Thermo & Stat Mech - Spring 2006 Class 17 1 Thermodynamics and Statistical Mechanics Entropy

Thermo & Stat Mech - Spring 2006 Class 172 Thermodynamic Probability

Thermo & Stat Mech - Spring 2006 Class 173 Distribution N = 4U = 3  k123 3  2  1   w4124

Thermo & Stat Mech - Spring 2006 Class 174 Combining Systems Consider two systems. System A: Number of arrangements: w A System B: Number of arrangements: w B Combined systems: w A × w B

Thermo & Stat Mech - Spring 2006 Class 175 Entropy S = k ln w S A = k ln w A S B = k ln w B S A+B = k ln(w A × w B ) = k ln w A + k ln w B S A+B = S A + S B

Thermo & Stat Mech - Spring 2006 Class 176 Wave Equation

Thermo & Stat Mech - Spring 2006 Class 177 Boundary Conditions

Thermo & Stat Mech - Spring 2006 Class 178 Energy of Particles

Thermo & Stat Mech - Spring 2006 Class 179 Density of States The allowed values of k can be plotted in k space, and form a three dimensional cubic lattice. From this picture, we can see that each allowed state occupies a volume of k space equal to,

Thermo & Stat Mech - Spring 2006 Class 1710 Density of States All the values of k that have the same magnitude fall on the surface of one octant of a sphere in k space, since n x, n y, and n z are positive. The volume of that octant is given by,

Thermo & Stat Mech - Spring 2006 Class 1711 Density of States Then, the volume of a shell that extends from k to k + dk can be obtained by differentiating the expression for V k,

Thermo & Stat Mech - Spring 2006 Class 1712 Density of States If we divide this expression by the volume occupied by one state, we will have an expression for the number of states between k and k + dk.

Thermo & Stat Mech - Spring 2006 Class 1713 Density of States  is the number of states with the same k, or the number of particles that one k can hold.

Thermo & Stat Mech - Spring 2006 Class 1714 Density of States In terms of energy of a particle:

Thermo & Stat Mech - Spring 2006 Class 1715 Free Electrons