By L. Hopf

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**Extra resources for Introduction to the Differential Equations of Physics**

**Example text**

Postulates for Entropy and Relative Entropy Von Neumann used arguments of phenomenological thermodynamics to deduce certain properties of the entropy of a state described by a statistical operator. His argument (reproduced also in the Introduction) supports the mixing condition S(Aço + (1 — A S(ço) = 1 0 and disjoint states cp and if p and the property is pure. 2) determine a unique functional S(w) = on states of a finite quantum system. ) Creating the foundations of information theory in 1948, Shannon derived from certain desiderata the formula H(pi ,P2,.

2) determine a unique functional S(w) = on states of a finite quantum system. ) Creating the foundations of information theory in 1948, Shannon derived from certain desiderata the formula H(pi ,P2,. . 3) log Pk = for the amount of information contained in a single observation of a random variable which takes the different values x1, x2,. Prob = Xk) (k = 1,2,.. , n). , with probabilities Pk = In the present chapter Shannon's axiomatic point of view will be adopted and we shall deduce the von Neumann entropy and the relative entropy from certain postulates.

20) was obtained in [Hiai, Ohya and Tsukada 1981] under the strong condition that restricted to the subalgebra B, be tracial. The form presented here is from [Petz 1986a]. 15 is due to [Hiai, Ohya and Tsukada 1981]. 16. 22). 17 is a very special case of well-known ergodic theorems in operator algebras, our formulation is not more than a functional analytic formulation of the simplest law of large numbers. More sophisticated limit theorems may be found in [Jajte 1985]. 18 is the analogue of the so-called fundamental theorem of information theory, it is from [Ohya and Petz 1991].

### Introduction to the Differential Equations of Physics by L. Hopf

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