11-19-25 Einstein Model of a Solid

Irreversibility

  • Many physical processes appear to only happen in one direction in time

    • Example: Ice left in the open in a warm room melts as energy from the warmer surroundings is spontaneously transferred to the ice. We never observe water left in a warm room losing energy to the surroundings and spontaneously freezing

  • Process that only happen in one direction are called irreversible

  • Irreversibility occurs when a process is much more likely to happen in one direction than it is to happen in the reverse direction. even though both directions are possible

  • The fundamental interactions are all reversible

Einstein Model of a Solid

  • In the Einstein model of a solid, each atom is connected to the rest of the solid by three spring-like interatomic bonds. Each of these bonds can be modeled as a quantum oscillator, with quantized energy levels.

Microstate vs. Macrostate

  • The total energy in collection of bonds specifies the macrostate. Each distinct way of distributing that total amount of energy among the bonds is called a microstate. 

  • Example: N=3N=3 oscillators with q=4q=4 total quanta of energy

Fundamental Assumption of Statistical Mechanics

  • Given enough time, every microstate of a system is as equally likely to appear as any other

  • Some macrostates contain more microstates than others, which makes the system more likely to be in them

  • Example: Microstates A-D are equally likely. The macrostate {system looks kind of like A} is more likely than the macrostate {system looks kind of like C}

Consider 4 quantized oscillators that share 3 quanta of energy. List all the ways you can arrange these 3 quanta of energy among the 4 oscillators. How many arrangements are there?

N=4N=4, q=3q=3

20 total variations

(3)()()()(3)()()() ()(3)()()()(3)()() ()()(3)()()()(3)() ()()()(3)()()()(3)

(2)(1)()()(2)(1)()() (2)()(1)()(2)()(1)() (2)()()(1)(2)()()(1) 

(1)(2)()()(1)(2)()() ()(2)(1)()()(2)(1)() ()(2)()(1)()(2)()(1) 

(1)()(2)()(1)()(2)() ()(1)(2)()()(1)(2)() ()()(2)(1)()()(2)(1) 

(1)()()(2)(1)()()(2) ()(1)()(2)()(1)()(2) ()()(1)(2)()()(1)(2) 

(1)(1)(1)(1)(1)(1)(1)(1) (1)(1)()(1)(1)(1)()(1) (1)()(1)(1)(1)()(1)(1) ()(1)(1)(1)()(1)(1)(1)

Counting Microstates

  • To find the total number of arrangements, Ω\Omega, of some total number of quanta energy, q , among some number of oscillators, NN, we can use the following combinatorial formula:

Ω(q,N)=(q+N−1)!q!(N−1)!\Omega (q,N) = \frac {(q+N-1)!}{q!(N-1)!} where n!=n⋅(n−1)⋅(n−2)…3⋅2⋅1n! = n\cdot(n-1)\cdot (n-2)… 3\cdot 2\cdot 1

Ex:Ex: Ω(3,4)=6!3!3!=6×5×4×3×2×1(3×2×1)(3×2×1)=20\Omega(3,4) = \frac {6!}{3!3!} = \frac {6×5×4×3×2×1}{(3×2×1)(3×2×1)}=20

If q=300q=300 and N=100N=100 what is the value of Ω\Omega? 

(300+99)!300!(99!)\frac {(300 + 99)!}{300!(99!)} == 399!300!99!=5.605×1095\frac {399!}{300!99!} = 5.605 × 10^{95}