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(2006-09-29)  
Maximizing under one constraint, or several constraints.

Consider a smooth function  S  of   variables: S ( x₁, x₂, ... , x ).

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   to such a constraint and introduces a new function  L :

L   =   S  +  F

The key point is that the constrainedmaximum we seek  (assuming there is one) occurs at a saddlepoint of  L  (i.e.,dL = 0)  for a specific value of  .

At the constrained maximum, any displacement which maintainsthe constraint entails a vanishingvariation of  S  (i.e., dF = 0    dS = 0).

dx1 , ... ,dx i	F	dxi   = 0  {  i	S	dxi   = 0
	xi		xi

Thus, any  -dimensionalvector which is perpendicular to [F/x_i] is also perpendicular to [S/x_i]. Therefore, these two are proportional:

,   i ,	S	+	F	=   0
	xi		xi

The parameter   thus obtained is called a Lagrange multiplier. One such Lagrange multiplier  corresponds to each of several  simultaneous constraints. Any constrained saddlepoint (possiblya maximum)  of  S  is an unrestricted saddlepoint ofthe following function  L ,  and vice-versa.

L   =   S  + _n n F_n

...   i ,	S	+  n n	F	=   0
	xi		xi

The (constant) value of each  F_n can be retrieved as  L / _n.

(2006-09-29)  
For an isolated system, entropy is maximal withequiprobable states.

Let's apply theabove to ClaudeShannon's definition ofstatistical entropy in termsof the respective probabilities of the   possible states:

S ( p1, p2, ... , p )  =

n = 1

k pn  Log (pn)

The basic constraint of completeness  ( p₁ + p₂ + ... + p =  1 )  is the only  constraintfor the probabilities in a completely isolated  system.

L   =   S + F  =   S + ( p₁ + p₂ + ... + p )
 
0   =  L / p_i  =      k [ 1 + Log(p_i) ]

Therefore, all values of  p_i  are equal to  exp( /_k-1)  =  ¹/

Plugging this equiprobability into the expression of  S,  yieldsBoltzmann's relation for a microcanonical ensemble  (i.e., an isolated system).

Boltzmann's Relation  (1877)

S   =   k  Log()

(2013-02-21)  
Every degree of freedom gets an equal share  (½ kT)  of thermal energy.

The particular forms of the formulas in classical mechanics are such thatthe total energy of every component in a large systemis the sum of the energies corresponding to all its degrees of freedom: Each of those is proportional either to the square of a velocity orto the square of a displacement (using thenonrelativistic expression of kineticorrotational energyand the approximation of Hooke's law for potential energy).

(2006-09-29)  
In a heat bath, probabilities are proportional to Boltzmann factors.

Let  E_i  be the energy of state i. Putting the system in thermal equilibrium with a "heat bath"makes its average  energy  p_i E_i constant.  This can be viewed as an additional "constraint" correspondingto a new Lagrange multiplier .

L   =   S  +   p_i  +   p_i E_i

  turns out to be inversely proportionalto the temperature of the bath.

Canonical: Average energy p_i E_iis 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 equipartionof energy between loosely connected degrees of freedom.

(2006-09-29)  
Taking into account the possibility of chemical exchanges.

(2012-07-17)     (1924)
Many particles (bosons) may occupy thesame state.

For masssless bosons  (photons)  at thermal equilibrium,the occupation number per quantum state is:

1 exp ( h / kT ) 1

Elementary particles with whole-integer spins are called Bosons  because they obeyBose-Einstein statistics (the term was coined by Paul Dirac).

(2012-07-17)     (1926)
All particles (fermions) are in different states.

Anelementary particle  whose spin isn't a whole multipleof the quantum pf spin  must have half-integer spin. Such particles  obey the Fermi-Dirac statistics described below and they're known as Fermions.

As fermions obey the Pauli exclusion principle, no two identical fermions can occupy the same quantum state of energy .  When there many fermions, the average number found in a given state of energy   is:

1 exp ( [] / kT )+ 1

(2006-09-29)    (foreither bosons or fermions)
The low occupancy limit  applies when almost all  states are unoccupied.

(2006-09-30)  
Boltzmann statistics applied tothe molecules in a classicalperfect gas.

(2006-09-29)  
Thermal summary of a distribution.

(2014-03-24)  
Fock basis  for the tensor product of many identical Hilbert spaces.