Autumn Statistical Flashcards
How many ways are there of choosing n from N distinct objects?
What is C(N,n)?
The number of ways to choose n from N distinct objects
What is a macrostate?
a thermodynamic state of a system defined by macroscopic functions of state e.g. temp, pressure etc
What is a microstate?
A quantum state of our system, specifying all relevant details about all particles e.g. position, velocity, spin
What does PEAPP stand for?
Principle of Equal A Priori Probability
What is the PEAPP?
When an isolated system reaches equilibrium, all microstates accessible to it are equally probable
How is entropy defined statistically?
S = kBlnP where P is probability
How is ΔS defined statistically? (two ways to write constants)
ΔS = NkBlnP1/P2 or = nRln2#P1/P2
What is S=kBln2 also known as?
Boltzmann’s entropy process
How is S written in terms of W (statistical) what is W?
(entropy of a macrostate) S = kblnW
where W is number of microstates available for that macrostate
If a macrostate has a larger W, it is…?
More probable
What is W?
Number of microstates in a macrostate
How are P and W related?
Proportional, probability is proportional to number of microstates
What W should we seek to find equilibrium state? Why?
The maximum, because the equilibrium state is the most probable.
What is the equation for statistical temperature?
What is Stirling’s approximation?
How is Stirling’s approximation derived?
What is the law of large numbers?
What does the Law of large numbers effectively mean (qualitatively)?
Uncertainties/fluctuations are essentially supressed by large numbers
What is the central limit theorem?
Sum of independent variables tends to a normal distribution even if the original variables are not normally distributed
Define an ensemble.
a collection of identical systems, whose statistical fluctuations are cancelled out upon averaging
What is the micro-canonical ensemble?
Systems with fixed number of particles N, total energy E, and volume V
What is the Canonical ensemble?
Systems with fixed V, N and T as they are in contact with a heat bath
By the Boltzmann distribution, what is the probability of finding a system in a particular microstate of the canonical ensemble is?
What are the ‘differences between PEAPP and Boltzmann?
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
What is Gibb’s entropy? Why might it be used over Boltzmann’s?
Gibbs applies to both canonical and micro-canonical ensembles so is more general
What is Shannon’s theorem?
the equation for (when maximised) the most likely distribution of a set of mutually exclusive probabilities pi
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?
For the two level system, how can the number of particles on a level be found using probability from the Boltzmann distribution?
What ‘type’ of ensemble does Boltzmann distribution use?
Canonical ensemble
For the two level system, we calculate probability of energies using the Boltzmann system so we use the __________ formula for the entropy.
Gibbs
How does Z relate to F?
