By R. S. Ingarden, A. Kossakowski, M. Ohya (auth.)

This booklet has an extended background of greater than twenty years. the 1st try to write a monograph on information-theoretic method of thermodynamics was once performed through one of many authors (RSI) in 1974 while he released, within the preprint shape, volumes of the ebook "Information conception and Thermodynamics" referring to classical and quantum details idea, [153] (220 pp.), [154] (185 pp.). regardless of the encouraging feedback by way of a number of the readers, the actual a part of this booklet used to be by no means written with the exception of the 1st bankruptcy. Now this fabric is written thoroughly anew and in a lot larger quantity. many years past, in 1970, moment writer of the current ebook, (AK), a doctoral scholar and collaborator of RSI in Toruli, released in Polish, additionally as a preprint, his habilitation dissertation "Information-theoretical choice scheme in quantum statistical mechanics" [196] (96 pp.). This small monograph provided his unique ends up in the actual a part of the speculation constructed within the Torun tuition. regrettably, this preprint was once by no means released in English. the current e-book includes these kinds of leads to a way more smooth and constructed form.

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**Example text**

P) ::; 0. p) = 0. For complete distributions p and q, Point 2 says nothing than this since then from }Jj ::; qj, (j = 1, 2, ... , n) it follows that Pi= qi, (j = 1, 2, ... , n). 3. 149) CLASSICAL ENTROPY Remark: It follows from 2 and 3 that (0 = log 2 p-q, S(eqiep)

103) nicely presents the dependence on the measure m, so the Boltzmann entropy appears as a relative entropy, in a sense (actually, its generalization with the opposite sign). , 1 em): r (dm dp, dfl) log a dm . x2- 2(}' 2 ) l a x2 log [ --exp(--. -) dx. 106) which is physically correct as being homogeneous with respect to units. , as a constant of nature or a parameter of experiment. s. ), and is given in symplectic coordinates, while X is a symplectic space representing phase space in physics (positionmomenta space {(q,p)} with the volume element dnqdnp), we may take as our constant the Planck constant in the n-th power, hn, since the physical dimension (positionx momentum]= [action].

We first quote few sentences from [329], p. 289, which explain our philosophical and mathematical position: "As it was pointed out in [136], information is seen intuitively as a much simpler and more elementary notion than that of probability. It gives a cruder and more global description of some physical situations than probability does. Therefore, information represents a more primary step of knowledge that that of cognition of probabilities. It should be noted that the axioms proposed in [136] and [137], in particular that which gives a connection between information of rings and their subrings, are very difficult to grasp in their intuitive meaning.