(2006-09-29) The method of Lagrange multipliers
Maximizing under one constraint, or several constraints.
Consider a smooth enough 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).
" dx1 , ... ,
dxW
{ å i
¶F
dxi = 0
} Þ
{ å i
¶S
dxi = 0
}
¶xi
¶xi
Thus, any W-dimensional
vector which is perpendicular to
[¶F/¶xi]
is also perpendicular to
[¶S/¶xi].
Therefore, these two are proportional:
$ l ,
" i ,
¶S
+ l
¶F
= 0
¶xi
¶xi
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
$ l1, l2 ...
" i ,
¶S
+ ån
ln
¶F
= 0
¶xi
¶xi
The (constant) value of each Fn
can be retrieved as
¶L / ¶ln.
(2006-09-29) Micro-Canonical Distribution
For an isolated system, entropy is maximal with equiprobable states.
Let's apply the above to Claude
Shannon's definition of
statistical entropy in terms
of the respective probabilities of the
W possible states:
S ( p1 , p2
, ... , pW )
=
W
å
n =1
- k
pn Log (pn )
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)
S = k Log(W)
(2013-02-21) Equipartition of Energy ( Newtonian mechanics )
Every degree of freedom gets an equal share (½ kT) of thermal energy.
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).
(2006-09-29) Canonical Distribution
In a heat bath, probabilities are proportional to Boltzmann factors.
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.
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.
(2006-09-29) Grand-Canonical Distribution
Taking into account the possibility of chemical exchanges.
(2012-07-17) Bose-Einstein Statistics (1924)
Many particles (bosons) may occupy the same state.
Satyandra
N. Bose
For masssless bosons (photons) at thermal equilibrium,
the occupation number per quantum state is:
