Week 4 (16) Flashcards
why is using the taylor expansion especially helpful when working out probabilities?
As long as the reservoir is sufficiently large, we can neglect the terms in E_i^2 and higher
how do you normalise the probabilities of being in an energy state?
The sum of all the probabilities of being in each state must be equal to 1
What is Z, the partition function?
Z is the sum of the states. (not just energy levels)
What is degeneracy?
How does it affect the partition function?
When different states can have the same energy levels (i.e different configurations but result in the same energy)
If there are degenerate states, then the partition function needs to include the degeneracy factor.
What is the probability of being in a particular state for a microcanonical system?
The propability is the same for all micro states.
P_i ∝ 1 ∕ Ω_i
What is the probability of being in a particular state for a Canonical ensemble system?
The probability is given by the Boltzmann factor
What is the general expression for the entropy, related to the probability of being in a particular state?
S = Boltzmann - Gibbs entropy
= -k_B Sum,i(P_i * ln(P_i))
For microcanonical distribution, reduces to (as only 1 state):
S = k_B ln(Ω)
What is the classical equipartition theorem?
Each degree of freedom of a classical system adds 1/2 k_BT to the total energy
- limit, as long as the DoF adds an extra term to the energy which varies as the variable squared (e.g 1/2 mv^2)
When a system has two+ contributions to the energy, how do you combine partition functions, Z?
The energy is the sum of the sums for energies of both variables. Subbing in E_i,j = E_i^x + E_j^y instead of E, across the sum of i and j, results in:
Z = Z_x*Z_y
so the partition functions from independent contributions multiply
How do you calculate the internal energy, from the energies?
U =
How do you calculate the average of a variable, x, using distribution functions?
If X is discrete values only:
= Sum,i(X_i*P_i)
If x is continuous:
= int,-∞, ∞ (x*P(x))
