statistical mechanics How the overall behavior of a system of many particles is related to the Properties of the particles themselves. It deals with the overall particles as whole and not with the individual particle. It tells the probability that the particle has a certain amount of energy at a certain moment.

statistical distributions – general considerations Maxwell-Boltzmann Bose-Einstein Fermi-Dirac Maxwell-Boltzmann statistics Maxwell-Boltzmann distribution energies in an ideal gas equipartition of energy quantum statistics fermions and bosons Bose-Einstein and Fermi-Dirac distribution comparison of the three statistical distributions applications Planck radiation law specific heats of solids free electrons in a metal

general considerations central question: how does the behavior of a many-particle system depend on the properties of the single particles? therefore: look at probabilities for particle properties but: too many single particles to describe them one by one example: a room filled with air number of particles >> mainly two kinds of particles (N 2 and O 2 ) impossible to know all coordinates and kinetic energies but: sm allows to calculate the probability of each particle to e.g. have a certain amount of kinetic energy at a time t

statistical distributions most easy setting: system of N particles in thermal equilibrium at temperature T question: how is the total energy E distributed over the particles? or: how many particles have the energy      etc. particles interact “weakly” with one another and the container walls thermal equilibrium but no correlation more than one particle may have a certain energy 

statistical distributions most easy case: thermal equilibrium constant energy (E=const.) constant number of particles (N=const., for “classical” particles) n(  )=g(  )f(  ) number of particles with energy  number of states with energy  (statistical weight) probability of occupancy of each state with energy  (distribution function) or average Number of particles in each state of Energy 

statistical distributions classical system:  d , etc. g(  )d  Maxwell Boltzmann Fermi Dirac Bose Einstein identical particles “far” apart (no overlap of  ) distinguishable identical particles integral spin (bosons) close together (overlapping  ) indistinguishable identical particles odd half-integral spin (fermions) close together (overlapping  ) indistinguishable

statistical distributions Maxwell Boltzmann Fermi Dirac Bose Einstein e.g. molecules in a gase.g. photonse.g. electrons

Maxwell-Boltzmann distribution Maxwell-Boltzmann distribution function: f MB (  )=A e -  /kT k=1.381 x J/K=8.617 x eV/K (Boltzmann constant) N(  )d  is the number of particles whose energy lie between  And  +d 

energies in an ideal gas ideal gas: PV=RT N is large translational motion, quantization is irrelevant (number of molecules between  and  + d  ) (energy distribution)

Since each momentum magnitude p corresponds to energy 

Total number of molecules is N THIS IS MOLECULAR ENERGY DISTRIBUTION

Most probable energy

Molecular speed distribution

1. Mean velocity: 1. Root mean square velocity: 1. Most probable velocity:

fermions and bosons distinguishable particles (non overlapping wavefunctions) indistinguishable particles (overlapping wavefunctions) bosons: integral spin (0,1,2,…) symmetric wave function (exchange of two bosons does not change the system) all bosons can be in the same quantum state Photons, Phonons Wave function of system of boson is not affected any exchange of any pair of particle

FERMIONS odd half integral spin (1/2,3/2,5/2,…) antisymmetric wave function –(exchange of two fermions changes symmetry of the system) only one fermion can be in a quantum state –(exclusion principle, Pauli principle) probability for two particles in one state: 0!

Consider a system of two particles, 1 and 2 one of which is in state a and the other is in state b. when all particles are distinguishable there are two possibilities for occupancy of the states as When two particles are indistinguishable we can not tell which of them is in which state, and wavefunction must be a combination of the both wavefunction  I AND  II.

FOR BOSONS- SYMMETRIC WAVEFUNCTION For fermions-ANTI SYMMETRIC WAVEFUNCTION

Now let both the particles are in same state a then both wave function will become Probability density For bosons wave function

For fermions SO two particles can not be in the same quantum state.

Bose-Einstein / Fermi-Dirac distribution bosons: one boson of a system in a certain state increases the probability of finding another boson in this state! fermions: one fermion of a system in a certain state prevents all other fermions from being in that state! A describes the system and may be a function of T  >>kT f BE and f FD converge into f MB  F is the Fermi energy

comparison of the distributions

Consider the F-D distribution at T = 0 K AT T=0 K 0 ∞ From this we conclude that all energy states above  F are Empty (f FD ) and all energy states below  F are occupied (f FD = 1). So  F gives the energy of the highest filled state at T = 0.

f FD  T = 0 FF 0 1 KT = 0.1  F FF KT = 1.0  F 0.5 FF f FD

comparison of the distributions Maxwell Boltzmann Fermi Dirac Bose Einstein identical distinguishable classical particles any spin  don’t overlap e.g. gas molecules unlimited number of particles per state indistinguishable no Pauli principle bosons spin 0,1,2 …  overlap  symmetric e.g. cavity photons (laser) liquid He at low T unlimited number of particles per state, more than in MB aproaches MB for high T indistinguishable, Pauli principle fermions spin 1/2,3/2,5/2 …  overlap  antisymmetric e.g. free electrons in metals electrons in white dwarfs never more than 1 particle per state, less than in MB aproaches MB for high T

free electrons in a metal (number of electron states)

free electrons in a metal (electron energy distribution) : electron density Fermi energy

free electrons in a metal T=0 T>>0 EFEF

Total internal energy AVERAGE ELECTRON ENERGY