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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Statistical Physics

 Joseph-Louis Lagrange 
 1736-1813  James Clerk Maxwell 
A scientist's aim  [...]  is not to persuade, but to clarify.
Leó Szilárd  (1898-1964) 

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Physics 301Thermal Physics   by   Ed. Groth  (Princeton University).
Equipartition of Energy   |   The Ergodic Hypothesis
Ergodic Theory  by  Cosma Rohilla Shalizi, Ph.D.  (CMU).
Einstein's Random Walk  by Mark Haw  (Physics World, January 2005).

Statistical Physics, Thermal Physics

(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  }
vinculum vinculum
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     QED
vinculum vinculum
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
vinculum vinculum
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 )   =    
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).

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Wikipedia :   Equipartition of 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 pEi  constant.  This can be viewed as an additional "constraint" corresponding to a new Lagrange multiplier b.

L   =   S  +  l å pi   +  b å pEi

b  turns out to be inversely proportional to the temperature of the bath.

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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.

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

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(2012-07-17)   Bose-Einstein Statistics   (1924)
Many particles (bosons) may occupy the same state.
   Satyandra N. Bose (1925)
Satyandra N. Bose

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

exp ( hn / kT ) - 1

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Bose-Einstein statistics   |   Satyendra Nath Bose (1894-1974)
Test of Bose-Einstein statistics for photons (animation)

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

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Fermi-Dirac statistics   |   Fermi energy   |   Fermi level

(2006-09-29)   Boltzmann's Statistics   (for either bosons or fermions)
The low occupancy limit where almost all states are unoccupied.

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(2006-09-30)   Maxwell-Boltzmann distribution of speeds
Boltzmann statistics applied to the molecules in a classical perfect gas.

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

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

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Wikipedia :   Fock Space  (Konfigurationsraum, 1932)   |   Vladimir Aleksandrovich Fock (1898-1974)

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