Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

PHYS 359 > Post Midterm > Flashcards

Post Midterm Flashcards

1
Q

Some examples of stuff we can study using stat mech

A
  • white dwarves and astrophysical plasmas
  • magnetizations and heat capacities in condensed matter systems
  • blood oxygenation
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Key formula for finding averages

A

Xavg = sum(states, s) x(s)p(s)

P(s) is probability of x being in state s

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Microcanonical ensemble

A
  • Formal description for a closed or isolated system

- no contact with outside world (all variables fixed)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Thermodynamic entropy of a closed system

A

S = k*log(omega)

Omega = number of accessible microstates

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Canonical ensemble

A
  • system that can exchange energy with a much larger system (thermal reservoir/bath)
  • reservoir is much larger than system such that reservoir variables are fixed (T)
  • treat system + reservoir as a closed system (can focus on reservoir)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Key equation for canonical ensemble

A

P(s2)/P(s1) = exp(-betaE2) / exp(-betaE1)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Partition for N independent systems

A
  • ex. Ideal gas

- Zn = (Z1)^n

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Grand canonical ensemble

A
  • system in contact with reservoir that can exchange both energy and particles
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Chemical potential

A
  • symbol is mu
  • particle flow from the reservoir in analogy to how the temperature characterizes the energy flow from the reservoir
  • denied as the change in energy of the system to add a new particle without changing system entropy
  • usually negative since adding an extra particle increases # microstates so you have to lower the energy to keep microstate number constant
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Thermodynamic relation (for deriving partition functions)

A

dU = TdS - pdV + mu*dN

  • set dV=0
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Key formula for the grand canonical ensemble

A

P(2)/P(1) = exp(-beta(E2-muN2)) / exp(-beta(E1-muN1))

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

How can we find the averages of any function of N, avg(f(N)) of the grand canonical ensemble?

A

Derivatives wrt mu, in the same way that derivatives wrt beta of the partition function lead to averages avg(f(E))

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

How is quantum stat mech different from classical?

A
  • identity of particles (fermion vs boson) is important

- need to anti-symmetrize or have no constraint

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

How do we consider quantum systems?

A
  • consider energy level as fundamental object (as opposed to a particle)
  • total energy of energy level is E = epsilon*N, where epsilon is the energy of the level
  • simplifies Gibbs factor for this energy level to exp(-beta(epsilon - mu)N)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Common procedure in quantum stat mech

A
  • sum distribution function over all energy levels while leaving mu unknown
  • then you can solve for mu given the total number of particles
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Adding interactions

A
  • lose independence of particles
  • weak interactions: expend partition function into smth that looks like independent particles * correction factor that’s close to 1
  • weakly interacting gas: cluster expansion
  • first term will dominate
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Adding interactions: quantum version

A
  • lose assumption that energy per particle of a state is unaffected by the number of particles in the state
  • this is way too hard for us
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Low T expansion

A

Expand in Boltzmann factors bc e(-beta*E) becomes small

- find largest term with some beta dependence

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Somerfeld expansion

A
  • integrate by parts to turn the fermi-Dirac distribution into its derivative (sharply peaked at Fermi energy)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Density of states

A

Number of states you’ll find per unit energy in a small interval of energies
- property of confinement (not particles in it - same for bosons and fermions)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

How do we find the density of states?

A
  • count in n space

- change to variables in epsilon using spectrum of the confining potential

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

How to build a mean-field approximation in 3 easy steps

A

1) assume all particles have identical behaviour
2) select one particle to analyze and derive its behaviour as a function of the other particles
3) using the assumption that your selected particle is identical to all others, derive a self-consistent equation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Ising model (MFT)

A
  • B = 0
    2) calculate expected magnetization of a single spin as a function of average neighbour spins
    3) set average magnetization equal to average magnetization of neighbours
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Gross-Pitaevskii equation (MFT)

A

1) assume product form for many-body wave function
2) consider single atom to find the energy shift it would feel from all other atoms
3) said our chosen particle should follow a schrodinger eqn with that extra energy, results in self-consistent description of wave function phi

25
Q

Diagrammatic expansion

A
  • for the cluster expansion
  • purpose is to organize a large collection of ugly integrals into a more intuitive fashion
  • dot labelled i stands for 1/V * integral dr^3
  • line labelled by i and j at ends stands for Mayer function fij = exp(-beta*u(int)(ri, rj))-1
  • shape of diagram encodes combinatorial and number-counting factors through symmetry factors and counting down N, N-1, … for the dots
26
Q

Energy spectrum of simple harmonic oscillator

A

E = hfn

27
Q

Interactions + Einstein solid

A

Debye model

- corrects low temperature behaviour while maintaining high temperature limit behaviour

28
Q

Paramagnet

A
  • baby’s first spin system
  • each spin is independent
  • spin is at lower energy when aligned with external magnetic field
29
Q

Paramagnet + interactions

A

Ising model

  • interactions are between spins
30
Q

Energy of classical ideal gas

A 
KE = p^2/2m
PE U(x), x is vector
31
Q

Partition function of classical ideal gas

A

Z(ext) = 1/h^3 * integral( exp(-beta(KE + PE)) dx^3 dp^3)

  • h in denominator to cancel units
  • Z = Z(ext)*Z(int)
32
Q

Energy spectrum for particle in a box

A

En = h^2/(8mL^2) n^2

33
Q

Maxwell-Boltzmann distribution

A
  • probability distribution for speeds (as opposed to velocities)
34
Q

Classical ideal gas + weak interactions

A

Cluster expansion

35
Q

Degenerate fermi gas

A
  • Pauli exclusion says no two identical fermions in the same state
  • at T=0 the fermions all stack up one per state until they’re all placed
36
Q

Fermi energy

A

Chemical potential at T=0

- large system: approximate as either highest filled energy level or lowest infilled energy level

37
Q

Degenerate fermi gas at low but nonzero temperatures

A
  • n(FD)(epsilon) is modified in a region around the Fermi energy with a width of roughly kT
38
Q

Fundamental relations that allow us to understand the degenerate fermi gas

A
  • N = integral( g(epsilon) n(FD)(epsilon) d(epsilon) )
  • U = integral( epsilong(epsilon)n(FD)(epsilon) d(epsilon) )
  • g is density of states
  • n(FD)(epsilon) is fermi-Dirac distribution
39
Q

Fermi-Dirac distribution at T=0

A

Step function with edge at the Fermi energy

- makes math easy!

40
Q

Energy of a degenerate fermi gas in a box

A

U = 3/5 N

41
Q

Fermi gas degeneracy pressure

A

P =- dU/dV (partials)

42
Q

What is degeneracy pressure important for?

A

Stability of white dwarves and neutron stars

43
Q

Important relations for a degenerate Bose gas

A
  • same as fermi gas but with Bose-Einstein distribution instead of FD
44
Q

Degenerate Bose gases at low temperatures

A
  • bosons prefer to occupy the same state
45
Q

Bose-Einstein condensate

A
  • set mu=0 in formula for N
  • solve for T -> this is critical temperature Tc
  • at T < Tc, chemical potential is 0
  • BEC is where a macroscopic number of particles all occupy the ground state of the potential
46
Q

Degenerate Bose gas + interactions

A
  • no longer an ideal gas

- mu of BEC becomes positive and depends on N

47
Q

Blackbody radiation

A
  • ideal gas of massless bosons
48
Q

Planck spectrum

A

u(epsilon) = d/d(epsilon) (U/V) (partials)

- tells us the energy density per unit energy of Blackbody radiation

49
Q

Energy relation for massless particles

A

E = pc

50
Q

Chemical potential for Blackbody radiation

A

Mu = 0 bc we can always add a massless particle of arbitrarily low energy to a system

51
Q

Stefan’s law

A
  • summarizes radiating of a Blackbody
  • P = sigmaeAT^4
  • sigma = Stefan-Boltzmann constant
  • A = surface area
  • e = enissivity (quantifies how close object is to being a perfect Blackbody)
52
Q

Emissivity of an object

A
  • specifies how close object is to being a perfect Blackbody
  • e=1 absorbs all radiation
  • e=0 reflects all radiation
  • can be a function of the frequency of the radiation (ex. Greenhouse effect)
53
Q

Ising model

A

Spins positioned at the points of a lattice

  • spins can point either up (s=1) or down (s=-1)
  • spins interact with neighbours (E = -epsilon * sum(neighbours) sisj)
  • energy minimized when spins aligned with each other
54
Q

Lattices considered in class (ising model)

A
  • chain
  • square
  • cube
  • 1D, 2D, 3D
55
Q

Ising model in 1D

A

Solve directly by iteratively considering the last spin in the chain and finding the sum over its possible states

56
Q

Ising model in 2+D

A

Use mean field approximation to calculate critical temperature at which system makes a transition from disordered to oriented (magnetized)

57
Q

Ising model applications

A
  • explains ferromagnetic materials qualitatively
  • idea that a local preference for aligning can lead to a transition from disordered to ordered phase has lead to ising model appearing in physics and in biology and in social studies (very broad)
58
Q

Stirling’s approximation

A

N! ~ N^N * e^-N

PHYS 359 (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions