Autumn Statistical

Question 1

How many ways are there of choosing n from N distinct objects?

Answer 1

Question 2

What is C(N,n)?

Answer 2

The number of ways to choose n from N distinct objects

Question 3

What is a macrostate?

Answer 3

a thermodynamic state of a system defined by macroscopic functions of state e.g. temp, pressure etc

Question 4

What is a microstate?

Answer 4

A quantum state of our system, specifying all relevant details about all particles e.g. position, velocity, spin

Question 5

What does PEAPP stand for?

Answer 5

Principle of Equal A Priori Probability

Question 6

What is the PEAPP?

Answer 6

When an isolated system reaches equilibrium, all microstates accessible to it are equally probable

Question 7

How is entropy defined statistically?

Answer 7

S = k_BlnP where P is probability

Question 8

How is ΔS defined statistically? (two ways to write constants)

Answer 8

ΔS = Nk_BlnP1/P2 or = nRln2#P1/P2

Question 9

What is S=k_Bln2 also known as?

Answer 9

Boltzmann’s entropy process

Question 10

How is S written in terms of W (statistical) what is W?

Answer 10

(entropy of a macrostate) S = k_blnW
where W is number of microstates available for that macrostate

Question 11

If a macrostate has a larger W, it is…?

Answer 11

More probable

Question 12

What is W?

Answer 12

Number of microstates in a macrostate

Question 13

How are P and W related?

Answer 13

Proportional, probability is proportional to number of microstates

Question 14

What W should we seek to find equilibrium state? Why?

Answer 14

The maximum, because the equilibrium state is the most probable.

Question 15

What is the equation for statistical temperature?

Answer 15

Question 16

What is Stirling’s approximation?

Answer 16

Question 17

How is Stirling’s approximation derived?

Answer 17

Question 18

What is the law of large numbers?

Answer 18

Question 19

What does the Law of large numbers effectively mean (qualitatively)?

Answer 19

Uncertainties/fluctuations are essentially supressed by large numbers

Question 20

What is the central limit theorem?

Answer 20

Sum of independent variables tends to a normal distribution even if the original variables are not normally distributed

Question 21

Define an ensemble.

Answer 21

a collection of identical systems, whose statistical fluctuations are cancelled out upon averaging

Question 22

What is the micro-canonical ensemble?

Answer 22

Systems with fixed number of particles N, total energy E, and volume V

Question 23

What is the Canonical ensemble?

Answer 23

Systems with fixed V, N and T as they are in contact with a heat bath

Question 24

By the Boltzmann distribution, what is the probability of finding a system in a particular microstate of the canonical ensemble is?

Answer 24

Question 25

What are the 'differences between PEAPP and Boltzmann?

Answer 25

PEAPP: equal distribution, for the **micro-canonical ensemble**, counts all microstates including the heat bath Boltzmann: exp distribution, for the **canonical ensemble**, counts probability of the system only not the heat bath

Question 26

What is Gibb's entropy? Why might it be used over Boltzmann's?

Answer 26

Gibbs applies to both canonical and micro-canonical ensembles so is more general

Question 27

What is Shannon's theorem?

Answer 27

the equation for (when maximised) the most likely distribution of a set of mutually exclusive probabilities p_i

Question 28

For a two level system with possible energies 0 or epsilon, how are probabilities of the atom being in either level found using the Boltzmann distribution?

Answer 28

Question 29

For the two level system, how can the number of particles on a level be found using probability from the Boltzmann distribution?

Answer 29

Question 30

What 'type' of ensemble does Boltzmann distribution use?

Answer 30

Canonical ensemble

Question 31

For the two level system, we calculate probability of energies using the Boltzmann system so we use the __________ formula for the entropy.

Answer 31

Gibbs

Question 32

How does Z relate to F?

Answer 32