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

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].