ ȱȱ

ȱȱǰȱȱ ǯ

ȱȱȱȱ¢ȱ

ȱȱ

ȱȱ

ȱ¢ǯ

130 7KHUPRG\QDPLFV

**Ȋ** ȱ

**Ȋ** ȱ

ȱȱ

ȱ¢ǰȱȱ

**ŘǯŗǯȱȬȱȱȱǰȱȬ**

ǻ ȩ Ǽȱȱ

ǻ ȩ Ǽȱȱȱȱ

ǻ Ǽ ƽ

ǻ ǼƽŖǯ

ǻ ȩ ǼƾŖ

ȱȱȱ ȱȱȱȱ#ȱȱȱȱ*ȱǻ ȱȱȱǼ*ȱ

ȱȱǻŗǼȬǻřǼȱ ȱȱȱ**ȱ**#**ȱȂ**ȱȱǻ ȱȱȱȱȱȱȱ Ǽȱȱȱȱ ȱȱȱȱȱǰȱǰȱȱȂȱ**ȱȬ** ȱǻ¢ǰȱǼȱǯȱȱȱȱȱȱȱȱȱȱȱȱȱ

ǻ Ǽ ƽ

ǻ Ǽȱȱ ȱ¢ȱȱȱȱȱ

ȱȱ#ǯȱȱȱȱ¡ȱ¢ȱȱȱȱȱ¢ȱ

ǻ Ǽƺ

       ȩ Dz ȱȱȱƽȱȱȱ Dz ȩ ǻŘǼ

ȱ ȱ ȱ ǽŘřǾǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ *ȱ* 

     ȩƽ ȩ ǻŗǼ

ǻ ȩ Ǽǰȱǰȱ¢ǰȱȱȱ

    Dz ƽ Ŗȱȱȱ ƽª º <sup>ȩ</sup> ¬ ¼ ǻřǼ

ǻ ȩ Ǽȱȱȱǰȱȱǰȱȱȱȱ

ǻ Ǽǰȱȱ ȱȱ

ǻ ȩ Ǽȱȱȱ

ǻ ǼȱǰȱǰȱȱȂȱȱȱ

ǻ ȩ Ǽǯȱȱǰ

ȱȱȱȱȱȱȱȱȱ#ȱȱȱȱȱȬ ȱǻȱǰȱǰȱǼȱȱȱȱȱ¢ȱ'ȱǽŜǰȱŞǾǯȱȱȱ'ǔ # ȱ ǯȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȂȂ 'Ȟ ǔ #<sup>Ȟ</sup> ǯȱ ȱ '<sup>Ȟ</sup> ȱ ǰȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¢ȱ 'ȱ ȱ ȱ ȱ ǰȱ ȱȱȱ*ǰȱ*ȱȱǽŜǰȱŞǾǯ

ȱȱȱȱȱȱȱ*ȱȱ¢* ''<sup>Ȟ</sup> *ȱȱȱ ¢ȱȱȱȱǻȱ ¢* '<sup>Ȟ</sup> Ǽȱȱ ȱ ȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ'ǯȱ ȱȱ ǻ¢Ǽȱȱȱ ǻ ȩ ǼȱȱȱȱȱȱȱǻǼȱ ǻ'ǔ #ǼȱȱȂȂǯȱ*ȱ* '<sup>Ȟ</sup> ǔ #<sup>Ȟ</sup> *ȱȱȱ*  ǻ ȩ Ǽ *ȱȱȱȂȬ Ȃ*  ǻ ȝȝ ȩ ȝȝ Ǽ *ǽ*  ǻ ȝȝ Ǽ ƽǻ Ǽ *ȱȱȱ¢Ǿǯ*

ȱȱ̇ ǰȱ̇ȱȱ̇Ŗȱȱȱȱ̇ ȝȝ ǰȱ̇ȝȝ ȱȱ̇ ȝȝ Ŗȱ¢ǰȱȱ ȱȱȱȱȱȱ£ȱǻ¢¢Ǽȱ¢ȱȱ ȱ¢ȱ'ȱȱ¢ȱȱȱȱ¢ȱ'<sup>Ȟ</sup> ȱǻȱǼȱ¢ȱǻȱȱ¢ ''<sup>Ȟ</sup> ȱȱǼǰ

$$\begin{aligned} H(\mathbf{X}) &= \frac{\Delta Q\_W}{\mathbf{k} T\_W}, \text{ resp. } H\left(Y''\right) = \frac{\Delta Q\_W''}{\mathbf{k} T\_W''}\\ H(Y) &= \frac{\Delta A}{\mathbf{k} T\_W}, \text{ resp. } H\left(X''\right) = \frac{\Delta A''}{\mathbf{k} T\_W''}\\ H\left(X \mid Y\right) &= \frac{\Delta Q\_0}{\mathbf{k} T\_W}, \text{ resp. } H\left(Y'' \mid X''\right) = \frac{\Delta Q\_0''}{\mathbf{k} T\_W''} \end{aligned} \tag{4}$$

ȱȱȱȱȱȱ*£*ȱȱȱ ȱ̇<sup>Ș</sup> ȱȱȱȱȱ¢ȱ''<sup>Ȟ</sup> ǰ ̇<sup>Ș</sup> ǁ Ŗǯȱȱ ȱȱȱ£ȱȱ Ș ǻ <sup>Ș</sup> Ǽƽ̇<sup>Ș</sup> Ȧ ǁ Ŗǯ

ȱȱȱǰȱȱȱȱ*¡*ȱȱ <sup>Ȟ</sup> *¡*ȱȱȱȱȱ¢ȱ'ȱȱ'<sup>Ȟ</sup> ǻ ȱ ȱ ȱ ȱ <sup>ƽ</sup> ȝȝ ȱ ȱ <sup>Ŗ</sup> <sup>ƽ</sup> ȝȝ Ŗǰȱ ǃ<sup>Ŗ</sup> ǁ ŖǼǰȱ ȱ ȱ ȱ ȱ *¡* <sup>ǁ</sup> <sup>Ȟ</sup> *¡*ȱDzȱ ȱ ȱȱ¢ȱȱȱŘ

$$
\Delta^\* A = \Delta A - \Delta A'' \ge 0\\\left\lceil \Delta A'' = \Delta Q\_W'' - \Delta Q\_0'' \right\rceil \tag{5}
$$

Řȱȱ ȱȱȱȱ*¢ȱȱȱȱ¢*ȱȱȱȱȱ¢ȱȱȱȱ ȱȱ*ǯ*ȱǰȱ¢ȱǽŞǾǰȱȱȱ*ǯ*ȱ¢ǯ

ȱ̇<sup>Ŗ</sup> ƽ̇ ȝȝ Ŗȱ ȱ ȱ ǰȱ ȱȱ ȱ ȱ̇ ȝȝ ǀ̇ <sup>ҥ</sup>ǻ*¡* <sup>ǁ</sup> <sup>Ȟ</sup> *¡*Ǽ ȱ ȱ ȱ ȱȱȱ

$$\begin{aligned} \Delta A^\* &= \Delta Q\_W \cdot \eta\_{\text{max}} - \Delta Q\_W'' \cdot \eta\_{\text{max}}'' > 0 \text{ but} \\ \Delta Q\_W \cdot \eta\_{\text{max}} - \Delta Q\_W'' \cdot \eta\_{\text{max}}'' &= \Delta Q\_0 - \Delta Q\_0'' = 0 \end{aligned} \tag{6}$$

ȱ ȱ ȱ ȱ ̇<sup>Ș</sup> ǁ Ŗ *ȱ ȱ* ȱ ȱ ¢ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱȱȱ ȱȱ̇ ƺ̇ <sup>Ȟ</sup> ȱȱ ȱȱ¢ȱ''<sup>Ȟ</sup> ǯȱȱ*¡* ǀ*¡*ȱȱȱ ̇ ƺ̇ ȝȝ ȱȱȱȱȱȱȱ ȱȱȱŖȱȱȱȱ ȱ ȱ *¢*ǰȱ ȱ¢ȱȱ¢ȱȱȱ ǯȱȱȱȱ̇<sup>Ș</sup> ƽ Ŗ ȱȱ¢ǯ

ȱ ȱ ȱ ''<sup>Ȟ</sup> ȱ ȱ ȱ ȱ ǯȱ *ȱ* ȱ ȱ ȱ ȱ ǯȱ *¡* <sup>ƽ</sup> <sup>Ȟ</sup> *¡*ȱȱȱȱǻȱȱ ȱȱȱ ȱȱȱȱ ȱȱȬ ȱŖȱȱǼȱ

$$
\Delta \eta\_{\text{max}} = \eta\_{\text{max}}'' < 1 \quad \text{and} \text{ then } \Delta Q\_W = \Delta Q\_W'' \tag{7}
$$

ȱȱȱ ȱȱȱ*¡* <sup>ƽ</sup> <sup>Ȟ</sup> *¡* ǀ ŗȱȱ ȱȱ¢ȱȱȱȱ ȱȱǻǼȱȱ¢ȱ''<sup>Ȟ</sup> ȱȱȱȱȱȱȱ*¢ȱȱ*

$$\ln H(X) \cdot \eta\_{\text{max}} = \frac{\Lambda Q\_W}{\text{k}T\_W} \cdot \left(1 - \beta\right) > 0, \ \beta = 1 - \eta\_{\text{max}} = \frac{T\_0}{T\_W} \tag{8}$$

ȱȱȱȱȱȱȱȱȱ*¢ȱȱ* ƺ ǻ Ǽ ջ*¡* ƽ ƺ ̇ ջ ǻŗƺ*Ά*Ǽ ȱȱȱȱȱ¢ȱȱ*£*ȱ

$$H^\*(Y^\*) = \frac{\Delta A^\*}{\mathbf{k}T\_W} = H(\mathbf{X}) \cdot \eta\_{\text{max}} - H(\mathbf{Y}'') \cdot \eta\_{\text{max}}'' = H(\mathbf{X}) \cdot \left(\eta\_{\text{max}} - \eta\_{\text{max}}\right) = 0 \tag{9}$$

ȱ ȱȱǰȱȱȱ¢ȱ¢ȱ ȱȱ¢ȱ''<sup>Ȟ</sup> ȱǰȱ ȱ ¢ȱ''<sup>Ȟ</sup> ȱȱ ǰȱ ȱ ȱ ǰȱȱ ȱ*¢ȱ* ǯȱ ǻȱ ȱ ȱ ȱ ȱ ȱ ǰ ȱȱȱȱȱȱǯǼ

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**ǰȱȱȱȱȱȬȱȱȱǰȱȱǰȱȱȬ £ǰȱȱȱȱȱȱ***ȱǯ*ȱȱǰȱ¢ǰȱȱȱ ȱȱ*ǯ*ȱȱȱ¢ȱ ȱȱȱǻśǼȱȱǻŜǼȱȱǯ

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#### **Řǯřǯȱ ȱ¡ȱȱȬȱȱȱȱȱ**

¢ȱȱ¢ȱȱǻǼȱȂȂȱȱȱȱ¢ȱȱ¢ȱȱǽŘŖǾǰȱ ȱ¢ ȱȱȱ¢ȱǻǼȱǰȱȱȬ£ȱȱȱȱ¢ǰ ȱȱȱȱȂȂǰȱȱǯ

ȱȱȱȱ¢ȱ¢ȱȱȱȱ ȱȱ ȱȱȱȱȱȱ ȱ ȱ ¢ȱ ǯȱ ȱ*ȱ* ȱ ȱȱȱ ƽ̋ǯȱ ȱ ȱ ¢ ȱȱȱȱ¢ȱ ȱȱȱ ȱȱ ƽ̋ǯ

ȱȱȱȱȱǰȱȱȱȱȱ *ȱȱ¢ȱ*ȱǽȱȱǻǼ ȱ¡ȱȱȱ*Έ*ȱ ȱȱ¢ȱȱȱǾȱ ȱȱȱ*ǯ ȱȱ¢*ǰȱ*Έ* ƽ Ƹ

ȱȱǰȱȱȱ*ȱ* ̇ ƽ ̇ ̋ ǰ̇ <sup>ƽ</sup>̇̋ Ƹ ̋̇ ǰ̋ ǁ Ŗǰȱ ȱ

$$S = n \left\| \left( c\_v \frac{\mathbf{d}\Theta}{\Theta} + R \frac{\mathbf{d}V}{V} \right) \right\| = n \left( c\_v \ln \Theta + R \ln V \right) + S\_0 \langle n \rangle = \sigma \left( \Theta, V \right) + S\_0 \langle n \rangle \tag{10}$$

ȱȱȂȂȱȱȱ¢ȱȱȱȱȱ ȱȱȱȱȱ̋ǰȱǰȱȱǰ ȱ ȱȱ ȱǻǰȱȱ ȱȱǼȱǰȱȱȱȱ ȱ¢ȱȱ ȱǻǰȱȱǼǰȱ ȱȱǻȱ¢ǰȱǰȱȂȂȱ¢Ǽǰȱ ȱ ¢ȱ ȱ ȱ ȱ ¢ǰȱ ȱ ȱ ȱ ǰȱ ӇŗǰȀǰȱ ǃŗȱ ȱ ȱ ȱȱȱ ǯȱ¢ȱ ƽnjƽŗ ȱȱ ƽnjƽŗ ǯ

ȱ ȱŖǻǼƽŖȱȱŖ ǻ ǼƽŖȱȱȱǯȱȱȱȱ ȱȱȱȱ¢ǰȱ ȱȱȱ̇ǰȱ ȱȱ ǰ ȱȱ¡ȱȱȱȱǰȱȱȱ ƽ ǰ ̋ƽ̋ ȱȱ ȱȱȱȱ*Η* <sup>ƽ</sup> ǯȱǰȱȱ <sup>ƽ</sup>*Η* <sup>ƽ</sup> ǻ̋ Ƹ Ǽȱȱǰȱ ȱ 

$$\begin{aligned} \sum\_{i=1}^{m} \mathbf{S}\_{i} &= \sum\_{i=1}^{m} \sigma\_{i} = n c\_{v} \ln \Theta + R \ln \left( \prod\_{i=1}^{m} V\_{i}^{n\_{i}} \right) \\ \Delta \mathbf{S} &= \mathbf{S} - \sum\_{i=1}^{m} \mathbf{S}\_{i} = \sigma - \sum\_{i=1}^{m} \sigma\_{i} = \Delta \sigma = R \ln \frac{V^{n}}{n} = -n R \sum\_{i=1}^{m} \frac{\eta\_{i}}{n} \ln \frac{\eta\_{i}}{n} > 0 \end{aligned} \tag{11}$$

ȱȱȱȱȱȱȱȱȱǰȱ ǀ Ŗǯȱȱ¢ȱƺ ȱ¡ȱȱǻŗŗǼȱȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ŗǰŘǰ ȱ ȱ ¢ȱ Ȧ ƽŗ ǯȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱȱ ȱ ȱ ȱȱ ȱ ȱ ȱ¢ȱ ȱȱ¡ȱ ǯ

ȱȱǻŗŗǼǰȱ̇ ƽ ƺǰȱȱȱ*¡*ǰȱȱȱ ȱȱȱȱȱ ȱ¢ȱȱȱȱ¢ȱǰȱǰȱȱȱȱȱȱȱȱ¢ȱ ǻȱȱ¢ȱȱǼȱ*ȱ*ȱȱ¡ȱ¢ǯȱǰȱ¢ȱȱȱȱȱ ǰȱȱ*ȱ*ȱǯ

ȱȱȱȱ ȱȱȱȱȬ£ȱȱȱŖǻǼǰȱŖ ǻ Ǽǰȱ ȱȱ ¢ǰȱȱȱȱ̇ ƽ ǻ*<sup>Η</sup>* <sup>Ƹ</sup> ŖǼƺnjƽŗ <sup>ǻ</sup>*Η* <sup>Ƹ</sup> Ŗ ǼƽŖȱȱȱȱȱ¢ȱȱȱ ȱȱȱ¢ȱȱ <sup>Ŗ</sup> <sup>ǻ</sup> Ǽ ƽ <sup>ƺ</sup> ǻ <sup>Ȧ</sup> *·* Ǽǯ

ȱ Ԕ ǰȱȱ ȱ ȱȱȱ

$$\mathbf{S}^{\text{Class}} = \sum\_{i=1}^{m} \mathbf{S}\_i^{\text{Class}} = \sum\_{i=1}^{m} \eta\_i \mathbf{R} \ln \boldsymbol{\gamma}\_i = n \mathbf{R} \ln \boldsymbol{\gamma} \implies \boldsymbol{\gamma} = \boldsymbol{\gamma}\_i; \ \Delta \mathbf{S} = 0. \tag{12}$$

ȱȱȱȱȱǰȱ <sup>Ӓ</sup> ƽ ƽ *£* ƽ ƺ <sup>Ӓ</sup> ƽ ƺ ǯ

ǰȱȱȱ ȱȱȱȱ ȱ¡ǰȱȱȱȱŖȱȱȱǻ Ǽȱ¢ȱ̇ȱ ȱȱȱȱȱ¢ȱ*Η ȱȱ*ȱȱȱ¢ȱ ȱȱ ȱȱȱȱ¢ȱȱǻȱȱȱȱ ¢Ȭȱ¡Ǽȱȱȱ ̋ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ¢ȱ *Η*ȱ ȱ ǰ *<sup>Η</sup>* <sup>ƽ</sup> ƺ̇ǰ̇ <sup>ƽ</sup>Ŗǯ

ȱȱȱȱȱȱ ȱ

$$\begin{split} \Delta S &= \frac{\Delta Q\_0}{\Theta} = -nR \ln \frac{n}{\mathcal{Y}},\\ \ln \gamma &= \frac{\Delta S}{k n N\_A} + \ln n = \frac{\Delta S}{k N} + \ln N - \ln N\_{A'} \ \mathcal{Y} = N \Rightarrow \frac{\Delta S}{k N} = \ln N\_A. \end{split} \tag{13}$$

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ǻ̋ Ƹ

ǰȱȱȱȱȱȱȱȱȱȱ,ȱȱȱȱ #ȱ ȱȱ ǻ Ǽǰȱ ǻ Ǽǰȱ ǻ ȩ Ǽȱȱ ǻ ȩ ǼȱȱǻŚǼȱȱ ȱȱ¢¢Dzȱ ȱȱȱȱȱȱȱȱȱȱ¢ȱȱDZ

$$\begin{aligned} input \quad & H(\mathbf{X}) = \frac{\mathbf{S} \cdot \mathbf{s}}{kN} = \ln \gamma = -B \ast = \ln N = -rB(r) \\ output \quad & H(\mathbf{Y}) = \frac{\sigma}{kN} \triangleq -B^{\text{Gibbs}} = -B^{\text{Boltz}} = -B(r), \\ \text{loss} \quad & H(\mathbf{X} \mid \mathbf{Y}) = \frac{\mathbf{S}\_0}{kN}, \\ \text{noise } & H(\mathbf{Y} \mid \mathbf{X}) = 0 \quad \text{for the simplicity;} \\ & H(\mathbf{X} \mid \mathbf{Y}) = -rB(r) - \left[-B(r)\right] = -B(r) \cdot (r - 1) = \left(-B \ast \right) \cdot \frac{r - 1}{r}, r \ge 1; \frac{1}{r} = \eta\_{\max}. \end{aligned} \tag{14}$$

ȱȱȱȱȱȱȱȱȱȱȱȱ ȱ ȱǰȱǂ ȱȱȱȱ¢ȱȱ ȱȱȱ ȱȂȱ Ȃȱȱȱ ǻȱ ȱ ȱȱ ȱ ȱǼȱ ȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ¢ȱ Ӈŗǰƺŗ ǁǰȱ ȱȱȱ ƽ ǻ ƺŗǼ Ȧ ǯ

ȱȱȱȱȱ¢ȱǰȱȱȱ*ȱ*ȱȱ ¡ǰȱȱȱȱ¢ȱȱȱȱȱ ǰ#ǰ ȱ

$$H(\mathbf{X}) = \frac{S^{\text{Class}}}{kN},\ H(\mathbf{X} \mid \mathbf{Y}) = \frac{S\_0}{kN},\ H(\mathbf{Y}) = \frac{S^{\text{Class}}}{kN},\ H(\mathbf{Y} \mid \mathbf{X}) = \frac{\Delta S}{kN}.\tag{15}$$

ǻ ǰȱ ȱ ȱȱȱȱȱ ǻŗśǼǰȱȱ ȱ ȱȱ ȱ ƽ ŗǰȱ ȱǽŜǾǯǼ

ȱȱ¢ȱǰȱȱȱȱȱȱ¢ȱȱȱ̇ȱȱ Ӓ ǰȱȱ¢ȱ **ȱ¡ǰȱ***ȱȱ ȱ ¢ȱ*ǯȱȬ ȱȱ ¢ǰȱ**ȱȱȱȱȱȱ***£* **ȱȱȱȱȱ ȱȱ¢ȱ**ȱǽŜǾǯȱǰȱȱȱȱ ǯ

**ȱȱ¢**ȱȱȱȱȱȱ ȱ¢ȱȱȱȱȱ ƽ*Lj*ǯȱȱ ȱ ȱȱȱ ƽ Ŗȱǰȱȱȱȱǰȱƽ ŗǰȱ ƽ ŖǯȱȂȱȱȂȱ ȱȂȱȂȱȱȱ¢ȱǰȱȱȱȱȱ ȱȱȱȱȱȱǻȱȱȂȂ ȱȬȱȱȱȂȂȱ¢ȱŖȱȱȂȱ ȂȱȱȱȱǼǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¢¢ǰȱ ȱ ȱ ȱ ǻǼȱȱ¢ȱȱ ǻ Ǽ ƽ ƺ  ƽ ŖǯȱǰȱȱȱȱȱȂȂȱȱŖǰ ȱȱȱȱ¢ȱ

$$H(X \mid Y) = \frac{S \cdot \ast}{kN} = \ln N = H(X).$$

ȱ ȱȱǁ ŗȱȱȱȱȱȂ Ȃȱȱȱ ȱȱȱȱǻǼ ¢ȱ *ȱȱ*ǰȱȱ ȱƽ ŗȱȱȱ ƽ Ŗǰȱ ȱȱȱ¢ȱȱ¢ȱ ȱ¢ȱȱ¢ȱȱȱȱȱ¡ǯȱȱ ȱ¢ȱȱȱ ȱȂ Ȃȱȱȱ ȱȂ Ȃȱȱ ȱȱ¢ȱȱȱ ǰȱȱ¢ȱȱǰȱǯȱȱ¢ȱ ȱȂ ȂȱȱȂȂȱȱȱ¢ȱȱ¢ȱȱȱǰȱ¢ȱȱ ¢ǰȱȂ ȱ ȱȂȱ ȱǰȱȂ ȱȱȂȱȱ¢ȱǯȱȱȂȱȂȱ ȱȱȱ ǰ ȱȂȂȱȱǰȱȱȱȱȱȱȱȂ£Ȃȱ ȱ¢ȱȱ ȱȱ¢ȱȱ ǰȱȱȱȱ ȱȱȱȱǻ¢ȱǼǯȱȱȱȱȱ ¢ȱǻȱȱȱ Ǽȱȱȱȱȱ£ȱ ȱȱȱ ƽ*Lj*ǯȱȱ¢ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ  *ȱ ȱ ȱ ȱ ȂȬȂȱ ȱ*  ǻ Ǽ ƽ ǻ Ǽƺ ǻ ȩ ǼƽŖ ǯȱȱȱȱȱ ¢ȱ ȱȱȱȱȱǻȱȱǼȱǰ ȱ ȱȱȱȱȱȱǯȱȱȱȱȱȱ ȱȂȱȂ¢ȱ Ȃȱȱ ȱȱ¢ȱǻ ƾŖȱȱǼȱȱȱȱȱȱȂ Ȃȱǻǁ ŗǼǯȱȱ ȱȱȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȂȂȱ ȱ ¢ȱ ǰȱ ȱ ȱ ȱ ȱ ǰ ȱ ǻ ȱ ƽ ŖǼȱ ǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ǀ*Lj*ǯȱȂȱȱȂȱȱ¢ȱȱȱȱȱ ȱȂǰȱȱȱǰȱ ǰȱ ȱȱ ȱȂȱȱȱ¢ǯȱǻȱ ȱȱȱȱȱȱȱ¡ȱ ȱ ǯǼȱȱȬȱȱȱDzȱȱȱȱȱ¢ȱȱǻȱ ȱ**Ƿ**Ǽȱȱȱȱȱ£ȱȱȱȱǻ**Ƿ**Ǽȱȱȱ¢ȱȱǯř

ȱȱȱȱȱȱȱȱȱȱ ȱȱ ¢ȱȱ Ȧ ƽŗ <sup>ŗ</sup> ǯȱȱȱȱǻ Ǽȱȱ ǻ Ǽȱȱȱ ȱȱȱŖȱǰȱȱȱǰȱȱȱ£ȱ¢ȱȱȱȱ ȱ ȱ ƽ ŗȱ ȱ ƽ*Lj*ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ  ǻ ǼƽŖ  ǻ Ǽ ƽ*¡* ջ ǻǻ Ǽ ƽ Ŗǰ*¡* ƽŗȦ *Lj* ǯ

ȱȱȱ ȱȱȱȱȂȬȂȱȱȱ¢ȱȱ¡ȱ¢ȱȱȱ ǀ*Lj*ǰȱǰ ȱ ȱȱȱȱȱȂȬȂǰȱ ȱȱȱȱȱ ȱ¡ ǽȱ ȱȱȱȱȱȱȱȱȱ Ӈǻŗǰ*Lj* ȱȱȱ Ǿǯ

ȱȱȂȬȂȱȱ¡ȱ¢ȱȱȱ ƽ ŗǯ

**ȱ***¡***ȱ¢**ȱȱȱȂȂǰȱȱ¢ȱȱȱ ȱȱȱ¢ȱǰȱ ȱȱ ƽ ŗǯȱȱǻǼ ƽ Șȱȱȱȱǰȱȱȱȱȱȱȱ ȱȱȱ

$$H(Y) = H(X) = -B\*,\ H(X \mid Y) = 0$$

ȱ ȱȱȱȱȂȬȂȱȱȱ¢ȱǯȱȂȱȱȱȱȱȂȱ ȱȱ ȱȱ¢ȱȱȱȱȱ ȱȱȱȱȱ¢ȱȱǻȱ

řȱȱ¢ȱȱȱȱ¢ȱȱ¢ȱǰȱȱǰȱȱȱȱȱȱ¢ȱȱ¢ǯ

Ǽȱȱǻ¢Ǽȱȱȱȱȱȱ£ȱȱȱȱǽȱȱȱȱ £ȱȱȱȱȱȱǻ ȩ Ǽȱ ȱ Ӈǻŗǰ*Lj* ǯǾ

<sup>Ș</sup> ǻ ȩ Ǽ ƽ ƽ ƽ ǻ Ǽǯ 

ȱ ȱȱǁ ŗȱȱȱȱȱȂ Ȃȱȱȱ ȱȱȱȱǻǼ ¢ȱ *ȱȱ*ǰȱȱ ȱƽ ŗȱȱȱ ƽ Ŗǰȱ ȱȱȱ¢ȱȱ¢ȱ ȱ¢ȱȱ¢ȱȱȱȱȱ¡ǯȱȱ ȱ¢ȱȱȱ ȱȂ Ȃȱȱȱ ȱȂ Ȃȱȱ ȱȱ¢ȱȱȱ ǰȱȱ¢ȱȱǰȱǯȱȱ¢ȱ ȱȂ ȂȱȱȂȂȱȱȱ¢ȱȱ¢ȱȱȱǰȱ¢ȱȱ ¢ǰȱȂ ȱ ȱȂȱ ȱǰȱȂ ȱȱȂȱȱ¢ȱǯȱȱȂȱȂȱ ȱȱȱ ǰ ȱȂȂȱȱǰȱȱȱȱȱȱȱȂ£Ȃȱ ȱ¢ȱȱ ȱȱ¢ȱȱ ǰȱȱȱȱ ȱȱȱȱǻ¢ȱǼǯȱȱȱȱȱ ¢ȱǻȱȱȱ Ǽȱȱȱȱȱ£ȱ ȱȱȱ ƽ*Lj*ǯȱȱ¢ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ  *ȱ ȱ ȱ ȱ ȂȬȂȱ ȱ* 

 ȱ ȱȱȱȱȱȱǯȱȱȱȱȱȱ ȱȂȱȂ¢ȱ Ȃȱȱ ȱȱ¢ȱǻ ƾŖȱȱǼȱȱȱȱȱȱȂ Ȃȱǻǁ ŗǼǯȱȱ ȱȱȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȂȂȱ ȱ ¢ȱ ǰȱ ȱ ȱ ȱ ȱ ǰ ȱ ǻ ȱ ƽ ŖǼȱ ǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ǀ*Lj*ǯȱȂȱȱȂȱȱ¢ȱȱȱȱȱ ȱȂǰȱȱȱǰȱ ǰȱ ȱȱ ȱȂȱȱȱ¢ǯȱǻȱ ȱȱȱȱȱȱȱ¡ȱ ȱ ǯǼȱȱȬȱȱȱDzȱȱȱȱȱ¢ȱȱǻȱ

ȱȱȱȱȱȱȱȱȱȱ ȱȱ

ȱȱȱŖȱǰȱȱȱǰȱȱȱ£ȱ¢ȱȱȱȱ ȱ ȱ ƽ ŗȱ ȱ ƽ*Lj*ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ 

ȱȱȱ ȱȱȱȱȂȬȂȱȱȱ¢ȱȱ¡ȱ¢ȱȱȱ ǀ*Lj*ǰȱǰ ȱ ȱȱȱȱȱȂȬȂǰȱ ȱȱȱȱȱ ȱ¡

**ȱ***¡***ȱ¢**ȱȱȱȂȂǰȱȱ¢ȱȱȱ ȱȱȱ¢ȱǰȱ ȱȱ ƽ ŗǯȱȱǻǼ ƽ Șȱȱȱȱǰȱȱȱȱȱȱȱ

> ǻ Ǽ ƽ ǻ Ǽ ƽ ǰȱȱ ǻ ȩ Ǽ ƽ Ŗ

ȱ ȱȱȱȱȂȬȂȱȱȱ¢ȱǯȱȂȱȱȱȱȱȂȱ ȱȱ ȱȱ¢ȱȱȱȱȱ ȱȱȱȱȱ¢ȱȱǻȱ

<sup>ŗ</sup> ǯȱȱȱȱǻ Ǽȱȱ

ǻ Ǽȱȱȱ

ǻ ǼƽŖ

ȱ**Ƿ**Ǽȱȱȱȱȱ£ȱȱȱȱǻ**Ƿ**Ǽȱȱȱ¢ȱȱǯř

ǽȱ ȱȱȱȱȱȱȱȱȱ Ӈǻŗǰ*Lj* ȱȱȱ Ǿǯ

ǻ Ǽ ƽ

136 7KHUPRG\QDPLFV

ǻ Ǽ ƽ*¡* ջ

ȱȱȱ

ǻ Ǽƺ

¢ȱȱ Ȧ ƽŗ

ǻ

ǻ Ǽ ƽ Ŗǰ*¡* ƽŗȦ *Lj* ǯ

ȱȱȂȬȂȱȱ¡ȱ¢ȱȱȱ ƽ ŗǯ

řȱȱ¢ȱȱȱȱ¢ȱȱ¢ȱǰȱȱǰȱȱȱȱȱȱ¢ȱȱ¢ǯ

ǻ ȩ ǼƽŖ ǯȱȱȱȱȱ ¢ȱ ȱȱȱȱȱǻȱȱǼȱǰ

ȱȂȬȂȱȱȱ ȱȱȱȱ ȱ ȱȱ ƽ ƺŗǰ ƽ ŗǯȱ ȱ ȱ ¡¢ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ¢ȱȱǰȱ¢ȱȱȂȬȂȱȱǰȱ ȱȱȱȱȱȱ ǻȱ£Ǽȱȱȱȱǯȱȱȱȱȱȱ£ȱȱȱ ȱȱȱȱ¢ȱȱ ȱ£ȱȱȱ ǻ ȩ Ǽȱȱȱȱȱ ȱ¢ȱȱȱȱǻȱ ƾŗǼǰȱȱȱȱȱ£ȱǯ

ǽȱȱȱȱȱȱȱȱ£ȱȱȱȱȱ ȱ ȱȱȱȱ¢ȱȱȱȱȱȱ¢ȱȱȱǯȱȱȱȬ ¢ȱǻȱȱȱǽŞǾǼǯǾ

#### **řǯȱȱ¢ȱȱȱȬ**

ȱȱȬȱǰȱ*ȱ ȱȱȱ*''*<sup>Ȟ</sup> ȱȱȱ¢ȱȱȱȱǯ ȱȱȱ ȱȱȱȱȱǻǼ¢ȱȱȱ ȱȱǻǼȱ ȱȱȱǯȱ¢ȱȱȱȱȱȱȱȱȱȱ*̇*ȱDzȱǰȱ*̇*ĺ Ljǰ ȱȱȱǰȱǰȱǰȱ*̇*Ljǰȱ ȱȱȱȱȱDzȱȱȱȱȱȱȱȱ ǯȱȱȱ¡ȱȱȱǯȱȱȱ¢ȱȱȱȱǰ ȱ ȱ ȱ ǰȱ ȱ ǻśǼȱ ȱ ǻŜǼǰȱ ¢ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ¡ȱ ¢ǰȱ ȱȂȬȂȱ ȱȱ¢ȱ*'*ȱȱ*'*<sup>Ȟ</sup> ǰȱȱȱ ¢ȱǻȱǼȱȱȱǯ*

ǰȱ ȱ ȱȱ ȱȱȱ ȱȱ ǻǼȱ ǻ ȩ Ǽȱ ȱ ȱ ȱ ȱ ȱ'ȱ ǻȱ ȱ ȱ#Ǽǰȱ ȱ ȱ ȱ £ȱ ȱ̇<sup>Ș</sup> ǰȱ £ȱȱ Ș ǻ <sup>Ș</sup> Ǽƽ̇<sup>Ș</sup> Ȧ ǁ Ŗǯ

ǰȱ¢ǰȱ ȱǰȱ ȱȱȱ¢ȱ'<sup>Ȟ</sup> ȱ ǻȱ ȱȱ#<sup>Ȟ</sup> Ǽȱȱ ȱ ȱ ȱȱȂȬȂȱǻ¡ȱȱ¢Ǽȱȱȱȱ'ǔ#ǯȱ ȱȱȱȱȂȬȂȱȱȱȱȱ'ȱȱȱȱȱ'<sup>Ȟ</sup> ȱȱ£ ¢ȱȱȱ <sup>ƺ</sup> ȝȝ ǁ Ŗǯȱ ǰȱ ȱȱ¢ȱȱȱ ǰȱȱȱȱ ̇<sup>Ȟ</sup> ǀ̇<sup>Ȟ</sup> ȱȱ<sup>Ŗ</sup> <sup>ƽ</sup> ȝȝ Ŗǰȱ <sup>Ȟ</sup> Ԕ <sup>Ș</sup> Ŗ

$$
\Delta A'' = \Delta Q\_W'' \cdot \left(1 - \frac{T\_0}{T\_W''}\right) = \Delta Q\_W \cdot \frac{T\_W''}{T\_W} \cdot \left(1 - \frac{T\_0}{T\_W''}\right) \tag{16}
$$

ǰȱȱȱ ȱȱȱ̇<sup>Ș</sup> Ȧ ȱȱȱȱ¢ȱȱȱȱ

$$\frac{\Delta A^{\*}}{\Delta T\_{W}} = H(Y^{\eta}) - H(Y^{\eta}) \cdot \boldsymbol{\beta}^{\*} = H(Y^{\eta}) \cdot \left(1 - \frac{T\_{0}^{\*}}{T\_{W}}\right) = H(Y^{\eta}) \cdot \left[1 - \frac{H(X \mid Y)}{H(X^{\eta} \mid Y^{\eta})}\right] \tag{17}$$

ȱȱ ǻ ȩ ǼȱȱȱȱȱǻǰȱǼȱ'ǔ#ȱȱȱ ȱȂȬȂȱ¢ǰȱȱ ȱ¢ȱȱȱǻ <sup>ǁ</sup> <sup>Ȟ</sup> Ǽǯȱȱȱ

$$\frac{\Delta A^{\circ}}{\mathbf{k}T\_{W}} > 0, \left[\frac{\Delta A^{\circ}}{\mathbf{k}T\_{W}} \equiv f\left[H(\mathbf{X} \mid \mathbf{Y})\right] > 0\right] \tag{18}$$

ȱǻśǼǰȱǻŜǼȱȱǻşǼȱȱȬȱȱȱ

$$T\_W = T\_W'' \left[ \Rightarrow H(\mathcal{Y}) = 0 \right] \tag{19}$$

 ȱ ȱ ȱǰȱȱȱ*ȱ¢ȱ ¢*ǰȱȱȱȱȱȬ ȱȱȬȱȱȱ ȱ ȱȱȱȱȱǯ

ȱ ȱ ȱȱȱȱȱ ǻ ȩ Ǽȱȱȱȱȱȱȱ ǔ ȱ ȱ ȱȱ ȱȱȱȱ£ȱȱǻǼȱ̇<sup>Ș</sup> ȱǰȱ¢ǰȱ £ȱȱ Ș ǻ <sup>Ș</sup> Ǽǰ

$$\mathbb{E}\left(H^\*(Y^\*)\right) = \frac{\Delta A^\*}{\mathbb{k}T\_W} \ge 0\tag{20}$$

ȱ ȱȱȱȂȂǰȱ ȱȱȱ¢ȱ'<sup>Ȟ</sup> ǔ <sup>Ȟ</sup> ȱ ǻȱ ȱȱ ȱ #Ȟ Ǽȱ ȱȱȱȱȱȱ ¢ȱ ȱ ȱȱȂȬȂȱȱ ȱ ȱȱ#ȱ ȱǯȱǽȱȱȱȂȬȂȱȱȱȱȱǻ'<sup>Ȟ</sup> ǰȱ,<sup>Ȟ</sup> Ǽȱ ȱȱȱǻ'ǰȱ,ǼDzȱȱ ȱȱȱȱȱȱ*Ύ* <sup>ĺ</sup> ȱȱȱȱȬ ȱȱ <sup>ĺ</sup> ǰȱȱȱǰȱȱǾǯ

 ȱȱȱȂȬȂȱȱ£ȱ¢ȱȱȱȱ <sup>ƺ</sup> ȝȝ ǁ Ŗǯȱȱ ȱǰȱ ȱȱȱȱȱ¢ȱǰȱȱ̇<sup>Ȟ</sup> ǀ̇<sup>Ȟ</sup> ȱȱ<sup>Ŗ</sup> <sup>ƽ</sup> ȝȝ Ŗȱ ǰȱȱȱȱ̇<sup>Ŗ</sup> ƽ̇ <sup>Ȟ</sup> Ŗǰȱȱ ȱȱȱȱȱ¢ȱ'ǰȱ'<sup>Ȟ</sup> ȱȱ''<sup>Ȟ</sup> ȱ

$$\begin{split} \Delta \mathbf{Q}\_{\mathcal{W}}'' &= \Delta A'' + \Delta \mathbf{Q}\_0'' \\ &= \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \boldsymbol{\eta}\_{\text{max}}'' + \Delta \mathbf{Q}\_{\mathcal{W}} \cdot \left( 1 - \boldsymbol{\eta}\_{\text{max}} \right) \\ &= \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \boldsymbol{\eta}\_{\text{max}}'' + \Delta \mathbf{Q}\_{\mathcal{W}} - \Delta \mathbf{Q}\_{\mathcal{W}} \cdot \boldsymbol{\eta}\_{\text{max}} \end{split} \tag{21}$$

ȱȱȱȱȱ

$$\begin{split} \Delta \mathbf{Q}\_{\mathcal{W}} &= \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \boldsymbol{\eta}\_{\max}'' + \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \left( 1 - \boldsymbol{\eta}\_{\max}'' \right) \\ &= \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \boldsymbol{\eta}\_{\max}'' + \Delta \mathbf{Q}\_{\mathcal{W}}'' - \Delta \mathbf{Q}\_{\mathcal{W}}'' \cdot \boldsymbol{\eta}\_{\max}'' \end{split} \tag{22}$$

ȱȱȱ̇<sup>Ŗ</sup> ƽ̇ <sup>Ȟ</sup> ŖȱǽȱȱǻŘŗǼȱȱǻŘŘǼǾȱȱ̇ <sup>Ȟ</sup> ȱ ȱ

$$
\Delta Q\_{\mathcal{W}} \cdot \left(1 - \eta\_{\text{max}}\right) = \Delta Q''\_{\mathcal{W}} \cdot \left(1 - \eta''\_{\text{max}}\right) \tag{23}
$$

ȱȱȱ*<sup>Ά</sup>* <sup>Ԕ</sup>ǻŗƺ*¡*ǰ*<sup>Ά</sup>* <sup>Ȟ</sup> Ԕǻŗƺ <sup>Ȟ</sup> *¡*Ǽȱ ȱ

ȱȱ

138 7KHUPRG\QDPLFV

ǻ ȩ ǼȱȱȱȱȱǻǰȱǼȱ'ǔ#ȱȱȱ

Ǽǯȱȱȱ

ȱǰȱ¢ǰȱ

ǰȱ,<sup>Ȟ</sup> Ǽȱ

ǁ Ŗǯȱȱ

Ŗȱ

ǻŘŗǼ

<sup>ĺ</sup> ȱȱȱȱȬ

ȱȱ<sup>Ŗ</sup> <sup>ƽ</sup> ȝȝ

ȱ

ȱ ǻȱ ȱȱ ȱ

ǀ̇<sup>Ȟ</sup>

ȱȱ''<sup>Ȟ</sup>

ȱ ȱ

¬ ¼ ǻŗşǼ

ǻ ȩ Ǽȱȱȱȱȱȱȱ

' ǻŘŖǼ

ǔ <sup>Ȟ</sup>

Ŗǰȱȱ ȱȱȱȱȱ¢ȱ'ǰȱ'<sup>Ȟ</sup>

 K

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$$\begin{aligned} \Delta Q\_W \cdot \beta &= \Delta Q\_W'' \cdot \beta''\\ \frac{\Delta Q\_W}{\Delta Q\_W''} &= \frac{T\_W}{T\_W''} = \frac{\overline{T\_W''}}{\overline{T\_0}} = \frac{\beta''}{\beta} \\ &\implies\\ \Delta Q\_W'' &= \Delta Q\_W \cdot \frac{\beta}{\beta''} = \Delta Q\_W \cdot \frac{T\_W''}{T\_W}, \ \Delta Q\_W'' < \Delta Q\_W \end{aligned} \tag{24}$$

ȱȱ¢ȱ'ȱȱ'<sup>Ȟ</sup> ȱȱ ȱȱȱǰ

$$\begin{aligned} \frac{T\_W - T\_0}{T\_W} &> \frac{T\_W'' - T\_0}{T\_W''}, \ T\_W > T\_W''\\ \frac{T\_0}{T\_W''} &> \frac{T\_0}{T\_W} \text{ a } \text{tedy } \frac{Q\_0}{Q\_W''} > \frac{Q\_0}{Q\_W}, \ Q\_W > Q\_W''\\ &\xrightarrow{\implies}\\ \frac{H(\overline{Y' \mid \overline{X''}})}{H(\overline{Y'})} &> \frac{H(\overline{X \mid \overline{Y})}}{H(\overline{X})} \end{aligned} \tag{25}$$

**ȱŘǯȱ**ȱȱȱȱȱȬ

¢ȱǻŘřǼȱȱǻŘŚǼȱȱ̇<sup>Ȟ</sup> ȱȱȱȱȱȱ¢ȱ'ே ȱ

$$\begin{split} \Delta A'' &= \Delta Q''\_W \cdot \left( 1 - \frac{T\_0}{T''\_W} \right) = \Delta Q\_W \cdot \frac{T''\_W}{T\_W} \cdot \left( 1 - \frac{T\_0}{T''\_W} \right) = \\ &= \Delta Q\_W \cdot \left( \frac{T''\_W}{T\_W} - \frac{T\_0}{T\_W} \right) = \mathbf{k} \cdot H(\mathbf{X}) \cdot \left( T''\_W - T\_0 \right) \\ &= \mathbf{k} \cdot H(\mathbf{X}) \cdot T''\_W \left( 1 - \frac{T\_0}{T'\_W} \right) = \mathbf{k} \cdot H(\mathbf{X}) \cdot T''\_W (1 - \beta'') = \mathbf{k} \cdot T''\_W \cdot H(\mathbf{Y}'') \end{split} \tag{26}$$

ǰȱǰȱȱ̇ȱȱȱ¢ȱ'ȱ ȱ

$$
\Delta A = \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_{\text{W}}(1 - \beta) = \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_{\text{W}} \left(1 - \frac{T\_0}{T\_{\text{W}}}\right) \tag{27}
$$

ȱǰȱȱȱ¢ȱ'<sup>Ȟ</sup> ȱȱ'ȱȱȱȱ

$$\frac{\Delta A''}{\Delta T''}\_W = H(X) \cdot \left(1 - \frac{T\_0}{T''\_W}\right) = H(X) \cdot (1 - \beta'') = H(X) \cdot \eta''\_{\text{max}}\tag{28}$$

$$\frac{\Delta A}{\Delta T\_W} = H(\mathbf{X}) \cdot \left(1 - \frac{T\_0}{T\_W}\right) = H(\mathbf{X}) \cdot (1 - \beta) = H(\mathbf{X}) \cdot \eta\_{\max}$$

ȱȱ ȱ ȱ̇<sup>Ș</sup> ȱȱȱȱ¢ȱ''<sup>Ȟ</sup> ȱ ȱ

$$
\Delta A^\prime = \Delta A - \Delta A^\prime = \left[ k T\_W \cdot H(X) \cdot (1 - \beta) - k T\_W^\prime \cdot H(X) \cdot (1 - \beta^\prime) \right] \ge 0 \tag{29}
$$

ǰȱ ȱ ȱ *ȱ* ȱ ȱ ȱ ¢ȱ ¢ȱ ȱ ȱ ȱ ¢ȱ''<sup>Ȟ</sup> ǻȱȱȱȱ *¢*ǰȱǰȱǼȱȱȱȱȱȱȱȱ ȱȬ ȱ¢ȱ Șǻ ȘǼ Ԕǰȱ ȱȱȱǻŘşǼǰȱȱȱ

$$\begin{split} H^\*(Y^\*) = \frac{\Delta A^\*}{\mathbf{k}T\_W} &= H(\mathbf{X}) \cdot \left[ (1 - \beta) - \frac{T\_W^\*}{T\_W} \cdot (1 - \beta^\*) \right] \\ &= H(\mathbf{X}) \cdot \left( 1 - \frac{T\_0}{T\_W} - \frac{T\_W^\*}{T\_W} + \frac{T\_0}{T\_W} \right) = H(\mathbf{X}) \cdot \left( 1 - \frac{T\_W^\*}{T\_W} \right) \end{split} \tag{30}$$

ȱȱǰȱȱ̇<sup>Ș</sup> ȱȱȱ*ȱ* ȱȱȱ ȱ̇ȱȱȱȱȱȱ ǯȱ ¢ǰȱ ȱ ȱ ȱ ȱ ¢ȱ <sup>Ȟ</sup> ȱ ǻ ȱ ȱ ȱ ¢ȱ ''<sup>Ȟ</sup> ȱ ȱ ̇<sup>Ŗ</sup> ƽ̇ <sup>Ȟ</sup> ŖǼȱȱ¡ȱ¢ȱȱ¢ȱ<sup>Ŗ</sup> Șǰȱ ȱȱǰȱȱȱ ȱ ȱȱ ȱ¢ȱ''<sup>Ȟ</sup> ȱȱ ȱȱ <sup>Ȟ</sup> ƽ<sup>Ŗ</sup> ȘǯȱȱȱǻřŖǼȱ¡ȱȱȱȱȱ ¢ȱ''<sup>Ȟ</sup> ȱȱȱȱȱ¢ȱȱ ȱȱ ȱȱ <sup>ǁ</sup> <sup>Ȟ</sup> ƽ<sup>Ŗ</sup> Șǯȱȱ ȱ¢ȱ''<sup>Ȟ</sup> ȱȱȱȱ

$$\begin{aligned} \boldsymbol{\beta}^{\sigma} &= \frac{\Delta \mathbf{Q}\_{0}^{\sigma}}{\Delta \mathbf{Q}\_{\text{W}}^{\sigma}} = \frac{\frac{\Delta \mathbf{Q}\_{0}^{\sigma}}{T\_{\text{W}}^{\sigma}}}{\frac{\Delta \mathbf{Q}\_{\text{W}}^{\sigma}}{T\_{\text{W}}^{\sigma}}} = \frac{H(\mathbf{Y}^{\sigma} \mid \mathbf{X}^{\sigma})}{H(\mathbf{Y}^{\sigma})} = \frac{T\_{0}}{T\_{\text{W}}^{\sigma}}, \ T\_{\text{W}}^{\sigma} = T\_{0}^{\sigma}, \text{ cyklux } \mathcal{O}^{\sigma} \\ \boldsymbol{\beta} &= \frac{\Delta \mathbf{Q}\_{0}}{\Delta \mathbf{Q}\_{\text{W}}} = \frac{T\_{\text{W}}}{T\_{\text{W}}} = \frac{H(\mathbf{X} \mid \mathbf{Y})}{H(\mathbf{X})} = \frac{T\_{0}}{T\_{\text{W}}}, \text{ cykus } \mathcal{O}^{\sigma} \\ \frac{\partial}{\partial \mathbf{Z}} &= \frac{T\_{\text{w}}^{\sigma}}{T\_{\text{W}}} = \frac{T\_{0}^{\sigma}}{T\_{\text{W}}} \triangleq \boldsymbol{\beta}^{\*} \end{aligned} \tag{31}$$

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$$\frac{\Delta A^{\*}}{\Delta T\_{W}} = H(X) \cdot \left(1 - \beta^{\*}\right) = H(X) \cdot \left[1 - \frac{H(X \mid Y) \cdot H(Y^{\*})}{H(Y^{\*} \mid X^{\*}) \cdot H(X)}\right] > 0\tag{32}$$

ȱȱȱ¢ȱȱȱ <sup>ǁ</sup> <sup>Ȟ</sup> ǰȱ <sup>Ȟ</sup> <sup>Ŗ</sup> ƽŖȱȱȱ¢ȱȱǰȱȱȱȱ¢  <sup>ǻ</sup> <sup>ȩ</sup> Ǽȱȱȱȱȱ¢ȱȱȱ̇<sup>Ŗ</sup> ƽ̇ <sup>Ȟ</sup> Ŗǯȱȱ¢ȱȱȱ¢ȱ''<sup>Ȟ</sup> ȱȱ ȱȱ

$$H(X) = \frac{\Delta Q\_W}{\mathbf{k}T\_W} = \frac{\Delta Q\_W''}{\mathbf{k}T\_W''} = H(Y'') \left[ = \frac{\Delta Q\_W''}{\mathbf{k}T\_0^{"\prime}} \right] \tag{33}$$

ȱ¢ȱȱȱȱǻŚǼȱ ȱ

$$\frac{H(X \mid Y)}{H(Y'' \mid X')} = \beta^\* < 1\tag{34}$$

ȱ ȱ ȱ ȱ ¢ȱ ̇<sup>Ș</sup> Ȧ ȱ ǻȱ ȱ ¢ȱ ¢ȱ +ȱ ȱǼȱȱ¢ȱ ȱȱȱȱȱȱȱȱ

$$\begin{split} \frac{\Delta A^{\*}}{\Delta T\_{W}} &= H(Y'') - H(Y'') \cdot \beta^{\*} = H(Y'') \cdot \left(1 - \frac{T\_{0}^{\*}}{T\_{W}}\right) \\ &= H(Y'') \cdot \left[1 - \frac{H(X \mid Y)}{H(X'' \mid Y'')}\right] \end{split} \tag{35}$$

ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ #ȱ ǻ¡ȱ ¢ȱ ȱ ¢  ǻ ȩ ǼǼȱȱȱ¢ȱȱȱ Ș ǻ <sup>Ș</sup> ǼȱȱǻŘŖǼǰȱǻřŘǼȱȱǻřśǼǯȱ¢¢ǰȱ ȱȱ ǰ ȱȱ ȱȱ ǰ

$$\mathbb{E}\left(H^{\circ}(Y^{\circ})\right) = \frac{\Delta A^{\circ}}{\mathbb{k}T\_{\mathcal{W}}} \cong f\Big[H(X \mid Y)\Big] > 0\tag{36}$$

ȱ¢ȱ'ǰȱ'<sup>Ȟ</sup> ȱȱ''<sup>Ȟ</sup> ȱȱȱȱ¢ȱȱǰȱȱȱȱȱ ¢ȱǰȱ¢Ś ȱȱǰȱȱȱ¢Dzȱȱȱȱȱȱ ȱ *ȱȱȱ¢*ȱǻȱǽŗŚǾǼȱȱȱȱ¢ǰ

$$H\left(X^{\{\boldsymbol{l}\}};Y^{\{\boldsymbol{l}\}}\right) = H\left(X^{\{\boldsymbol{l}\}}\right) - H\left(X^{\{\boldsymbol{l}\}}\mid Y^{\{\boldsymbol{l}\}}\right) = H\left(Y^{\{\boldsymbol{l}\}}\right) > 0 \quad \text{and} \quad \Delta S^{\{\boldsymbol{l}\}}\_{\mathcal{L}} = 0 \tag{37}$$

ȱȱȱȱ¢ȱ''<sup>Ȟ</sup> ȱȱȱȱ£ȱȱȱ*ȱ¢* 'ȱȱDz ȱȱȱȱȱȬȱȱ''<sup>Ȟ</sup> ȱȱȱ ȱȱ¢ȱ

$$T\_W = T\_{W'}^\bullet \left[ \left[ T\_W = T\_W^\bullet \right] \Longrightarrow \left[ H^\circ(Y^\circ) = 0 \right] \right] \tag{38}$$

ǻǰȱ ȱ ȱ¢ȱ ȱ¢ȱȱȱȱ£ȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱǰȱ ¢ȱ ȱ ¢ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱȱ¢ǯǼ

ȱȱȱȱȱ'ȱ ȱȱȱȱȱȱȱȱȱ£ȱȱ ȱȱ*Ύ* <sup>ĺ</sup> ȱȱȱȱȱȱ ȱȱȱȱȱ, ȱȱȱ ȱ#ȱȬȱȱȱȱȱȱȱ¡ȱ¢ȱȱȱȱȱ¢  ǻ ȩ Ǽǰȱȱ ȱȱ ȱ ¢ȱ ¢ȱ ȱȱ ȱ ȱ ȱȂȬȂȱ ȱ ȱȱ'ȱȬǰȱ ȱȱȱȂȂȱ'<sup>Ȟ</sup> ǔ,<sup>Ȟ</sup> ǰȱ ȱȱȱȱȱ ȱ¡ȱ¢ȱȱȱȱȱ¢ȱ ǻ Ȟȩ <sup>Ȟ</sup> Ǽǯ

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