scholarly journals The Shannon–McMillan Theorem Proves Convergence to Equiprobability of Boltzmann’s Microstates

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 899
Author(s):  
Arnaldo Spalvieri
Keyword(s):  
Information Theory ◽  
Statistical Mechanics ◽  
Identical Particles ◽  
Entropy Formula ◽  
Number Of Particles

This paper shows that, for a large number of particles and for distinguishable and non-interacting identical particles, convergence to equiprobability of the W microstates of the famous Boltzmann–Planck entropy formula S = k log(W) is proved by the Shannon–McMillan theorem, a cornerstone of information theory. This result further strengthens the link between information theory and statistical mechanics.

ON THE EXTENSIVITY OF THE ENTROPY Sq FOR N ≤ 3 SPECIALLY CORRELATED BINARY SUBSYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1727-1738 ◽  
Author(s):  
YUZURU SATO ◽  
CONSTANTINO TSALLIS
Keyword(s):  
Statistical Mechanics ◽  
Stationary States ◽  
Nonextensive Statistical Mechanics ◽  
Entropy Formula ◽  
Gibbs Entropy ◽  
Statistical Mechanical

Many natural and artificial systems whose range of interaction is long enough are known to exhibit (quasi)stationary states that defy the standard, Boltzmann–Gibbs statistical mechanical prescriptions. For handling such anomalous systems (or at least some classes of them), nonextensive statistical mechanics has been proposed based on the entropy [Formula: see text], with [Formula: see text] (Boltzmann–Gibbs entropy). Special collective correlations can be mathematically constructed such that the strictly additive entropy is now Sq for an adequate value of q ≠ 1, whereas Boltzmann–Gibbs entropy is nonadditive. Since important classes of systems exist for which the strict additivity of Boltzmann–Gibbs entropy is replaced by asymptotic additivity (i.e. extensivity), a variety of classes are expected to exist for which the strict additivity of Sq (q ≠ 1) is similarly replaced by asymptotic additivity (i.e. extensivity). All probabilistically well defined systems whose adequate entropy is S1 are called extensive (or normal). They correspond to a number W eff of effectively occupied states which grows exponentially with the number N of elements (or subsystems). Those whose adequate entropy is Sq (q ≠ 1) are currently called nonextensive (or anomalous). They correspond to W eff growing like a power of N. To illustrate this scenario, recently addressed [Tsallis, 2004] we provide in this paper details about systems composed by N = 2, 3 two-state subsystems.

scholarly journals THE HATSOPOULOS–GYFTOPOULOS RESOLUTION OF THE SCHRÖDINGER–PARK PARADOX ABOUT THE CONCEPT OF "STATE" IN QUANTUM STATISTICAL MECHANICS

2006 ◽  
Vol 21 (37) ◽  
pp. 2799-2811 ◽  
Author(s):  
GIAN PAOLO BERETTA
Keyword(s):  
Information Theory ◽  
Quantum Theory ◽  
Statistical Mechanics ◽  
Quantum Statistical Mechanics ◽  
Quantum Information Theory ◽  
Individual State ◽  
Fundamental Difficulty ◽  
Quantum Statistical ◽  
Theory Interpretation

A seldom recognized fundamental difficulty undermines the concept of individual "state" in the present formulations of quantum statistical mechanics (and in its quantum information theory interpretation as well). The difficulty is an unavoidable consequence of an almost forgotten corollary proved by Schrödinger in 1936 and perused by Park, Am. J. Phys.36, 211 (1968). To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss a possible resolution proposed in 1976 by Hatsopoulos and Gyftopoulos, Found. Phys.6, 15; 127; 439; 561 (1976).

scholarly journals Application of information theory in atoms, nuclei and atomic clusters

2019 ◽  
Vol 10 ◽  
pp. 209
Author(s):  
C. P. Panos ◽  
S. E. Massen
Keyword(s):  
Information Theory ◽  
Single Particle ◽  
Functional Form ◽  
Mean Field ◽  
Atomic Clusters ◽  
Fermion System ◽  
Electron Distributions ◽  
Particle Potential ◽  
Single Particle Potential ◽  
Number Of Particles

The position- and momentum-space information entropies of the electron distributions of atomic clusters are calculated using a Woods-Saxon single particle potential. The same entropies are also calculated for nuclear distributions according to the Skyrme parametrization of the nuclear mean field. It turns out that a similar functional form S = α + Μη Ν for the entropy as function of the number of particles Ν holds approximately for atoms, nuclei and atomic clusters. It is conjectured that this is a universal property of a many-fermion system in a mean field.

scholarly journals Volitional Motion Theory

2021 ◽  
Author(s):  
Masaru Kondo
Keyword(s):  
Mathematical Model ◽  
Information Theory ◽  
Statistical Mechanics ◽  
Behavioral Economics ◽  
Computational Neuroscience ◽  
Behavioral Science ◽  
Classical Mechanics ◽  
The Public ◽  
Observable Data ◽  
Animal Spirit

We propose a mathematical model for quantifying willpower and an application based on the model. Volitional Motion Theory (VMT) is a mathematical model that draws on classical mechanics, thermodynamics, statistical mechanics, information theory, and philosophy. The resulting numbers are statistical theoretical values deduced using observable data. VMT can be applied to a variety of fields, including behavioral science, behavioral economics, and computational neuroscience. For example, "What is animal spirit in economics?" VMT is one proposal to answer this question. In addition, a scheduling application has been created to validate VMT. This application is open to the public for anyone to use.

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