Consider a smooth function S of W variables: S ( x1 , x2 , ... , xW ).
We seek to maximize S subject to the constraint that some other function F of those same variables is a given constant. Lagrange's method associates a parameter l to such a constraint and introduces a new function L :
L = S + l F
The key point is that the constrained maximum we seek (assuming there is one) occurs at a saddlepoint of L (i.e., dL = 0) for a specific value of l.
Proof: At the constrained maximum, any displacement which maintains the constraint entails a vanishing variation of S (i.e., dF = 0 Þ dS = 0).
Thus, any W-dimensional vector which is perpendicular to [¶F/¶xi] is also perpendicular to [¶S/¶xi]. Therefore, these two are proportional:
The parameter l thus obtained is called a Lagrange multiplier. One such Lagrange multiplier corresponds to each of several simultaneous constraints. Any constrained saddlepoint (possibly a maximum) of S is an unrestricted saddlepoint of the following function L , and vice-versa.
L = S + ån ln Fn
The (constant) value of each Fn can be retrieved as ¶L / ¶ln.
Let's apply the above to Claude Shannon's definition of statistical entropy in terms of the respective probabilities of the W possible states:
The basic constraint of completeness ( p1 + p2 + ... + pW = 1 ) is the only constraint for the probabilities in a completely isolated system.
L = S + l F
= S + l
( p1 + p2 + ... + pW )
0 =
¶L / ¶pi
= l
- k [ 1 + Log(pi ) ]
Therefore, all values of pi are equal to exp( l/k-1) = 1/W
Plugging this equiprobability into the expression of S, yields Boltzmann's relation for a microcanonical ensemble (i.e., an isolated system).
Boltzmann's Relation (1877)The particular forms of the formulas in classical mechanics are such that the total energy of every component in a large system is the sum of the energies corresponding to all its degrees of freedom: Each of those is proportional either to the square of a velocity or to the square of a displacement (using the nonrelativistic expression of kinetic or rotational energy and the approximation of Hooke's law for potential energy).
[画像: Come back later, we're still working on this one... ]
Wikipedia : Equipartition of energy
Let Ei be the energy of state i. Putting the system in thermal equilibrium with a "heat bath" makes its average energy å pi Ei constant. This can be viewed as an additional "constraint" corresponding to a new Lagrange multiplier b.
L = S + l å pi + b å pi Ei
b turns out to be inversely proportional to the temperature of the bath.
[画像: Come back later, we're still working on this one... ]
Canonical: Average energy å pi Ei is constant for the system in contact with a heat bath. Lagrange multiplier is inversely proportional to temperature.
Micro-canonical: Given energy for the system... Special case is equipartion of energy between loosely connected degrees of freedom.
For masssless bosons (photons) at thermal equilibrium, the occupation number per quantum state is:
[画像: Come back later, we're still working on this one... ]
Elementary particles with whole-integer spins are called Bosons because they obey Bose-Einstein statistics (the term was coined by Paul Dirac).
Bose Audio : A younger relative of Satyendra Bose was Amar Gopal Bose (1929-2013) the electrical engineer who became a billionnaire after founding Bose corporation in 1964. Amar Bose was a graduate of MIT where he kept teaching from 1956 to 2001. He donated a majority of his company to MIT in 2011, in the form of non-voting shares.
Bose-Einstein statistics
|
Satyendra Nath Bose (1894-1974)
Test of Bose-Einstein statistics for photons
(animation)
An elementary particle whose spin isn't a whole multiple of the quantum pf spin must have half-integer spin. Such particles obey the Fermi-Dirac statistics described below and they're known as Fermions.
A composite particle containing an even number of fermions (possibly none) is a boson. Otherwise, it's a fermion. The electron is an elementary fermion, the proton and the neutron are composite fermions.
As fermions obey the Pauli exclusion principle, no two identical fermions can occupy the same quantum state of energy e. When there many fermions, the average number found in a given state of energy e is:
[画像: Come back later, we're still working on this one... ]
Wikipedia : Fock Space (Konfigurationsraum, 1932) | Vladimir Aleksandrovich Fock (1898-1974)
