 By Ilya Prigogine

One hundred and one PAGES fresh PLASTIC LAMINATION OVER PAPERJACKET

Best thermodynamics and statistical mechanics books

kinetic theory of vehicular traffic

A hundred and one PAGES fresh PLASTIC LAMINATION OVER PAPERJACKET

Aerothermodynamik

Dieses Buch wendet sich an Studenten der Ingenieurwissenschaften und Ingenieure der Raumfahrtindustrie und der Energieverfahrenstechnik. Es verkn? pft die klassischen Gebiete der Aerodynamik mit der Nichtgleichgewichts-Thermodynamik hei? er Gase. Am Beispiel des Wiedereintritts einer Raumkapsel in die Erdatmosph?

Extra info for kinetic theory of vehicular traffic

Sample 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.