Statistical Mechanics

1. Statistical Mechanics David K. Ferry and Dragica Vasileska Arizona State University Tempe, AZ 85287-5706

2. Objective:To treat an ensemble of particles or system in a statistical or probabilistic fashion, in which we are concerned with the most probable values of the properties of the ensemble without investigating in detail what the value of that properties may be for a particular particle at any time. • We limit our discussion to an assembly of identical particles that are independent of each other and can only interact via instantaneous processes that conserve energy and momentum. • Phase space: A six dimensional phase space is defined by the coordinates (x,y,z,kx,ky,kz) where x,y,z refer to real space and kx,ky,kz refer to momentum space. • Basic Postulate :The apriori probability for a system to be in any quantum state is the same for all quantum states of the system. This is true only when there are no dynamical restrictions. The apriori probabilities will be modified by external constraints, such as the number of particles is a constant and the total energy is constant. • Different types of systems considered: • Distinguishable particles (Fermions when spin is not considered) • Indistinguishable particles that obey Pauli exclusion principle (Fermions) • The role of indistinguishability of elementary particles (Bosons) Outline of Statistical Mechanics

3. Thus the type of distribution function depends upon whether the number of particles is constant or not whether particles are indistinguishable whether particles are quantum like or not The Distribution Function and DOS Definitions: f(E) is the probability that a state at energy E is occupied by a particle g(E)dE is the number of available states in the energy range E and E+dE Number of particles between E and E+dE is given by N(E)dE=f(E)g(E)dE The average value of any quantity α is given by In general α can be a function of the system coordinates qi (x,y,z for i=1,2,3) and pi (px,py,pz). For example the Hamiltonian of a system is in general,

4. Calculation of the 3D Density of states function We solve the 3D SWE for free particles confined in a box with dimensions x0,y0,z0.The TISE is then of the form The Boundary conditions imposed cause the values of kx, ky and kz to be quantized as there is confinement. Thus this results in quantization of energy: Only discrete values of energy are allowed. The allowed values are :

5. If we draw another sphere E+dE then the volume of the shell between these two spheres will be The number of quantum states found in this volume of momentum space is In momentum space, this Energy can be represented by a sphere, whose radius is given by For free particle, Therefore, Including corrections for spin, we see that since only the volume appears, the same DOS would be obtained for a 3D system of any shape.

6. N1 N2 The Maxwell Boltzmann Distribution E1 E2 E3 ..………….. EN Energy levels N1 N2 N3 ………………… NN # of particles N1E1 N2E2 N3E3 ……………. NNEN Energy in each level The distribution which has the maximum probability of occurrence is one which can be realized in a maximum number of statistically independent ways. This is analogous to putting numbered objects into a set of numbered containers. Start with two boxes #1, #2.Let us denote Q(N1,N2) as the number of statistically independent ways of putting N1+N2 objects in two boxes such that one of them contains N1 and the other N2 objects.

7. Suppose now that the second container is divided into 2 compartments containing v1 and v2 objects. Thus N2=v1+v2. We may think of this as having 3 distinct distributions of N1,V1 and V2 objects. In that case, Generalizing this expression, we get, Modeling DegeneracyNow if each of the containers were actually a group of containers, say gi, then there will be an additional giNi ways of distributing these Ni particles among the gi containers. Thus the total number of ways will be modified according as, We now apply, the other two assumptions, namely conservation of particles and energy. Method of Lagrangean multipliers:If one wants to find the maximum of f under the restriction that some other functions which remain constant, independent of the choice of n equations

8. Total of (n+2) equations. We have from the definition of Q Using Stirling’s approximation, we have: Then, This particular energy distribution obtained under the classical assumption of identifiable particles without considering Pauli principle, is called Maxwell Boltzmann Distribution.α, can be expressed in terms of total number of particles. Now applying the method of Lagrangean multipliers to lnQ we have, If the energy levels are packed very closely, we can replace the summation by an integral and so,

9. N1 N2 The Fermi Dirac Distribution Fermi Dirac Distribution:In this case since the particles are indistinguishable, there is only one way of distributing them among the two boxes. Therefore, Q(N1,N2)=1. Now along with this there are other constraints like the conservation of particles and Total energy. Consider now the i th energy level with degeneracy gi. For this level, the total #of ways of arranging the particles is: Now the Ni particles can have Ni! Permutations and yet not give rise to any new arrangement as they are indistinguishable. Therefore we have to divide the number of possible ways of distributing the particles by this amount. Now we impose the other restrictions like conservation of particles and total energy of the system and obtain the other two functions to apply the Lagrange method. Now we proceed in the standard fashion, by applying Stirling’s approximation to lnQ, and then using the method of Lagrange multipliers to maximize Q. Now if the energy levels crowd in a continuum, then The Fermi energy is a function of temperature and also depends upon the DOS of the system

10. The Bose Einstein Distribution Bose Einstein Distribution: Consider an array of Ni particles and (gi-1) partitions needed to divide them into gi groups. The number of ways of permuting Ni particles among gi levels equals the number of ways of permuting objects and partitions. i.e. (Ni+gi-1)!. Now the particles and the partitions are indistinguishable, the number of ways of permuting them is Now we impose the other restrictions like conservation of particles and total energy of the system and obtain the other two functions to apply the Lagrange method. Now we proceed in the standard fashion, by applying Stirling’s approximation to lnQ, and then using the method of Lagrange multipliers to maximize Q. The solution for the case where the total number of particles is not conserved, but only the energy is conserved can easily be obtained by setting α=0

11. Applications:Maxwell Boltzmann statistics of an ideal gas We now discuss the properties of an ideal gas of free particles, for which For this particular case, and the integral is evaluated as a gamma function as, To summarize, the MB distribution function for an ideal gas is The distribution of particle density with energy is given by Now the total internal energy of the system can be got as follows Thus the average energy of the particles of the system is given by,

12. Applications, contd :Our aim is to derive the equations of state for an ideal Boltzmann gas from the dynamical properties and from the distribution. For this purpose, we need to convert the energy distribution function into a velocity distribution, using the relation, ,for free particles and parabolic dispersion. Now we have, Since, We have, This distribution expresses the # of particles whose velocities lie in the range v and v+dv. The above derived distributions are equilibrium distributions : If we apply an external field, they will change shape. Applications Fermi Dirac Distribution 3D SYSTEM The DOS is given by The total internal energy of the system is given by, The Electron density is then

13. The average internal energy per particle is then, For energies much greater than the Ef, the FD Distribution can be written as At low temperatures, the Fermi Dirac Distribution may be represented as a sphere in Momentum space in which all or most of the states of energy less than Ef are filled , while those greater than Ef are empty. The equation of the sphere is If all the energies of the system are such that E-Ef>>KBT, the system is called Non degenerate and the FD system reduces to a MB system • Note on Fermi Energy: • As Temperature increases, . This means that under high temperature limit, the Fermi Dirac statistics reduces to Maxwell Boltzmann. • At lower temperatures, the above will occur in gases where the masses are large • For dense gas of very light particles (free electrons in a metal), the Fermi Energy is very large and the condition, E-Ef>>KBT is practically never satisfied. • In semiconductors, due to peculiar form of the DOS function, the MB distribution is virtually always a good approximation to the FD distribution Applications Fermi Dirac Distribution 2D SYSTEM The DOS is given by ,A being the area of the 2D container. The total electron number is given by, Sheet electron density