By Arieh Ben-Naim

The critical message of this ebook is that thermodynamics and statistical mechanics will make the most of exchanging the unlucky, deceptive and mysterious time period entropy with a extra ordinary, significant and applicable time period akin to details, lacking info or uncertainty. This alternative could facilitate the translation of the motive force of many tactics by way of informational adjustments and dispel the secret that has continually enshrouded entropy.

it's been one hundred forty years in view that Clausius coined the time period entropy ; virtually 50 years considering Shannon built the mathematical conception of data consequently renamed entropy. during this booklet, the writer advocates changing entropy by means of info, a time period that has turn into standard in lots of branches of technology.

the writer additionally takes a brand new and impressive method of thermodynamics and statistical mechanics. details is used not just as a device for predicting distributions yet because the primary cornerstone idea of thermodynamics, held previously through the time period entropy.

the subjects lined contain the basics of likelihood and data thought; the final idea of data in addition to the actual idea of data as utilized in thermodynamics; the re-derivation of the Sackur Tetrode equation for the entropy of an incredible gasoline from in basic terms informational arguments; the basic formalism of statistical mechanics; and plenty of examples of straightforward strategies the driver for that's analyzed by way of details.

**Contents: **

- Elements of likelihood idea;
- Elements of knowledge conception;
- Transition from the final MI to the Thermodynamic MI;
- The constitution of the rules of Statistical Thermodynamics;
- Some easy functions.

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**Example text**

Let F : B → C = ∨k Ck be a ϕ-preserving conditional expectation. Then (C, ϕ|C , {Ck }k , F ◦ E) is an abelian model for (A, ϕ, {Ck }k ). Since F ◦ E is the identity map on Ck ⊂ C, the entropy of this model is the same as the entropy of the model (C, ϕ|C , {Ck }k , idC ) for (C, ϕ|C , {Ck }k ). The latter entropy is S(ϕ|C ). Thus Hϕ (C1 , . . 2(vii). On the other hand, we have Hϕ (C1 , . . , Cn ) ≤ Hϕ (A1 , . . , An ) ≤ Hϕ (B) ≤ S(ϕ|B ). Hence the above inequalities are in fact equalities. 2.

Nn be ﬁnite dimensional subalgebras of N , and let ENk : N → Nk denote the τ -preserving conditional expectation. Then the mutual entropy of N1 , . . in )) − τ η ENk (xik ) , Hτ (N1 , . . in . in N , where In the case of one algebra B ⊂ N , the deﬁnition says that Hτ (B) is the supremum of 34 3 Dynamical Entropy η(τ (xi )) − i τ (xi )τ (η(EB (τ (xi )−1 xi ))). 1) i We shall see soon that this supremum is just S(τ |B ). This deﬁnition does not make sense if τ is not a trace, since in general a state preserving conditional expectation N → Nk may not exist.

I Proof. Consider a decomposition ϕ = k λk ϕk of ϕ into a convex combination of pure states. 2(iii) we have λk S(ϕk , ϕ) ≥ S(ϕ) = k λk S(ϕk |B , ϕ|B ) k = S(ϕ|B ) − λk S(ϕk |B ) = S(ϕ|B ) − λk S(ϕk |Bi ). i k k The restriction of ϕk to Bi = ei Aei is a scalar multiple of a pure state, whence S(ϕk |Bi ) = η(ϕk (ei )). By concavity of η we then get λk S(ϕk |Bi ) = k λk η(ϕk (ei )) ≤ η k λk ϕk (ei ) = η(ϕ(ei )), k which completes the proof of the lemma. In Chap. 9 we shall use the following form of positivity of relative entropy, called the thermodynamic inequality.