Information (physics)

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In statistical physics , the missing information of a system is the information that is needed to describe the state in which a system is. It is the weighted sum of the logarithms of the state probabilities

{\ displaystyle I = -k \ sum _ {i = 1} ^ {n} p_ {i} \ ln (p_ {i}) \,}

where which are the probabilities of the states of the system. ${\ displaystyle p_ {i} \,}$ ${\ displaystyle n}$

As information entropy , the missing information is referred to, which describes in what state a randomly chosen example of a representative to the ensemble is located. ${\ displaystyle S}$

If one sets , then the information content results from the information theory in the unit Shannon . In statistical physics, the Boltzmann constant is used as a proportionality factor because the information entropy of an ensemble then corresponds to the thermodynamic entropy . ${\ displaystyle k = \ ln (2) ^ {- 1}}$ ${\ displaystyle k}$ ${\ displaystyle k = k _ {\ mathrm {B}}}$

The vestibular system is in this usage the system with the maximum lack of information.

Micro-canonical ensemble

In the micro-canonical ensemble, all states are represented equally frequently. Is there for the macroscopic state in which z. If, for example, the energy, the number of particles and the volume are a number of microscopic states, the energy levels are consequently represented by the probabilities . ${\ displaystyle (E, N, V, \ dots)}$ ${\ displaystyle E}$ ${\ displaystyle N}$ ${\ displaystyle V}$ ${\ displaystyle W (E, \ dots)}$ ${\ displaystyle E_ {n} \,}$ ${\ displaystyle p_ {n} = 1 / W (E, \ dots)}$

The information entropy is then

{\ displaystyle S \; = \; - k _ {\ mathrm {B}} \ sum _ {n} {\ frac {1} {W (E_ {n})}} \, \ ln \ left ({\ frac {1} {W (E_ {n})}} \ right) \,}

or for a certain energy ${\ displaystyle E_ {n}}$

{\ displaystyle S \; = \; - k _ {\ mathrm {B}} \ ln \ left ({\ frac {1} {W (E_ {n})}} \ right) \; = \; k _ {\ mathrm {B}} \ ln (W) \.}

Canonical ensemble

For the canonical ensemble, the probabilities are for different energies . And the information entropy is ${\ displaystyle p_ {n} = {\ frac {1} {Z}} \ exp \ left (- \ beta E_ {n} \ right)}$

{\ displaystyle S \; = \; - k _ {\ mathrm {B}} \ sum _ {n} p_ {n} \, \ left (- \ ln (Z) - \ beta E_ {n} \ right) \ ; = \; - k _ {\ mathrm {B}} \ left (- \ ln (Z) - \ beta \ sum _ {n} p_ {n} \, E_ {n}) \ right) \; = \; - {\ frac {FU} {T}} \,}

where are the free energy and the internal energy . ${\ displaystyle F}$ ${\ displaystyle U}$

literature

Kerson Huang : Introduction to Statistical Physics . CRC Press, Boca Raton 2010, ISBN 978-1-4200-7902-9 .
W. Dieterich, Oliver Schlotterer: Statistical Mechanics. (PDF; 2.4 MB) Lecture in WS 2004/5. Physics Department, University of Konstanz, February 2009, pp. 57–61 , accessed on January 21, 2012 .
Horst Völz : That is information. Shaker Verlag, Aachen 2017. ISBN 978-3-8440-5587-0 .
Horst Völz : Description of the world. Space, time, temperature and information - aspects, positions, debates. Shaker Verlag, Aachen 2018, ISBN 978-3-8440-6323-3 .