PHYS 172: Modern Mechanics Lecture 24 – The Boltzmann Distribution Read 12.7 Summer 2012

Summary: Foundations Einstein Model of Solids # microstates (N oscillators, q quanta) Fundamental assumption of statistical mechanics Over time, an isolated system in a given macrostate (total energy) is equally likely to be found in any of its microstates (microscopic distribution of energy).

If the initial state is not the most probable, energy is exchanged until the most probable distribution is reached. Summary: Entropy and Temperature

Summary: Specific Heat Al Pb

Derivation: Choose Your System Wisely Today: The Boltzmann Distribution Simple Applications: Many Systems or Many Observations? Probabilities for Atomic Excitations Block of Lead Biological Physics

Boltzmann Distribution example: a macroscopic volume of water in a glass The Boltzmann distribution comes about by cleverly picking our system. Say that we want to analyze a cup of water. What system do we pick? A tiny volume of the water! Ω, E Ω res, E res E tot = E res + E = constant

CLICKER QUESTION q1q1 q2q2 Ω1Ω1 Ω2Ω2 Ω 1 Ω atom 1 (3 oscillators) atom 2 (3 oscillators) For a macrostate of 4 quanta distributed among 2 atoms (6 oscillators), what is the probability that the first atom has 1 quanta? A) 1/2B) 3/30C) 30/126 D) 3/126E) 60/126 Total = 126

Recall the Pb Nanoparticle (3 atoms) from Ch 11, p : q ln S k B k B k B qT(K) ABC What if we wanted to know what is happening on one of the three atoms? For example, with q=6, how often is atom A found with q A =2?

We can simply count the states with 2 quanta on atom A: What if we wanted to know what is happening on one of the three atoms? For example, with q=6, how often is atom A found with q A =2? ABC

Now, a more interesting question: What is the probability of finding 2 quanta on atom A if it is in contact with a large Pb block? (Assume same T as before.) This is a property of any small system in thermal equilibrium with a large reservoir at a fixed temperature.

We are now going to show WHY this is true

VIEWER WARNING! The following slides may contain Derivations or Derivation Byproducts. Do not operate heavy machinery while viewing the following slides. Also, do not go swimming for at least one hour after viewing these slides. Derivations and Derivation Byproducts are known to the State of Indiana to have the following possible side effects: Furrowed Brows, Increased Logical Abilities, and in rare cases, Brain Explosions.

Probabilities “Reservoir” A ginormous system with a constant temperature tiny system WARNING! This slide may contain Derivations or Derivation Byproducts.

Probabilities 0 E E tot (E tot - E) What is the probability of finding the system in this state? Reservoir (Ginormous) System (Tiny) WARNING! This slide may contain Derivations or Derivation Byproducts. Reservoir (Ginormous) System (Tiny)

Probabilities E (E tot - E) WARNING! This slide may contain Derivations or Derivation Byproducts. We can find the probability from the number of ways of arranging the energy. Reservoir (Ginormous) System (Tiny) Total number of ways of arranging energy in combined system: Tot = Res + Sys Number of ways of having E sys = E

Now Do Some Math... WARNING! This slide may contain Derivations or Derivation Byproducts. We can find the probability from the number of ways of arranging the energy.

More Math... WARNING! This slide may contain Derivations or Derivation Byproducts. Spot The Temperature! Use a Taylor Expansion for the Ginormous Reservoir: (E << E res )

More Math... WARNING! This slide may contain Derivations or Derivation Byproducts. Use a Taylor Expansion for the Ginormous Reservoir: (E << E res ) Spot The Temperature!

More Math... WARNING! This slide may contain Derivations or Derivation Byproducts. Exponentiate Everything... BOLTZMANN FACTOR Very very important!

VIEWER WARNING! The previous slides did contain Derivations and Derivation Byproducts. So, you should not go swimming for at least one hour after class. The Derivation is now OVER. We now return you to your regularly scheduled physics learning experience. Also known as lecture.

Boltzmann Distribution The probability of finding energy E in a small system in contact with a large reservoir is The exponential part, e -E/kT, is called the “Boltzmann factor.” Ω(E) is the number of microstates in the small system at energy E. In many circumstances, Ω(E) changes so slowly compared to e -E/kT that it is essentially constant: Ω, E Ω res, E res We just showed WHY this is true

Boltzmann Distribution The Boltzmann distribution comes about by cleverly picking our system. Say that we want to analyze a cup of water. What system do we pick? A tiny volume of the water! Ω, E Ω res, E res E tot = E res + E = constant Example: a macroscopic volume of water in a glass of water

Application: Atomic Excitations How likely is an atom to be in 1 st excited, compared to odds of being in ground state? NOTE: kT at room temp = 1/40 eV. For the above atom, odds of being in first excited state are Typical atomic energy gaps are big compared to room temp. A room-temperature box of neon doesn’t glow! (Unless you add energy).

Application: Block of Lead BOLTZMANN FACTOR Very very important! At room temperature, the Boltzmann factors for exciting vibrations are: T=300K at room temperature Think of adding q quanta of energy to one lead atom... Most probable state Watch out! These are relative probabilities

Application: Block of Lead The actual probabilities use a normalization factor “Z” (partition function). T=300K at room temperature Think of adding q quanta of vibrational energy to one lead atom... Most likely to find the vibrations are excited. On average, q=5.

Derivation: Choose Your System Wisely Today: The Boltzmann Distribution Simple Applications: Many Systems or Many Observations? Probabilities for Atomic Excitations Block of Lead Biological Physics

Speed Distribution in a Gas Next Time: Boltzmann Applications Energy Equipartition and Specific Heat Pressure and the Ideal Gas Law