Term 1

Question 1

What is a Bernoulli trial?

Answer 1

An experiment with 2 outcomes

Question 2

What is the equation for the Binomial Probability distribution?

Answer 2

ⁿC_kP^k(1-P)^n-k

where ⁿC_k = n!/k!(n-k!)

Question 3

What is the standard deviation for the Binomial Distribution?

Answer 3

Std. Dev. = (np(1-p))^1/2

Question 4

What is Bayes Theorem?

Answer 4

P(A|B) = P(B|A) x P(A)
P(B)

For dependent events

For Independent events, P(A|B) = P(A) P(B)

Question 5

Please state the 3 main laws of Thermodynamics

Answer 5

1. dU = TdS - PdV
2. dS >= 0
3. S –> 0 as T–> 0

Question 6

What are the approaches to Thermal Physics used in Thermodynamics?

Answer 6

• assumes thermal equilibrium
• 4 laws relating P, T, U, S & V
• Laws are empirical & justified experimentally

Question 7

What are the approaches to Thermal Physics used in Statistical Mechanics?

Answer 7

• Defines themal equilibrium and entropy in terms of microscopic probabilities.
• Gives the probability of a particular state.
• Determines measurable properities using averaged over states.

Question 8

Microstates Vs Macrostates

Answer 8

Micro:
- Microscopic configuration of the system.
- Very difficult / impossible to measure for large systems.

Macro:
- Any configuration with particular measurable properties.
- Each macrostate can correspond to very many microstates.

Question 9

What are the 3 main assumptions of Statistical Mechanics?

Answer 9

1. In thermal equilibrium, each possible microstate is equally likely (isolated system).
2. Microstate is constantly changing.
3. Ergodic hypothesis: Given enough time a system will explore all possible microstates and spend equal time in each.

Question 10

What are the consequences of the assumptions of S.M.?

Answer 10

• Systems choose the macrostate with the most number of microstates.
• The measured value of an observable is equal to the average over all accessible microstates.

Question 11

What are the three types of ensemble? Give a brief explanation of each

Answer 11

• Microcanomical: Isolated system, energy E.
• Canomical: Exchanges energy with a large reservoir –> fixes T.
• Grand - Canomical: Exchanges energy & particles with a large reservoir –> fixes T & μ, (Chemical Potential)

Question 12

What is the equation for Entropy in S.M.?

Answer 12

S = k_Bln(Ω) = -k_BΣ_iP_iln(P_i)

Ω is the number of microstates in the given macrostate

Question 13

Thermodynamics Vs Statistical Mechanics: 2nd Law

Answer 13

• T.D.: Isolated system in equilibrium is a state of maximum entropy.
• S.M.: Isolated system in equilibrium will adopt the state with the highest number of microstates.

Question 14

What is the Statistical definition of Temperature?

Answer 14

1/k_BT = dln(Ω)/dE –> 1/T = dS/dE

Question 15

Please give the equation for the Boltzmann distribution

Answer 15

P(E_i) = e^-βE_i/z

Question 16

Please give the equation for the partition function, z.

Answer 16

z = Σ_ie^-βE_i

Question 17

Please state the Equipartition Theorem

Answer 17

For a system in contact with a reservoir at temperature, T, each independent quadratic energy mode contributes 1/2 k_BT to the average energy.
C_V = (dU/dT)_V = n/2 k_B

Question 18

Please state the 4 functions of state in T.D.

Answer 18

• Internal Energy: dU = TdS-pdV
• Helmholtz free energy: F = U - TS
• Enthalpy: H = U + pV
• Gibbs Free Energy: G = H - TS

Question 19

Please state the 4 functions of state in S.M.

Answer 19

• Internal Energy: U = - d(ln(z))/dβ = Σ_i E_i P_i
• Entropy: S = k_B(βU + ln(z)) = -k_B Σ_iP_iln(P_i)
• Helmholtz free energy: F = -k_BTln(z)
• Specific heat Capacity: C_V = k_B(βΔ/2)²sech²(βΔ/2)

for a system of 2 energy levels Δ/2 and - Δ/2

Question 20

Please give the definition of degeneracy

Answer 20

When a quatised system has two or more states with the same energy

Question 21

Please give the new partition function, accounting for degeneracy

Answer 21

z = Σ_ig(E_i) e^-βE_i

where g(E_i) is the number of distinct states associated with energy level, i.

Question 22

Please give the equation for average Energy and standard deviation

Answer 22

• ⟨E⟩ = 1/z Σ_iE_ie^-βE_i = - 1/z dz/dβ
• σ_E = T√k_BC_V
• σ_E/⟨E⟩ ∝ 1/⟨N⟩

Question 23

Please define Intensive and Extensive properties of a system

Answer 23

• Extensive: scales with the system, e.g. U
• Intensive: doesn’t scale with the system e.g. T