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1
Q

State principle of equal a priori probabilities

A

At equilibrium, an isolated system can be found in any of its microstates with equal probability.

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2
Q

What is the macrostate?

A

A collection of microstates with a common property; same value of that property (combinations of boxes with a common property).

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3
Q

Define temperature as a partial derivative

A

Use fundamental relation of thermodynamics (1st law with chemical potential).

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4
Q

What is meant by microstate?

A

A description of a system where all values of the dynamical variables are exactly specified

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5
Q

Why fluctuations in system properties are rarely observed in macroscopic systems?

A
  • standard deviation about modal macrostate goes like 1/Sqrt(N)
    • so as N-> infinity, standard dev. ->0
    • hence system rarely strays from the modal macrostate
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6
Q

What is meant by an irreversible process?

A
  • System and surroundings cannot be returned to original conditions when process is reversed
    • due to WD against dissipative forces
  • Entropy is a state function, so the system entropy change is zero, although not of the surroundings (not of the surroundings what?
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7
Q

Boltzmann entropy of a system

A
  • S = klnΩ
  • Ω = number of microstates of a macrostate (i.e. microstate multiplicity)
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8
Q

Gibbs entropy

A
  • Pi is prob. of occupation of a macrostate
  • generalisation of Boltzmann entropy
    • Boltzmann only applies to isolated systems where energy is conserved
    • e.g. microcanonical ensemble
    • or the total sum of canonical & grand canonical ensembles i.e. the reservoir
  • Gibbs -> Boltzmann
    • pi = 1/Ω for all in i
      • hence SB = klnΩ
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9
Q

What does the Fermi-Dirac function represent?

A
  • describes a distribution of particles over energy states in systems consisting of many identical particles
  • particles obey the Pauli exclusion principle (half integer spin in thermodynamics eqm)
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10
Q

Conditions between system and environment that allow us to employ a microcanonical ensemble

A
  • System is isolated with environment
  • hence system cannot exchange energy or particles with its environment
    • energy of system remains exactly known as time goes on
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11
Q

Conditions between system and environment that allow us to employ a grand canonical ensemble

A
  • System can exchange energy particles
  • hence states of system can differ in both total energy E & total number of particles N
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12
Q

What is a degenerate Fermi gas?

A
  • Single particle states are occupied with probability 1 up to max. single particle energy (EF = Fermi energy)
  • whilst all single particle states with E > EF are empty
  • All energy levels from 0 to EF are fully occupied
  • occurs at T=0 & approximately kT << EF
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13
Q

What happens when the temperature is much less than the Fermi temperature?

A
  • Degenerate Fermi gas is formed when T << TF (TF = fermi temp. =EF/k)
  • and also formed at absolute zero.
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14
Q

State the canonical partition function

A
  • s = index for microstates of the system
  • β = 1/kT
  • T = temp. of reservoir
  • Es = total energy of system in the corresponding microstate
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15
Q

What is the canonical partition function of N indistinguishable particles given single particle state Z1?

A
  • ZN = Z1N/N!
    • is the exception for identical particles
  • generally the parition fn of an entire system is the product of the individual parition fns
    • so if subsystems have same physical properties then
      • ZN = Z1N <=> lnZN = NlnZ1
    • non-interacting & distinguishable
    • more generally
    • Z = product of ZJ <=> lnZ = sum of ln ZJ
      • for Debye model, the sum with ln ZJ is integral over frequencies
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16
Q

What is meant by thermal equilibrium?

A
  • 2 systems are in thermal equilibrium if they are in thermal contact
  • i.e. have the same temperature.
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17
Q

What is the meaning of extenstivity? Which state variables in the expression for Gibbs free energy G = E + pV -TS are extensive?

A
  • Extensive variable depend on N
  • hence is proportional to the amount of the substance
  • State functions are extensive
    • so E, V and S are extensive
    • T, p, μ are intensive
      • since they stay constant with changing N
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18
Q

What is dG? And hence show that G = μN

A
  • Gibbs free energy is the “free” energy available in a system to do useful work
  • related to chem. pot. μ=dG/dN
  • if μ is non-zero,
    • particles will enter or leave system
    • so G changes
    • this can be shown by using 1st law at const. T and p.
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19
Q

What is meant by a particle bath?

A
  • a surrounding reservoir in which the chemical potential is constant
  • so there maybe exchange of particles in the system with the surroundings (reservoir)
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20
Q

Give an example of physical property of a degenerate fermi gas that is different from classical expectation.

A

.

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21
Q

What is the microstate multiplicity if Q indistinguishable objects are shared between N distinguishable boxes? What is the multiplicity if the boxes are indistinguishable?

A
  • number of ways of placing Q indistinguishable items in N distinguishable boxes
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22
Q

Why is it important to take into account particle indistinguishability when modelling stats therm properties of a system?

A
  • All permutations of the particles give the same state of the system
    • so leads to over-counting and degeneracy**
    • for correct sum over the states, must divide by N!
  • If it is a lattice => indistinguishable
  • distinguishability is revelant in counting configuration
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23
Q

State the grand canonical partition function

A
  • i = index of microstate
  • Ni = total particle number in the ith microstate
  • Ei = total enegy in the ith microstate
  • μ = chemical potential of reservoir
  • T = temp of reservoir
  • related to Grand potential (Φ = -kTlnZG)
    • whereas canonical parititon is related to F
      • F = -kTlnZ
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24
Q

What is the grand canonical partition function for fermions?

A
  • no more than 1 fermion can occupy the same single-particle state
    • N0 = 0 for E = 0
    • N1 = 1 for E = ε
      • so summation series ends at 1
  • Z = 1 + e-(E-μ)/kT
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25
Q

How do you derive Maxwell equations?

A
  • Write fundamental relations in differential form
  • Rewrite as a total differential eqn
  • Compare eqns
  • Find the 2nd mixed derivatives (which are equivalent)
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26
Q

Give the equation of the heat capacity at constant volume. How do you find entropy from this?

A
  • Cv = (dQ/dT)v = (dE/dT)v
    • derived from 1st law in differential form
    • dE = dQ + dW = TdS - pdV
    • so at constant volume
      • dE = dQ
  • dQ = Cv dT
  • dS = dQ/T = Cv/T dT
  • Integrate dS to find S
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27
Q

What is meant by the thermodynamic limit? What stats property of a system exposed to heat bath is expected to vanish in this limit?

A
  • the macroscopic limit, i.e. limit in which N –> infinity
  • fluctuations in mean energy -> 0
  • generally, fluctuations in system variables -> 0 in the thermodynamic limit of a particular ensemble
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28
Q

Thermodynamic potential and when are they used to identify an eqm state of a system?

A

Is this how to maximise entropy by minimising the thermodynamic potentials?

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29
Q

What is meant by a Bose-Einstein condensate?

A
  • boson gas at very low temp.
  • finite ratio of total number of particles occupy the single-particle ground state (hence a macroscopic number of particles in the thermodynamic limit)
  • phase transition to BEC occurs at critical temp. Tc of order n2/3(ħ/mk)
    • ​when mean interparticle distance is comparable to the thermal deBroglie wavelength
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30
Q

What are the different states of matter observed when particle density lies above, below or at the quantum concentration?

A
  • above: BEC ( maybe also degenerate fermi gas)
  • Quantum effects become noticeable when the particle density ≈ quantum concentration nq
  • below: classical limit
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31
Q

What is a Boltzmann factor? Which stats ensemble employs this factor and what does the ensemble represent?

A
  • Type of probability distribution
  • employed by the canonical ensemble
  • which represents a system that is able to exchange energy with its surroundings
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32
Q

How do you find the Fermi energy of N electrons from the density of states?

A
  • by looking for number of electrons N, the Fermi energy EF can be extracted
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33
Q

How do you find the mean energy of a gas of N electrons?

A
  • = mean energy of a Fermi gas
  • also given by finding total energy (since reference energy is 0 hence total gives average) by integrating EF wrt dN from 0 to N
34
Q

How does chemical potential determine how a partition might move?

A
  • Particles move from a region of high μ to a region of low μ
  • if particles move to the other side through the partition
    • pressure on that side (side with more particles) now increases
    • pushing the partition in the opposite direction
  • hence partition should move to the side which initially had a higher μ
35
Q

Deduce the Clausius-Clapeyron eqn for slope of coexistence curve on a p-T diagram.

A
  • Given the eqn for dμ
  • know that along the coexistence line, dμ1=dμ2 for the diff. phases
    • at critical point Tc and pc, system equilibrates in state with min G => (dG/dN1)T,p = 0
    • total number of particles, N = N1 + N2, is conserved
    • therefore dN1 = -dN2
    • so (dG/dN1)T,p = 0 = μ1- μ2 => μ1=μ2
    • dμ1 = -s1dTc + v1dpc & dμ2 = -s2dTc + v2dpc​
    • s = S/N & v = V/N
  • equate the differentials, and do some algebra to arrive at:
36
Q

What is an appropriate dispersion relation for EM wave modes?

A

ω=ck

37
Q

How do you calculate mean energy of radiation in frequency range ω -> ω + dω using density of EM wave modes g(ω)?

A
  • Integrate E g(ω) dω from 0 to infinity
    • E = mean energy of the QHO oscillators excluding zero point energy
    • (can’t do the expectation sign here)
38
Q

What is the mathematical difficulty if zero point energy was included in the calculation of total mean energy in the cavity?

A
  • integrand for total mean energy would contain a term that is proportional to the integral of ω3 dω which goes to infinity as frequency (integral upper limit) goes to infinity
    • giving infinite mean energy
39
Q

What is Einstein’s Cv in limit T-> 0?

A
  • T->0, ex -> infinity
    • so ex>>1 so (ex-1)2 ≈ e2x
  • so in the limit that T->0, Cv = (3Nkx2)/ex
  • which is exponentially decreasing
    • this does not accurately represent reality as Cv doesn’t go to zero that fast
    • Debye model gives a better representation
40
Q

What is Einstein’s Cv in limit T-> infinity?

A
  • T-> infinity, x->0 so sinhx ≈ x (small angle approximation)
  • then Cv = 3Nk
  • agrees with classical
41
Q

State Clausius and Kelvin statements of 2nd law.

A
  • Clausius: Heat cannot spontaneously flow from cold -> hot without external work being performed on the system.
  • Kelvin: No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work. (don’t get this sentence)
42
Q

How does Boltzmann’s entropy account for an increase in entropy of an isolated system when a constraint is released?

A
  • When a constraint is released, the number of system microstates now available increases
  • therefore the system entropy also increases
43
Q

What is the probability associated with a microstate in a microcanonical ensemble of Ω microstates?

A

1/Ω

because of equal a priori prob., an isolated system can be found in any of its microstates with equal probability

44
Q

How do you calculate Gibbs entropy of a system described by a microcanonical ensemble?

A

Use Pi = 1/Ω in the Gibbs entropy formula

  • because by equal a priori prob., each microstate is populated with prob. 1/Ω
45
Q

State the Wiedemann - Franz law.

A

k = (sigma)*L*T The ratio of the thermal and electrical conductivities in metals is proportional to the temperature, through a constant L, which is independent of any microscopic or thermodynamical variable of the metal, and called the “Lorentz number”

Do we need to kno

46
Q

How can photons be described as bosons?

A
  • When you find the expression for the average occupancy of a single-particle state , you find that it is the same as that of a boson gas with chemical potential zero.
47
Q

What is the Stefan-Boltzmann Law?

A
  • gives the flux coming from a black-body at temp. T
  • Flux = σT4
48
Q

Describe the Einstein model and how to arrive at the heat capacity Cv.

A
49
Q

What are phonons?

A
  • Lattice vibrations which are excited modes of acoustic standing waves in solids
50
Q

What are the factors included to account for degeneracy in the density of states?

A
  • Multiply by spin multiplicity (2S +1) for any quantum gas
  • For photons, multiple by 2 to account for the 2 polarisations ( 1 trans. + 1 long.) with the same wave-vector
  • For phonons, times by 3. There are 3 polarisations (2 trans. + 1 long.)
51
Q

What are the 2 assumptions the Einstein model is based on?

A
  1. Each atom in a lattice is an independent 3D QHO
  2. All atoms oscillate with the same frequency
52
Q

How does the Debye model improve on the Einstein model?

A
  • heat capacity found using the Einstein model (using canonical ensemble) fails at low temp. T -> 0
  • Debye assumed lattice vibrations can be modelled as phonons and derives their density of states
53
Q

Describe how the ultraviolet catastrophe arises in photon gases? How is it remedied?

A
  • energy of the QHO’s contains the zero-point energy which is also included in the canonical partition function
    • when finding the mean energy, infinites arise from the zero - point energy
    • since integral is over frequencies which tend to infinity
  • remedied by assuming the zero-point energy is of no thermodynamic interest and neglecting it
  • alternatively, a cut-off frequency can be introduced
54
Q

What is necessity of the Debye frequency?

A
  • (Similar to the ultraviolet catastrophe)
  • number of phonon modes depends on frequency, which goes to infinity with frequency
  • since there are 3N QHOs (clearly finite) this cannot happen
  • Debye introduces cut-off frequency to keep the number of modes finite
55
Q

When does the grand canonical ensemble become equivalent to the canonical case?

A
  • chemical potential of the particle bath = zero i.e. μ=0
  • there is no longer particle exchange
  • only energy exchange = canonical ensemble
56
Q

Explain why the canonical partition function of a system depends on whether the particles are fermions or bosons?

A
57
Q

Describe how the entropy decreases as the temperature is reduced at constant volume.

A
58
Q

Describe how the occupation of the lowest energy single particle state of the system evolves as the temp. is reduced from Tc to zero.

A
59
Q

Deriving the density of states for

  1. fermions
  2. relativisitc regime
  3. photons
A
  • k = nπ/L
    • each spatial direction has wavefn quantised as k
  • so number dN of single particle states in a wavevector interval k -> k + dk is:
    • dN = (L/2π)d​
      • d = no. of dimension
      • 2 because only +ve
      • each length contributes 1/π
  • for d = 3
    • dN = (V/8π3) dkxdkydkz
    • switch to polar coordinates then dkxdkydkz = |k|2dk sinθdθ dφ
    • integrated over domain [0,π/2] which spans the octant of +ve kx, ky, kz
    • so dN = (V/2π2) |k|2dk (why 4π? - elemental volume)
  • then dN = g(k) dk
  1. sub E = ħ2k2/2m and multiply by spin multiplicity (2S+1)
  2. sub E = khc
  3. sub ω = kc and multiply by 2 for polarisation vectors (trans & long)
  • Calculate the number of single-particle states between k -> k + dk : Elemental volume 4Pi*k^2 *dK, divided by volume of each lattice point, (delk)^3 = (Pi/L)^3
    1. Divide elemental volume by 8 since considering positive quadrant
    1. Equate g(k)dk=g(E)dE –> g(E) = g(k)dk/dE
    1. Sub. in expression for E and calc. dk/dE
    1. Multiply by the degeneracy factor - in this case, spin multiplicity.
60
Q

What is meant by thermal equilibrium?

A

System and surroundings, perhaps another system, are at the same temperature, where temeperature is the defining feature of thermal equilibrium. This is precisely examined in the zeroth law.

61
Q

Aside from changing its temperature, how might a degenerate fermi gas be made to behave classically?

A
  • EF grows with particle density, n
  • so a degenerate fermi gas may arise at any temperature provided that n is high enough
  • In this case, lowering n lowers TF (so it should be provided that n is low enough?)
  • so it can be forced to become classical
62
Q

What are the minimum and maximum values for average N in a single-particle state?

A
  • min = 0
  • max = N (isn’t it 1?)
63
Q

What is the mean energy in the ensembles and how do you derive it?

A
  • (1/Z) Σe-βEi = 1 because of the way the partition function is defined in terms of probabilities
    • Pi = (1/Z) Σe-βEi
      • probability of specified microstates = sum of microstates/parition fn (which represents all microstates)
    • parition fn Z plays the role of a normalising constant to ensure probabilties sum up to 1
  • and sum of all probabilities = 1
64
Q

What is the variance in energy?

A
65
Q

What are the 4 laws of thermodynamics?

A
  • Zeroth: if 2 systems are in thermal eqm with a 3rd then they are all in thermal eqm with each other
    • state fn T may be defined
  • First: There exists a state fn E that is conserved in isolated systems
  • Second: State fn S that either grows or stays constant in isolated systems
  • Third: S is 0 at absolute zero of T
66
Q

What is the mean number of particles/oscillators in system over grand canonical ensemble?

A
  • (1/kT)T partial derivative means kT is constant so comes out of differentiation
  • derivative of lnZ wrt x always give (1/Z)(dZ/dx)
67
Q

How to switch variables (ones that differentials depend on) from the fundamental relation of thermodynamics (S,V,N)?

A
  • Write other thermodynamics potentials in differential form
    • dF = -SdT-pdV+μdN
    • dH = TdS+Vdp+μdN
    • dG = -SdT+Vdp+μdN
    • dφ = -SdT-pdV-Ndμ
68
Q

How to maximise S if the constraints on the system are

  1. T & V
  2. S & p
  3. T & p?
A
  • a thermodynamic process starts with a system under certain constraints
    • system evolves to new eqm state that maximises total entropy change when constraints are changed
    • different thermodynamic potentials allow different expressions of maximising entropy depending on the contraints

By minimising

  1. F (T,V,N)
  2. H (S,p,N)
  3. G (T,p,N)
69
Q

What are the characteristics of Debye solids?

A
  • Heat capacities
    • = 3Nk (high T regime)
    • proportional to T3 (low T regime)
  • similarities with photon gas in expressions for E & S in terms of T
70
Q

How can you tell from the mean number of particles integral involving density of states whether it is a boson or fermion?

A
  • g(E) gives density of 1-particle states in E->E+dE
    • N<span>E</span> gives average number of particles in this interval
    • N is the average of these averages
  • for bosons, due to -1 in denominator, it is required that μ<e>E &gt; 0</e>​ i.e. physical
    • otherwise NE < 0 is unphysical
  • (Ns should have expectation brackets around them)
71
Q

What does mechanical equilibrium imply? If gases are also in thermal equilibrium, what does this imply?

A
  • their pressures are the same
    • so p can be treated as a constant
  • their temperatures are the same
    • so T can be treated as a constant
72
Q

What is internal entropy generation?

A
  • In non-quasistatic adiabatic processes
    • ΔSres = 0 but ΔStotal > 0
    • means that ΔSsys > 0
      • this is due to internal entropy generation
73
Q

What does a reduction in entropy imply?

A
  • less uncertainty in the position microstate for gases
    • better idea of where to look for them & more likely to
    • decrease uncertainty => entropy decreases
74
Q

What is the entropy change for quasistatic processes (system in thermal contact with reservoir)? And work done? What is the entropy and WD if the process was non-quasistatic?

A
  • ΔStot = 0 = ΔSres + ΔSsys => ΔSres = -ΔSsys
    • ΔE = 0 (thermal contact) => ΔW = -ΔQ
    • ΔSres = -ΔQ/Tres
    • hence ΔW = TresΔSres​
  • For non-quasistatic, total entropy change must be > 0
    • & ΔSsys will not change because ΔSsys is a state fn
    • so ΔSres has to increase
      • => ΔW has to increase
75
Q

Define the microcanonical ensemble (describes isolated system)

A
  • all microstates with a certain given energy taken with same probability
  • if E is not specified
    • take a “microcanonical average” whih means taking all possible microstates with same probability
76
Q

What is the Gibbs entropy of a canonical ensemble?

A
77
Q

State the 1st and 2nd law in differential forms

A
  1. dE = dQ + dW
  2. ΔStot = ΔS + ΔSres
78
Q

What is an adiabatic process?

A
  • one where no heat is transferred
    • ΔQ = 0
79
Q

What is a quasistatic transformation?

A
  • one where system & reservoir are always infinitely close to eqm
    • so all state fns may be defined along transfomration
  • or where ΔStot = 0
  • N.B. It is always true that ΔSr = -ΔQ/Tr
    • even if it is not quasistatic
80
Q

What are the 4 laws of thermodynamics?

A
  • Zeroth: 2 systems thermal contact with 3rd all in thermal eqm
    • define state fn T
  • First: state fn E conserved in isolated systems
  • Second: state fn S either grows or stays constant in isolated systems
  • Third: S (T=0) = 0
81
Q

Why 1/N!?

A
  • particle indistinguishability
    • so all permutations of these particles give same state
    • = overcounting

Physics Completed (8 decks)

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