Temperature and Thermal Energy

• Class is now in Agyros Forum 202.

Office Hours

• Mondays 4pm-5pm
• Wednesdays 9am-10am
• Killefer Conference Room: https://maps.app.goo.gl/FJGbZzKD7XNxPYrR8
• Reading quiz due on Wednesday at noon for T3 chapter, for Thursday

Temperature and Thermal Energy

• Increasing temperature increases thermal energy
• Small changes result in a linear relationship: dU = mcdT or c = 1/m * dU/dT
• Change in thermal energy is proportional to mass
• Specific heat (c) changes slowly with temperature and depends on the system's composition

Example Calculation

• A 100g lead pipe's temperature increases from 25°C to 37°C
• The specific heat of lead is 0.128 J/g·K (approximately constant)
• Thermal energy increase is calculated: ΔU = mcΔT
• Temperature differences are the same in Kelvin and Celsius
• ΔU = (100 g) * (0.128 J/g·K) * (37-25) K = 153.6 J

Specific Heat Formula

• Specific heat is usually determined empirically
• For simple systems, statistical mechanics can be used
• A formula can be used: c ≈ (nNkB)/(m*2)
• Boltzmann's constant is kB = 1.38 x 10^-23 J/K
• N is the number of molecules in the system
• m is the mass of the system
• n is an integer depending on the system, approximately 6 for solids or 3 for low density gas at room temperature

Combining Formulas

• Combining dU = mcdT, c ≡ 1/m * dU/dT, and c ≈ (nNkB)/(m*2)
• Results in dU ≈ (n/2 * kB) * N dT
• In turn du/dT ≈ nN(kB/2)
• This can be simpler to work with than specific heat

Einstein Model of a Solid

• Statistical mechanics simplifies interacting systems
• Neglecting interactions simplifies the dynamics
• Each atom oscillates independently
• Atoms only oscillate a short distance
• Treating them as harmonic oscillators is a good approximation

PHYS 101 Concepts

• Potential energy of a harmonic oscillator is ½ksr²
• Angular frequency of a harmonic oscillator is ω = √(ks/m)

Energy in the Einstein Model

• The total energy of a single atom: E = Kinetic Energy + Potential Energy

• Given by the equation: E = ½m|v|^2 + ½ks|r|^2 = ½m(vx² + vy² + vz²) + ½ks(x² + y² + z²)

• Which then expands: (½mvx² + ½ksx²) + (½mvy² + ½ksy²) + (½mvz² + ½ksz²)

• This is equivalent to three independent 1D harmonic oscillators

• In total, 3N independent 1D oscillators exist with 3 for each noninteracting atom

Quantum Mechanics

• The textbook uses quantum mechanics, but not studied later in the course
• The energy levels is discrete
• Equation dictates energy: E = (n + ½)ħω
• Planck's constant is denoted by: ħ = 1.055 × 10-34 J s^-1

Need for Quantum Mechanics

• Discretizing the system (coarse-grain)
• It is necessary to count the energy distributions between 3N oscillators
• The possible energies of an oscillator must form a continuum unless you use many ways of doing this
• So bin the energies and call it the energy to do work

Coarse-Graining Scale

• ⅓ ∈ if 0 ≤ E < ∈

• 3/2 ∈ if ∈ ≤ E ≤ 2∈

• 5/2 ∈ if 2∈ ≤ E ≤ 3∈

• (n + ½) ∈ if n∈ ≤ E ≤ (n + 1)∈

• Providing a tiny amount of energy compared to the total energy of the solid does not make a big difference

• In fact, you can often take the limit e → 0 at the end of the calculation

• It is an alternative motivation for using a discrete set of energies

• QM suggests energy scale where: ∈ = ħω

• But it isnt needed, we need to discretize energy

• Quantum systems dont need to be in states that have a definite energy value so you can count states by counting the number of justification

Total Energy

• There are no interactions with the total energy of the solid which is the sum of the energies of the independent oscillators
• Etot = Σ (ni + ½) € = Σ ni€ + Σ ½ = Σ ni€ +
• If we are using discrete values for the energy equation

Meaningful Changes

• Only changes to energy are physically meaningful
• Assuming that the number of atoms number is fixed for arbitrary potential
• The equation 3𝑁/2 e is a constant
• E tot = Σn i e

Macrostates and Microstates Distinction

• Microstate is the specification of a state
• The system must specify physical properties to measure and calculate
• A Gas contains position & velocity of every single molecule
• The total energy only gives total energy
• Magnets go in different directions which is its microstate

Macrostate specification

• Is the bulk properties of a system (the things directly experienced)
• Interested in using statistical model to calculate them
• Gas Macrostate: Pressure Volume Temperature
• Einstein Solid macrostate: Total Energy
• Magnet Macrostate: Total magnetization
• In many cases, properties arent independent
• e.g.: temp determined by pressure & volume
• Macrostate is determined when given its pressure and volume

Multiplicity

• Huge number of microstates that correspond to each macrostate
• Multiplicity Ω(𝑀) of a microstate 𝑀: the number of microstates that correspond to the macrostate
• Einstein solid with total energy number, and the number of atoms
• Omega(U,N) = # of ways distriburing energy (U) among 3N oscillators

Computing Ω(𝑈, 𝑁)

• q = U/∈. This is how many unit of our energy scale we possess at the total energy of U
• The Ω(𝑈, 𝑁) is how many ways you distribute q pieces of energy among 3 𝑁 oscillators.
• This is determined by combinatorics

The general idea

• Distribute k objects among (n) slots
• k = q and n = 3 N

Combinatorics

• Suppose there are n objects, and in the example 1,2,3:
• 1,2,3 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1 - which are 6 possible ways
• 3 numbers to choose, and it goes down from there
• So the total number of ways to choose 3 numbers is 3 x 2 x 1 = 3! = 6.

Combinatorics Background Cont.

• Number of ordering n objects n! = n × (n - 1) × (n − 1) x･･･ × 2 × 1
• The goal is the total number of possible ways distributing k objects among n slots.
• Mark the division between two slots = |
• Mark an object = *
• There are n - 1 = 2 bars and k stars
• So a total of 7 symbols

The number of objects

• (n + k − 1): ways and if you
• Swapping doesn't change
• So the stars also dont change
• (n 1)! (n 1)! k!

Einstein Solid Multiplicity

• (𝑛 + 𝑘 − 1)! (𝑛 − 1)! 𝑘!

• Omega Function: (3𝑁 + 𝑞 − 1)! (3𝑁 − 1)! 𝑞!

Thermal Contact

• There exists two subsystems 𝐴 and 𝐵
• These have Ω𝐴 and Ω𝐵 for shorthand
• total energy: Ω𝐴𝐵 = Ω𝐴 × Ω𝐵

Macropartitions

• We composed of subsystems 𝐴 and 𝐵
• macrostate of 𝐴𝐵 will be designated as 𝑀𝐴𝐵
• macropartition is a specification for the state 𝐴 and 𝐵
• So (𝑈𝐴 , 𝑁𝐴 ) that is (𝑈𝐵 , 𝑁𝐵

Equations for macropartitions

• if U = UA + UB and N = NA + NB 3𝑁+𝑞−1 !
• Ω(U, 𝑁) = (3𝑁−1)! 𝑞| we can calculate the number of microstates that are compatible with a macropartition

Fundamental Assumption

• A system’s accessible microstates are the microstates that its marcropartition supports
• An isolated system’s accessible microstates are all equally likely in the long run
• Initial conditions in nature lead to dynamics causing a system to spend equal time in its microstates