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Principles of statistical thermodynamics

Principles of statistical thermodynamicsknowing 2 atoms and wishing to know 1023 of them – part I

Marcus Elstner and Tomas Kubar

November 11, 2012


Introduction

“. . . ensemble generated by a simulation does not representa canonical ensemble” – what does this mean?

The phase space may be sampled (walked through) in various ways– just what is the correct way?

We will start with the microcanonical ensemble,and turn to the canonical after that,to derive the canonical probability distribution function.


Introduction

QM systems – discrete states – derivation is simplerMM systems – continuum of statesour development – with discrete states,

results (density of probability in phase space)will be valid for all kinds of systems, generally


Microcanonical ensemble

Terms and conditions

microstate – particular distribution of energy among particles

microcanonical ensemble – every microstate contains the sameenergy, and every microstate occurs with equal probability

configuration = macrostate – particular occupation of energylevels by the individual indistinguishable particles

probability of configuration = number of relevant microstates



Example – counting the microstates

system of 3 particles that possess 3 identical E quanta altogetherIn how many ways can these 3 E quanta be distributed?




One particle obtains l quanta, the next m, and the last one nquanta of energy, so that l + m + n = 33 possibilities: (3,0,0), (2,1,0) and (1,1,1) – configurations A, B, CFor every configuration:

pick a first particle and give it the largest number of E quantain the configuration – 3 choices

assign the second-largest number of quanta to anotherparticle – 2 choices

one particle left to accommodate the smallest number ofquanta – 1 choice (= no choice)

in total: 3 · 2 · 1 = 3! = 6 choices – generally n! choices

This way – 6 microstates for every configuration.But – if there are energy degeneracies – some microstates are equal




Energy degeneracies for each configuration:

Cfg. A: particle α has 3 quanta → following situations are equivalent:β gets 0 quanta first and γ gets 0 quanta next, or γ get 0quanta first and β gets 0 quanta nexttwo identical assignments – however they were both countedto obtain the right number – divide by 2 · 1 = 2!number of microstates = 3!

2! = 3

Cfg. B: each particle – a different number of quanta – no degeneracy

Cfg. C: trivially – only one microstate for this configurationformally – assign three identical numbers of energy quanta tothe particles, thus divide the number of microstates by 3!(permutations): 3!

3! = 1

3 + 6 + 1 = 10 microstates in total



Number of microstates

N particles: N ways to pick the 1st, N − 1 ways to pick the 2nd. . .– N! ways to build the system

if na particles have to carry the same number of energy quanta,then we have to divide by na! to get the real number of microstatesthe number of microstates W for this system:

W =N!

na! · nb! · . . .ni – occupation number of energy level iW – extensively large for large N → logarithm

lnW = lnN!

na! · nb! · . . .= lnN!− ln na!− ln nb!− . . . = lnN!−

∑i

ln ni !

Stirling: ln a! = a · ln a− a

lnW = N · lnN −∑i

ni · ln ni



BTW: James Stirling

Scottish mathematician



Number of microstates

fraction of particles in state i – probability that particle in state i

pi =niN

lnW =∑i

ni · lnN −∑i

ni · ln ni = −∑i

ni · lnniN

= −N ·∑i

pi · ln pi

important – which config. is most probable – largest weight= config. with the largest number of microstates

– find the maximum of W as the function of ni



Method of Lagrange’s multipliers

find the maximum of a function f (~x) so thatconstraints yk(~x) = 0 are fulfilled

Lagrange: search for the maximum of function f −∑

k λkyk :

∂

∂xi

(f (~x)−

∑k

λk · yk(~x)

)= 0

∂

∂λi

(f (~x)−

∑k

λk · yk(~x)

)= 0

xi are the components of ~x



Most probable configuration

find the configuration with maximum weight of a system of Nparticles distributed among levels εi , with the total energy Esubject to maximization is the weight

lnW = N · lnN −∑i

ni · ln ni

under the conditions of constant N and E∑i

ni − N = 0∑i

ni · εi − E = 0




apply Lagrange’s method:

∂

∂ni

lnW + α ·

∑j

nj − N

− β ·∑

j

nj · εj − E

= 0

∂ lnW

∂ni+ α− β · εi = 0

−α and β are Lagrange’s multipliers; note also

∂ lnW

∂ni=

∂

∂ni

(∑i

ni · ln∑i

ni −∑i

ni ln ni

)=

= lnN + N · 1

N−(

ln ni + ni ·1

ni

)= − ln

niN




solution:

niN

= exp [α− β · εi ]

parameter α may be obtained from the condition with α as

expα =1∑

j exp[−β · εj ]

so that

niN

=exp[−β · εi ]∑j exp[−β · εj ]

β might be obtained from the other condition

E

N=

∑j εj · exp[−β · εj ]∑

j exp[−β · εj ]



Dominating configuration

important observation for a huge number of particles N:one config. has a much larger weight than all of the others

dominating configuration– occupation numbers ni obtained before– determines the properties of the system


Entropy

Microscopic definition of entropy

extreme cases of configurations:

pi = 1N ,

1N ,

1N , . . .: W = N · lnN is maximal

pi = 1, 0, 0, . . .: W = 0 is minimal (for large N)

define the microscopic entropy

S = −kB · lnW

kB – universal Boltzmann constantentropy tells us something about the travel of the system

through the configuration (phase) spacesmall entropy – few states are occupiedlarge entropy – many states are visitedrequirement of maximal number of microstates

– effort to reach maximal entropy


Entropy

BTW: Ludwig Boltzmann

Austrian physicist


Entropy

Microscopic definition of entropy

Entropy can be related to the order in the systemlow entropy – only a small part of configuration space accessible

– ordered systemhigh entropy – extended part of the configuration space covered

– less ordered system

Example – pile of books on a deskJan Cerny (Charles University in Prague, Dept Cellular Biology):

anthropy – “entropy of human origin”

another route to entropy – information entropyminimal entropy = perfect knowledge of a system (pi = 1)maximal entropy = no information at all – all states equally likely


Canonical ensemble

Closed system – canonical ensemble

system in thermal contact with the surroundings– temperature rather than energy remains constant– Boltzmann distribution of pi applies:

pi =exp[−β · εi ]∑j exp[−β · εj ]

ln pi = −βεi − lnQ

Q =∑j

exp[−β · εj ]

Q – canonical partition function (Zustandssumme)derive the meaning of β – fall back to basic thermodynamics. . .


Canonical ensemble


energy – basic thermodynamic potential – dependence on:

E = E (S ,V ,N) = TS − pV − µN

thermodynamic temperature – comes into play with entropy

∂E

∂S= T or

∂S

∂E=

1

T

use these identities. . .∑i

∂pi∂β

=∂

∂β

∑i

pi =∂

∂β1 = 0

∂

∂β

∑i

pi ln pi =∑i

∂pi∂β

ln pi +∑i

∂pi∂β

=∑i

∂pi∂β

ln pi


Canonical ensemble


. . . and apply the microscopic definition of S :

1

T=

∂S

∂E=

∂S∂β

∂E∂β

=−kBN ∂

∂β

∑i pi ln pi

N ∂∂β

∑i piεi

=−kBN

∑i∂pi∂β ln pi

N∑

i∂pi∂β εi

=

= −kB ·∑

i∂pi∂β · βεi +

∑i∂pi∂β · lnQ∑

i∂pi∂β · εi

= kB · β (1)

Finally, we have estimated the factor β as

β =1

kBT


Canonical ensemble

Continuous systems

continuous energy levels – infinitesimally narrow spacing– introduce the occupation density of energy states ρ

ρ(~r , ~p) =1

Q· exp

[−E (~r , ~p)

kBT

]– this is the phase space density that we wish to obtain froman ergodic MD simulation performed with a correct thermostat!

canonical partition function is then

Q =

∫exp

[−E (~r , ~p)

kBT

]d~r d~p


Canonical ensemble

Canonical partition function

Q – seems to be purely abstract. . .but – to characterize the thermodynamics of a system

Q is completely sufficient,all thermodynamics observables follow as functions of Q

once Q is obtained from the properties (energy levels) of singlemolecules or similar – thermodynamic properties of macroscopicsystems can be calculated

partition function connects microscopic and macroscopic world

example – mean total energy of a system

〈E 〉 = −∂ lnQ

∂β= kBT

2 · ∂ lnQ

∂T

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