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statistical physics > definitions > Flashcards

definitions Flashcards

1
Q

what are the two basic assumptions of statistical mechanics

A

1 - if we wait long enough the initial conditions become irrelevant (state of no memory= equilibrium)
2-for a system at equilibrium all possible quantum states are equallly likely

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2
Q

what is the physical meaning of entropy

A

way of expressing the number of choices in a system

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3
Q

what is the equation for entropy

A

S(E,N,V)=k_B log(Γ)

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4
Q

equation to define temperature using energy

A

T=(∂E/∂S)|N,V

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5
Q

equation to define chemical potential using energy

A

μ=(∂E/∂N)|S,V

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6
Q

equation to define pressure using energy

A

P=−(∂E/∂V)|S,N

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7
Q

differential equation for energy

A

dE=TdS+μdN−PdV

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8
Q

what are the conjugate varible pairs and which are extensibe vs intensive

A

T & S, μ & N, P & V

S,N,V-extensive

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9
Q

what is an intensive varibale

A

They do not scale with the system size

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10
Q

what is an extensive variable

A

They scale with the system siz

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11
Q

what is the euqation for the free energy

A

F(T,N,V)= E−TS= μN−PV

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12
Q

what is the euqation for the free energy (differential)

A

dF=−SdT+μdN−PdV

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13
Q

equation to define pressure using free energy

A

P(T,N,V)=−(∂F/∂V)|T,N

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14
Q

equation to define chemical potential using free energy

A

μ(T,N,V)=(∂F/∂N)|T,V

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15
Q

equation to define entropy using free energy

A

S(T,N,V)=−(∂F/∂T)|N,V

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16
Q

gibbs free energy equation

A

Φ(T,N,P)=F+PV=μN

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17
Q

equation to define entropy using gibbs free energy

A

S(T,N,P)=−(∂Φ/∂T)|N,P

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18
Q

equation to define volume using gibbs free energy

A

V(T,N,P)=(∂Φ/∂P)|T,N

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19
Q

equation to define chemical potential using gibbs free energy

A

μ(T,N,P)=(∂Φ/∂N)|T,P

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20
Q

equation for the landau potential

A

Ω(T,μ,V)=F−μN=−PV

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21
Q

what is the problem with an equation to describe a system using the variables T,μ,V

A

T,μ,V are all intensive so they dont know anything about the size of the system, they cannot describe a physical system

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22
Q

equation to define entropy using landau potential

A

S(T,μ,V)=−(∂Ω/∂T)|μ,V

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23
Q

equation to define pressure using landau potential

A

P(T,μ,V)=−(∂Ω/∂V)|T,μ

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24
Q

equation to define no. of particles using landau potential

A

N(T,μ,V)=−(∂Ω/∂μ)|T,V

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25
Q

equation for the landau potential (differential)

A

dΩ=d(F−μN)=−SdT−PdV−Ndμ

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26
Q

gibbs free energy equation (differential)

A

dΦ=d(F+PV)=−SdT+μdN+VdP

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27
Q

free energy equation (differential)

A

dF=d(E−TS)=μdN−PdV−SdT

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28
Q

when is it best to use the free energy

A

for a closed system

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29
Q

when is it best to use the landau potential

A

for an open system

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30
Q

how do we measure/control variables

A

we embed the subsystem of interest in a larger medium - subsystem and medium toghetehr form the total system which is isolated with const energy and N

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31
Q

what assumtions do we make as a result of assuming that the medium is always internally at equilubrium?

A

(∂E′/∂S′)|N′=T=const

(∂E′/∂N′)|S′=μ=const

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32
Q

in equilibrium each state of the total system has what probability of occuring?

A

w_eq= 1/Γ_0

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33
Q

If the subsystem has Γ choices and the medium choices Γ′ how many does the total system have - and how much entropy

A

Γ_t=Γ·Γ′

S_t=S+S′

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34
Q

what is probability that the

subsystem is in a state α

A

w_α=Γt,α/Γ0=Γ′α/Γ0

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35
Q

Thermodynamic average of a function

A

f_α: f ̄=∑w_αf_α

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36
Q

variance of a function

A

∑w_α(f_α− f ̄)^2

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37
Q

probability distribution for state alpha

A

(1/L)exp[(−E_α−μN_α)/kBT]

L= normalization constant - grand partition function

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38
Q 
Grand-Partition
function equation
A

L=∑exp[(−E_α−μN_α)/kBT]

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39
Q

what kind of system is the grand partition function applicable to

A

open system

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40
Q

what does the grand partition function depend on

A

T , V and μ (V is via E)

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41
Q

how is the grand partition function connected to the landou potential

A

−k_BTlogL=E−μN−ST=Ω

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42
Q

partition function equation

A

Z=∑_αexp[−Eα/kBT]

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43
Q

how is the partition function connected to the free energy

A

F(T,N,V)=−k_BTlogZ

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44
Q

how do you identify a many body state

A

identify how many particles occupy each single particle state

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45
Q

what is the total grand partition function of a many body state

A

L=∏_kL_k (the product of all the gpf)

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46
Q

what is an ideal gas

A

a perfect gas in which the probability of occupation of quantum states is very small

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47
Q

what is the ideal gas aproximation

A

μ<0, |μ|

48
Q

total landau potential equation

A

-PV~=Nk_BT

gives you the ideal gas euqation

49
Q

probability of energy E_

A

ρE_αw_α=ρE_α/Zexp[−E_α/kBT]

50
Q

partition function in terms of density of states

A

Z=∫dρ(ε)exp[−ε/kBT]

51
Q

normalization for partition function with denity of states

A

1 = ∫∞0dw(ε) =

∫∞0(dρ(ε)/Z)exp(−ε/k_BT)

52
Q

single particle energy dispersion (particle in a box)

A

ε=ℏ^2k^2/2m

53
Q

3d density of states for a massive particle final result

using particle in box dispersion

A

ρ(ε)=

(4πc2/(2πℏ )^3)m^3/2Vε^1/2

54
Q

2d density of states for a massive particle final result

using particle in box dispersion

A

ρ(ε)=L^2m/2π ℏ^2

55
Q

1d density of states for a massive particle final result

using particle in box dispersion

A

ρ(ℏ)=L√2m/2πℏ√ε

56
Q

method for finding density of states

A

from the dispersion relation find the relation btwn dl and dk (length and wave vector)
then place in integral (the sum over all dimensions of length) - change into polar coordinates

57
Q

what is temperature in terms of energy fot a single particle - words and equation

A

Temperature gives the average energy of a single particle in the gas
ε=3/2kBT

58
Q

what is the dulong and petit law

A

the heat capacity (at constant volume) of the ideal gas
C_V=(∂E/∂T)=3/2NkB
(factor 3 is bc 3d / degrees of freedom)

59
Q

energy with maximal probability equation

A

ε_max= k_BT(3N/2 −1)

60
Q

statistical averages are accurate when

A

the square root of variance is small compared to the average

61
Q

when are fluctuations supressed

A

when N (no of particles) is high

62
Q

dilute system

A

system where the average particle sepparation is much greater than particle size

63
Q

thermal debroglie wavelength equation

A

λ∼2πℏ /√(2mk_BT)

64
Q

what is the thermal debroglie wavelength

A

it is the typical particle “size” in your system - describes the wavepacket of the particles of your system

65
Q

what does it mean if the temperature becomes small enough that the thermal debroglie wavelength overlap

A

you are now in the quantum regime and will see quantum effects

66
Q

what are identical quantum particles

A

particles for which no new many body states are generated when two or more are interchanged

67
Q

indistinguishable quantum particles equation

A

ψ(1,2) = ±ψ(2,1)

68
Q

what particles obey fermi dirac stats

A

quantum particles forming antisymmetric states - fermions-particle cant occupy same state

69
Q

what particles obey bose einsten statistics

A

quantum particles forming symmetric states - bosons - particle can occupy same state

70
Q

Pauli exclusion principle

A

two (or more) identical fermions cannot occupy the same single-particle state

71
Q

Spin-statistics

“theorem”

A

Elementary particles with integer intrinsic spin (0, 1, 2, …) obey Bose-Einstein statistics -bosons
Elementary particles with half-integer intrinsic spin (1/2, 3/2, …) obey Fermi-Dirac statistics-fermions

72
Q

landau potential for the single particle state k

A

Ω_k=−k_BTlogL_k

73
Q

how to find the avearge occupation of state k

A

n_k=−∂Ω_k/∂μ

partial diferential of the landau potential wrt μ

74
Q

what is the fermi driac distribution

A

n_k= 1/exp [(ε_k−μ) /k_BT] + 1

probablity of occupation dep on energy

75
Q

what is the bose einsten distribution

A

n_k= 1/exp [(ε_k−μ) /kBT] - 1

76
Q

Boltzmann distribution

Ideal gas

A

n_k= exp[ε−k−μkBT]«1

77
Q

in a dilute system does it matter if we start with bososns or fermions

A

no both will convenrge to the boltzman distributiobn (ideal gas)

78
Q

what was did we initially forget when trying to calculate the max number of bosons in a state

A

that there exists a non excited state of energy 0 where the bosons will condense to where there is no more space for them. - bose einsten condensate

79
Q

critical temp - bosons

A

T_c= (1/k_B)(NV·2.3 c)^2/3

80
Q

when does a bose einsten condensate occur

A

occurs when μ→0

remebering that it is negative

81
Q

Number of particles in the single-particle ground state-BEC

A

N_0= N−N_∗= N[1 −(T/Tc)^3/2]

82
Q

heat capacity for a BEC (equation)

A

CV=(∂E/∂T)=

~1.9 Nk_B(T/Tc)^3/2

83
Q

what does it mean for something to be a super fluid

A

it has zero viscosity

84
Q

Landau criterion of superfluidity

A

v_c= min (ε_p/p) if it is not 0 it is a superfluid

85
Q

what do we mean by a coherent state

A

a state with well defined relative phases

86
Q

how to find the total number of particles in a system

A

integral of density of states times distribution over energy

87
Q

what is the fermi energy

A

E_f=ℏ^2k_f^2/2m

is the energy of the highest occupied single-particle state at T=0

88
Q

what happens when the chemical potential is less than the fermi energy

A

all states are unnocupied

89
Q

when do chemical potential and fermi energy coincide

A

when T=0

90
Q

fermi energy equation

A

E_F=(N/V·3/2·1c)^2/3

91
Q

fermi temperature equation

A

T_f=E_f/k_b

92
Q

what does temperature do the the fermi distribution

A

Temperature smears out the Fermi distribution in an energy window of size k_bT around the Fermi level

93
Q

who likes to party more ? bosons or fermions

A

bosons babey theyre the sociable ones

94
Q

what happens to particle number at fixed chemical potential as temp increases for a fermi gas

A

At a fixed chemical potential the particle number grows while increasing temperature (quadratically)

95
Q

what happens to chemical potential at fixed particle number temp increases for a fermi gas

A

it must decrease while temp increases

96
Q

Average energy per

particle at T=0 (fermi gas)

A

ε=3/5.E_F

97
Q

How can we discuss Electrons in metals as non-interacting particles in an empty box

A

The metal is globally neutral
Most collisions between electrons are blocked by
the Pauli principle
Most electrons wavelengths are huge w.r.t. the
lattice spacing a. Thus the lattice essentially does not affect the qualitative dispersion

98
Q

how does heat capacity grow w temperature at very low temperatures

A

c proportional to T^3

99
Q

How can we describe the properties of insulators?

A

In insulators electrons are tightly bound to their nuclei.
Mechanical vibrations of atoms dominate the thermodynamic response
Atoms interact with each other in elastic crystals

100
Q

crucial assumption of einsten model for heat capacity

A

the oscillation of each atom DOES NOT affect the neighboring ones - ignores compressibility

101
Q

why is the einsten model wrong (qualitative reason)

A

we did not account for the elasticity of the whole lattice
there arent enough states in the low temperatures
I.e. in a solid, the relevant low energy spacing is much
smaller than that of the single oscillator

102
Q

Comments on the Einstein model

A

The Einstein model correctly predicts the high T expansion of the heat capacity as well as its suppression at low temperatures If T is too low, the probability of occupying high vibrational states
is exponentially suppressed, and so is the heat capacity
(qualitatively correct)CV
BUT The exponential suppression at low T is too fast

103
Q

Phonons

A

The resulting collective excitations involving all atoms in the lattice are the proper eigenstates

104
Q

Phonon

dispersion

A

ω= 2√(k/m)sin( qa/2)

105
Q

how do phonons help improve the einstein model

A

The low frequency of collective phonons yields a larger C_v at low T and can compensate for the suppressed value of the Einstein model

106
Q

what does the debye model consider

A

considers an isotropic medium with 3 acoustic phonon branches up to a cutoff energy (Debye Energy)
It also assumes that the 3 phonon branches have the same (average) velocit

107
Q

number of phononic states in a band

A

3N=∫dερ(ε)

108
Q

General picture of the heat capacity in metals

A

In metals we have to consider both the phononic and electronic contribution to the heat capacity
As we have seen for degenerate Fermi gases the
electronic contribution to C_v is linear in T

109
Q

differences in Debye theory to allow us to treat the blackbody radiation

A

The 3 branches of phonons are replaced by 2 branches of photons (they are only transverse!) with constant velocity c
There is no upped Debye energy, since infinite photon modes exist

110
Q

Stefan-Boltzmann

law

A

Ephotons/V= π^2·k^4·T^4 /(ℏc)^3·15

111
Q

spin component energy

A

E_σ=−μ_BBσ

112
Q

magnetization equation

A

M=μ_BNtanh[μ_BB/k_BT]=−∂F/∂B

113
Q

dimensionless magnetization equation

A

M=M0tanh[Tc/T·M/M0]

114
Q

Selfconsistent equation

for the Magnetization

A

m=tanh[m/t]

115
Q

critical temp for magnetization

A

Tc=μ^2Nλ/k_B

116
Q

order parameter

A

is a quantity which characterizes the phases. It is zero in one phase and non-zero in the other

117
Q

what happesn to free energy across a phase transition

A

continous

statistical physics (1 decks)

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