Editing Concentration of measure (section)

== Concentration of measure in physics ==

All classical statistical physics is based on the concentration of measure phenomena:
The fundamental idea (‘theorem’) about equivalence of ensembles in thermodynamic limit ([[Josiah Willard Gibbs|Gibbs]], 1902<ref>{{cite book |last= Gibbs |first= Josiah Willard |date=1902 |title= Elementary Principles in Statistical Mechanics |url= https://www-liphy.ujf-grenoble.fr/pagesperso/bahram/Phys_Stat/Biblio/gibbs_1902.pdf |location= New York, NY |publisher= Charles Scribner's Sons |page= <!-- or pages= --> }}</ref> and [[Albert Einstein|Einstein]], 1902-1904<ref>{{cite journal | last = Einstein | first = Albert | title = Kinetische Theorie des Wärmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik [Kinetic Theory of Thermal Equilibrium and of the Second Law of Thermodynamics]| journal = Annalen der Physik |series=Series 4| volume = 9 | pages = 417–433| date = 1902| url = http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1902_9_417-433.pdf| doi = 10.1002/andp.19023141007| access-date = 21 January 2020 }}</ref><ref>{{cite journal | last = Einstein | first = Albert | title = Eine Theorie der Grundlagen der Thermodynamik [A Theory of the Foundations of Thermodynamics]| journal = Annalen der Physik |series=Series 4| volume = 11 | pages = 417–433| date = 1904 | url = http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1904_14_354-362.pdf | access-date = 21 January 2020 }}</ref><ref>{{cite journal | last = Einstein | first = Albert | title = Allgemeine molekulare Theorie der Wärme [On the General Molecular Theory of Heat]| journal = Annalen der Physik |series=Series 4| volume = 14 | pages = 354–362| date = 1904 | url = http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1904_14_354-362.pdf | doi = 10.1002/andp.19043190707| access-date = 21 January 2020}}</ref>) is exactly the thin shell concentration theorem. For each mechanical system consider the [[phase space]] equipped by the invariant [[Liouville measure]] (the phase volume) and conserving energy ''E''. The [[microcanonical ensemble]] is just an invariant distribution over the surface of constant energy E obtained by Gibbs as the limit of distributions in [[phase space]] with constant density in thin layers between the surfaces of states with energy ''E'' and with energy ''E+ΔE''. The [[canonical ensemble]] is given by the probability density in the phase space (with respect to the phase volume)
<math>\rho = e^{\frac{F - E}{k T}},</math>
where quantities F=const and T=const are defined by the conditions of probability normalisation and the given expectation of energy ''E''.

When the number of particles is large, then the difference between average values of the macroscopic variables for the canonical and microcanonical ensembles tends to zero, and their [[Thermal fluctuations |fluctuations]] are explicitly evaluated. These results are proven rigorously under some regularity conditions on the energy function ''E'' by [[Aleksandr Khinchin|Khinchin]] (1943).<ref>{{cite book |last= Khinchin |first= Aleksandr Y. |date=1949 |title= Mathematical foundations of statistical mechanics [English translation from the Russian edition, Moscow, Leningrad, 1943]|url= https://books.google.com/books?id=D7oEAAAAMAAJ |location= New York, NY |publisher= Courier Corporation  |page= <!-- or pages= --> | access-date = 21 January 2020}}</ref>
The simplest particular case when ''E'' is a sum of squares was well-known in detail before [[Aleksandr Khinchin|Khinchin]] and Lévy and even before Gibbs and Einstein. This is the [[Maxwell–Boltzmann distribution]] of the particle energy in ideal gas.

The microcanonical ensemble is very natural from the naïve physical point of view: this is just a natural equidistribution on the isoenergetic hypersurface. The canonical ensemble is very useful because of an important property: if a system consists of two non-interacting subsystems, i.e. if the energy ''E'' is the sum, <math>E=E_1(X_1)+E_2(X_2)</math>, where <math>X_1, X_2</math> are the states of the subsystems, then the equilibrium states of subsystems are independent, the equilibrium distribution of the system is the product of equilibrium distributions of the subsystems with the same T. The equivalence of these ensembles is the cornerstone of the mechanical foundations of thermodynamics.