Lecture 630.01.2019

recap of module I

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ti. 15. jan. Introduction/Thermodynamics laws, entropy

on. 16. jan. Thermodynamic potentials, response functions and Maxwell’s relations, thermodynamic stability

ti. 22. jan. Statistical ensembles, Phase space, Liouville’s theorem, microcanonical ensemble

on. 23. jan. Canonical ensemble

ti. 29. jan. Grand-canonical ensemble

Module I: Thermodynamics (review) and statistical ensembles

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Ensemble density of states

• Phase space !, # − 2d×( dim. space

• Representative volume element

,- = /012/01345ℏ 01 ≡⋕ of microstates in a unit volume

• Ensemble Probability density 9(!, #): is the probability

that a particular microstate is occupied by the system

• Liouville’s theorem for equilibrium ensembles <,= => → <(@, A) = <(=(@, A))

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Microcanonical Ensembles: equilibrium of isolated systems

Equilibrium ensemble density of states: ! ", $ = &'()) +(, ", $ − ))

• Microcanonical density of states: ' ), .,/ = ∫ 12 + , ", $ − ) , d4 = 56785679:;ℏ 67

• Describes a system at a fixed energy, volume and number of particles • Each possible state at fixed ) and / has an equal probability

• Phase space volume: = ), .,/ = ∫, ",$ >)12 + , ", $ − )

• Boltzmann’s formula (correspondence to thermodynamics) Entropy: ? @, A, B = C DE [= ), .,/ ]

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Microcanonical ensemble: Ideal gasMicrocanonical phase space volume of free particles

Ω ",$ =1$!

V)

2+ℏ -)

+-).

3$2 !

20"-). , Σ " =

2Ω "2"

Boltzmann’s formula

3 ",$ = 4 l6 Ω ",$ = 789:+ <=

>8

?@AB8ℏ:

A:

CD =EFC@ +

GFC> + HC8

• Temperature I

J=

KL

KM N,)=

-)O

.

I

M→ " =

-

.$4Q

• Pressure R

J=

KL

KN J,)=

)O

N→ ST = $4Q

U

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Canonical Ensembles: equilibrium with a thermal bath • Describes a system that is in thermal equilibrium with a heat bath at a fixed temperature ! in terms of the equilibrium density

" #, % ='

( !)*+, #,% , ∫ ./ " #, %, 0 = '

• Canonical partition function and Helmholtz free energy (thermodynamics correspondence)

( ! = ∫ ./)* +, #,% = )* +1 !

• Energy fluctuates around the average, equilibrium value 2 = ⟨4⟩, with a probability 6(4) =9

:;* <(=*>?)

( !, @ = ∫ .A )*+AB A = )* +1 ! , 1 = C − !E C , A = C = −FF+

G0 ((!)

• Energy Fluctuations

H4I =1KLI

LMIK −

1KLLM

KI

=LI

LMIlO K

Fluctuations are much smaller than the average in the thermodynamic limit: PAQ

A∼

'

S→ U

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Canonical ensemble: Ideal gas!" #, % =

'

"!

)*

+,*(.)=

'

0!!'", Λ =

2

3456.

Thermodynamic correspondence

7 #, %, 8 = −:# ln Z" #, % = − 8:# ln%

8Λ> #+ 1 ,

A7 = −BA# − CA% + DA8• B = −

EF

E. ),"= 8:[ln

)

"+,(.)+ 1 +

>"

3: =

HFIJ

.

• C = −EF

E) .,"=

"6.

)

• D =EF

E" .,)= −:# ln

)

"+, .→

)

"+, .= LHMN

Energy fluctuations for a macrostate at fixed #: C O =P(Q)

R(.)LHMQ ∼

Q,*

"!LHMQ

O = −T

TUlog ! = −

38

2

T

TUlog

2Z[

U=38

2:# ∼ 8

ΔO3 =T3

TU3log ! = −

38

2

T

TU

1

U=38

2:3#3 ∼ 8

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Grand-Canonical Ensembles: equilibrium with a particle and thermal reservoir

• Describes a system that is in thermal and chemical equilibrium with a reservoir at a fixed temperature ! and chemical potential "

# $, &, ' =)

* !, ")'! ,

-. / $,& -"' , 0'∫ 23 # $, &, ' = )

• Grand Canonical partition function and Landau free energy (thermodynamics correspondence)

Ξ 5, 6 = 0789

:;<=7

>! ∫ ?@ ;-<A(C,D) = 0'8F

:

,-.(G-"') = ,-.H(I,") , H(I, ") = J(I, K) − "M

• Particle number fluctuates around the average, equilibrium valueN = ⟨>⟩, with a probability Q > = R

S T,= ;- < U-=7 , ∑'W(') = )

' = M = X!YY" Z' *(!, ")

• Particle number fluctuations

[>\ = ]5 \ ^\

^6\ l> Σ(5, 6)

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Grand-Canonical ensemble: Ideal gas

Ξ ", $, % = '()*

+

,-.(/(", $, 1)

• Ξ ", $, % = ∑()*+ 4

(!

6

78,-.

(= ,9(:,.)6, z =

<=>

78(:)is the fugacity

? @, A, B = CDA = CEFG(@,A,B)

• Thermodynamiccorrespondence:Landau potential

Ω T, % = −^" _$ = −^"$,-.

Λa ", bΩ = −cb" − db$ − eb%

• Numberfluctuations d 1 ∼4

(!

6j

78j,-.(

Δnl = ^" l ml

m%ll1 Ξ = $

,-.

Λa(")

n = ^"m

m%l1 Ξ = $

,-.

Λa(")

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Monte Carlo algorithm• Evaluate multivariable integrals by random sampling from a known equilibrium distribution (e.g. canonical ensemble distribution—

Boltzmann statistics)

! (#) = &'(ℏ *+

∫ -*+.-*+/ ! .,/ 12 3! .,/

4(#)

• The goal of a simulation is sampling through a large enough subset of the microstates of the equilibrium ensemble, so that averages of observables (e.g. energy) approach their equilibrium values.

• After a simulation has run long enough for the system to approach equilibrium, the simulation then generates a sequence of 5 configurations, 67 789

:chosen from the set of microstates in the equilibrium ensemble and the corresponding sequence for the value

of the energy in each microstate, ;(67) 789:

• At equilibrium, the microstates are drawn from the canonical density of states <1/ / = 123! /

∑ /> 123!(/>)

• ! = ! ? ± A?

! ? = &?B

C8&

?!(/C) , A?' = !' ? − ! ?

' = &?B

C8&

?! /C − ! ?

'

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Metropolis Monte Carlo algorithm• Equilibrium condition for the ensemble density

!"#$%"("′)) "′ → " = $%" " !

"#)(" → "′)

• Detailed balance condition (stronger constraint)

$%" "#$%" "

= ) "→",) ",→" → %-./0 = ) "→",

) ",→"

• Metropolis rule for the transition probability

) " → "# = 11, ΔE ≤ 08-9:;, Δ< > 0

>

>#

>#

>#

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Metropolis Monte Carlo : Ideal gas • Detailed balance: !"# $%!"# $

= ' $→$)

' $)→$ = "*+,-, - = /01$

0

Metropolis rule for the transition probability

' $ → $% = 21, Δ5 ≤ 08*9:;, Δ5 > 0

• $="> = $?@A + C(0 EF=A − /)

• If 5IJK ≤ 5LMN, accept the move

• If 5IJK > 5LMN, accept the move with 8*9:;Δ5

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Metropolis Monte Carlo : Ideal gas

Δ"

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% for each trial generate a new velocityvnew = v+s*(2*rand(size(v))-1);% estimate the energy changeDE = m/2*sum(vnew.^2-v.^2);% estimate the jump probabilityW = exp(-beta*DE);r = rand;% accept the move if it lowers the energy or if the jumpprobability is larger than a random nr in [0,1]v = vnew.*(DE<=0)+...

(DE>0).*(vnew*(r<=W)+v.*(r>W));% collect particle speed for the accepted microstateV(n)=norm(v);

Metropolis Monte Carlo : Ideal gas

Δ"

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