Introduction to (Statistical) Thermodynamics

2

Molecular Simulations

Molecular dynamics: solve equations of motion

Monte Carlo: importance sampling

r1

MD

MC

r2rn

r1

r2

rn

3

4

5

Questions

• How can we prove that this scheme generates the desired distribution of configurations?

• Why make a random selection of the particle to be displaced?

• Why do we need to take the old configuration again?

• How large should we take: delx?

What is the desired distribution?

6

SummaryThermodynamics

– First law: conservation of energy– Second law: in a closed system entropy increase

and takes its maximum value at equilibrium

System at constant temperature and volume– Helmholtz free energy decrease and takes its

minimum value at equilibrium

Equilibrium:– Equal temperature, pressure, and chemical

potential

7

Entropy

Closed system:•1st law: total energy remains constant•2nd law: entropy increases

dE =TdS-pdV + μidNii=1

M

∑

T =∂E∂S

⎛⎝⎜

⎞⎠⎟V ,Ni

Temperature

E,V

dS ≥0

E =E(S,V,Ni )

or1

T=

∂S∂E

⎛⎝⎜

⎞⎠⎟V ,Ni

8

Helmholtz free energy

F ≡E −TS dE =TdS-pdV + μidNii=1

n

∑dF =−SdT−pdV + μidNi

i=1

n

∑

Pressure: p=-∂F∂V

⎛⎝⎜

⎞⎠⎟T ,Ni

Energy: E =∂F T∂1 T

⎛

⎝⎜⎞

⎠⎟V ,Ni

F =F T,V,Ni( ) dF ≤0

Chemical potential: μi =∂F

∂Ni

⎛

⎝⎜⎞

⎠⎟T ,V ,N j

9

Statistical Thermodynamics

System of N molecules: r1K r

N,v

1K v

N

But how do we obtain themacroscopic properties of this system from this microscopic information?

10

Outline (2)Statistical Thermodynamics

– Basic Assumption• micro-canonical ensemble• relation to thermodynamics

– Canonical ensemble• free energy• thermodynamic properties

– Other ensembles• constant pressure• grand-canonical ensemble

11

Statistical Thermodynamics:the basics

• Nature is quantum-mechanical• Consequence:

– Systems have discrete quantum states.– For finite “closed” systems, the number of states is

finite (but usually very large)

• Hypothesis: In a closed system, every state is equally likely to be observed.

• Consequence: ALL of equilibrium Statistical Mechanics and Thermodynamics

12

Basic assumptionEach individual

microstate is equally probable

…, but there are not many microstates that

give these extreme results

If the number of particles is large (>10)

these functions are sharply peaked

13

Does the basis assumption lead to something that is consistent with classical

thermodynamics?

1ESystems 1 and 2 are weakly coupled such that they can exchange energy.

What will be E1?

( ) ( ) ( )1 1 1 1 2 1,E E E E E EΩ − = Ω ×Ω −

BA: each configuration is equally probable; but the number of states that give an energy E1 is not know.

2 1E E E= −

14

( ) ( ) ( )1 1 1 1 2 1,E E E E E EΩ − = Ω ×Ω −

( ) ( ) ( )1 1 1 1 2 1ln , ln lnE E E E E EΩ − = Ω + Ω −

( )1 1

1 1

1 ,

ln ,0

N V

E E E

E

⎛ ⎞∂ Ω −=⎜ ⎟

∂⎝ ⎠

∂lnΩ1 E1( )∂E1

⎛

⎝⎜

⎞

⎠⎟

N1 ,V1

+∂lnΩ2 E −E1( )

∂E1

⎛

⎝⎜

⎞

⎠⎟

N2 ,V2

=0

( ) ( )1 1 2 2

1 1 2 1

1 2, ,

ln ln

N V N V

E E E

E E

⎛ ⎞ ⎛ ⎞∂ Ω ∂ Ω −=⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠ ⎝ ⎠

Energy is conserved!dE1=-dE2

This can be seen as an equilibrium

condition

( ),

ln

N V

E

Eβ

⎛ ⎞∂ Ω≡ ⎜ ⎟

∂⎝ ⎠

1 2β β=

15

Conjecture:

Almost right.

•Good features:

•Extensivity

•Third law of thermodynamics comes for free•Bad feature:

•It assumes that entropy is dimensionless but (for unfortunate, historical reasons, it is not…)

Entropy and number of configurations

lnS = Ω

16

We have to live with the past, therefore

With kB= 1.380662 10-23 J/K

In thermodynamics, the absolute (Kelvin) temperature scale was defined such that

But we found (defined):

( )lnBS k E= Ω

,

1

N V

S

E T

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠

( ),

ln

N V

E

Eβ

⎛ ⎞∂ Ω≡ ⎜ ⎟

∂⎝ ⎠

dE =TdS-pdV + μidNii=1

n

∑

17

And this gives the “statistical” definition of temperature:

In short:

Entropy and temperature are both related to the fact that we can COUNT states.

( ),

ln1B

N V

Ek

T E

⎛ ⎞∂ Ω≡ ⎜ ⎟

∂⎝ ⎠

Basic assumption:1. leads to an equilibrium condition: equal temperatures2. leads to a maximum of entropy3. leads to the third law of thermodynamics

18

How large is Ω?

•For macroscopic systems, super-astronomically large.

•For instance, for a glass of water at room temperature:

•Macroscopic deviations from the second law of thermodynamics are not forbidden, but they are extremely unlikely.

Number of configurations

252 1010 ×Ω ≈

19

Canonical ensemble

Consider a small system that can exchange heat with a big reservoir

iE iE E− ( ) ( ) lnln lni iE E E E

E

∂ ΩΩ − = Ω − +

∂L

( )( )

ln i i

B

E E E

E k T

Ω −= −

Ω

1/kBT

Hence, the probability to find Ei:

( ) ( )( )

( )( )

exp

expi i B

i

j j Bj j

E E E k TP E

E E E k T

Ω − −= =

Ω − −∑ ∑( ) ( )expi i BP E E k T∝ −

Boltzmann distribution

20

Thermodynamics

What is the average energy of the system?

( ) ( )( )

exp

exp

i iii ii

jj

E EE E P E

E

β

β

−≡ =

−

∑∑∑

( )ln exp iiEβ

β

∂ −= −

∂∑

, ,ln N V TQ

β∂

=−∂Compare:

1

F TE

T

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

Hence: , ,ln N V T

B

FQ

k T=−

Thermo recall (2)

d d dE T S p V= −

First law of thermodynamics

Helmholtz Free energy:

1

1 1

F T F FF F T

T T T T

⎛ ⎞∂ ∂ ∂= + = −⎜ ⎟∂ ∂ ∂⎝ ⎠

F E TS≡ −d d dF S T p V=− −

F TS E= + =

21

Remarks (1)We have assume quantum mechanics (discrete states) butwe are interested in the classical limit

( ) ( )2

3

1exp d d exp

! 2N N Ni

ii ii

pE U r

h N mβ β

⎧ ⎫⎡ ⎤⎪ ⎪− → − +⎨ ⎬⎢ ⎥

⎪ ⎪⎣ ⎦⎩ ⎭∑ ∑∫∫ p r

3

1

h→ Volume of phase space (particle in a box)

1

!N→ Particles are indistinguishable

332 2 22

d exp dp exp2 2

N N

N ii

i

p p m

m m

πβ ββ

⎧ ⎫ ⎡ ⎤⎡ ⎤ ⎧ ⎫ ⎛ ⎞⎪ ⎪− = − =⎨ ⎬ ⎨ ⎬⎢ ⎥⎢ ⎥ ⎜ ⎟⎪ ⎪ ⎝ ⎠⎩ ⎭⎣ ⎦ ⎣ ⎦⎩ ⎭

∑∫ ∫p

Integration over the momenta can be carried out for most systems:

22

Remarks (2)

Define de Broglie wave length:1

2 2

2

h

m

βπ

⎛ ⎞Λ ≡⎜ ⎟

⎝ ⎠

Partition function:

( ) ( )3

1, , d exp

!N N

NQ N V T U r

Nβ⎡ ⎤= −⎣ ⎦Λ ∫ r

23

Example: ideal gas( ) ( )3

1, , d exp

!N N

NQ N V T U r

Nβ⎡ ⎤= −⎣ ⎦Λ ∫ r

3 3

1d 1

! !

NN

N N

V

N N= =

Λ Λ∫ r

Free energy:

3

3 3

ln!

ln ln ln ln

N

N

VF

N

NN N N N

V

β

ρ

⎛ ⎞=− ⎜ ⎟Λ⎝ ⎠

⎛ ⎞≈ Λ + = Λ +⎜ ⎟⎝ ⎠Pressure:

T

F NP

V Vβ∂⎛ ⎞=− =⎜ ⎟∂⎝ ⎠

Energy:

3 3

2 B

F NE Nk T

ββ β

⎛ ⎞∂ ∂Λ= = =⎜ ⎟∂ Λ ∂⎝ ⎠

Thermo recall (3)

Helmholtz Free energy:

T

FP

V

∂⎛ ⎞ =−⎜ ⎟∂⎝ ⎠

d d dF S T p V=− −

1

F T FE

T

ββ

⎛ ⎞ ⎛ ⎞∂ ∂= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

Energy:

Pressure

24

Ideal gas (2)

Chemical potential: μi =∂F

∂Ni

⎛

⎝⎜⎞

⎠⎟T ,V ,N j

βF =NlnΛ3 + Nln

NV

⎛

⎝⎜⎞

⎠⎟

βμ =lnΛ3 + lnρ +1

βμIG =βμ0 + lnρ

25

Summary:Canonical ensemble (N,V,T)

Partition function:

Probability to find a particular configuration

Free energy

( ) ( )3

1, , d exp

!N N

NQ N V T U r

Nβ⎡ ⎤= −⎣ ⎦Λ ∫ r

( ) ( )expP Uβ⎡ ⎤Γ ∝ − Γ⎣ ⎦

, ,ln N V TF Qβ =−

26

Summary:micro-canonical ensemble (N,V,E)

Partition function:

Probability to find a particular configuration

Free energy

( ) 1P Γ ∝

, ,ln N V ES Qβ =

( ) ( )( )3

1, , d d ,

!N N N N

NQ N V E H E

h Nδ= −∫∫ p r p r

27

Thermodynamic properties

For many properties the momenta do not matter: only

the configurational part

Probability to find a configuration: { }1 2, , , NΓ = r r rK

( ) ( )expP Uβ⎡ ⎤Γ ∝ − Γ⎣ ⎦

Ensemble average:

( ) ( )1expA d A U

Qβ⎡ ⎤= Γ Γ − Γ⎣ ⎦∫

28

Other ensembles?In the thermodynamic limit the thermodynamic properties areindependent of the ensemble: so buy a bigger computer …

However, it is most of the times much better to think and to carefullyselect an appropriate ensemble.

For this it is important to know how to simulate in the variousensembles.

But for doing this wee need to know the Statistical Thermodynamicsof the various ensembles.

COURSE: MD and MC

different ensembles

29

Example (1): vapour-liquid equilibrium mixture

Measure the composition of the coexisting vapour and liquid phases if we start with a homogeneous liquid of two different compositions:– How to mimic this with the

N,V,T ensemble?– What is a better ensemble?

composition

T

L

V

L+V

30

Example (2):swelling of clays

Deep in the earth clay layers can swell upon adsorption of water:– How to mimic this in the N,V,T

ensemble?– What is a better ensemble to

use?

31

Ensembles

• Micro-canonical ensemble: E,V,N

• Canonical ensemble: T,V,N

• Constant pressure ensemble: T,P,N

• Grand-canonical ensemble: T,V,μ

32

Constant pressure simulations: N,P,T ensemble

Consider a small system that can exchange volume and energy with a big reservoir

( ) ( ),

ln lnln ln ,i i i i

V E

V V E E V E E VE V

∂ Ω ∂ Ω⎛ ⎞ ⎛ ⎞Ω − − = Ω − − +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

L

( )( )

,ln

,i i i i

B B

E E V V E pV

E V k T k T

Ω − −= − −

Ω

1/kBT

Hence, the probability to find Ei,Vi:

( ) ( )( )

( )( )

( ), ,

exp,,

, exp

exp

i ii ii i

j k j kj k j k

i i

E pVE E V VP E V

E E V V E pV

E pV

β

β

β

⎡ ⎤− +Ω − − ⎣ ⎦= =⎡ ⎤Ω − − − +⎣ ⎦

⎡ ⎤∝ − +⎣ ⎦

∑ ∑

,i iV E ,i

i

E E

V V

−−

p/kBTThermo recall (4)

d d d + di iiE T S p V Nμ= − ∑

First law of thermodynamics

Hence

,T N

S p

V T

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠

,

1=

V N

S

T E

∂⎛ ⎞⎜ ⎟∂⎝ ⎠

and

33

N,P,T ensemble (2)

In the classical limit, the partition function becomes

( ) ( ) ( )3

1, , exp d exp

!N N

NQ N P T dV PV U r

Nβ β⎡ ⎤= − −⎣ ⎦Λ ∫ ∫ r

The probability to find a particular configuration: ,N Vr

( ) ( )( ), expN NP V PV U rβ⎡ ⎤∝ − +⎣ ⎦r

34

Grand-canonical simulations: μ,V,T ensemble

Consider a small system that can exchange particles and energy with a big reservoir

( ) ( ),

lnlnln ln ,i i i i

N E

N N E E N E E NE N

∂ Ω∂ Ω ⎛ ⎞⎛ ⎞Ω − − = Ω − − +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

L

( )( )

,ln

,i i i i i

B B

E E N N E N

E N k T k T

μΩ − −= − +

Ω

1/kBT

Hence, the probability to find Ei,Ni:

( ) ( )( )

( )( )

( ), ,

exp,,

, exp

exp

i i ii ii i

j k j k kj k j k

i i i

E NE E N NP E N

E E N N E N

E N

β μ

β μ

β μ

⎡ ⎤− −Ω − − ⎣ ⎦= =⎡ ⎤Ω − − − −⎣ ⎦

⎡ ⎤∝ − −⎣ ⎦

∑ ∑

,i iN E ,i

i

E E

N N

−−

-μ/kBTThermo recall (5)

d d d + di iiE T S p V Nμ= − ∑

First law of thermodynamics

Hence

,

i

i T V

S

N T

μ⎛ ⎞∂=−⎜ ⎟∂⎝ ⎠

,

1=

V N

S

T E

∂⎛ ⎞⎜ ⎟∂⎝ ⎠

and

35

μ,V,T ensemble (2)

In the classical limit, the partition function becomes

( ) ( ) ( )31

exp, , d exp

!N N

NN

NQ V T U r

N

βμμ β

∞

=

⎡ ⎤= −⎣ ⎦Λ∑ ∫ r

The probability to find a particular configuration: , NN r

( ) ( ), expN NP N N U rβμ β⎡ ⎤∝ −⎣ ⎦r