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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
Principles of statistical thermodynamics
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.
Principles of statistical thermodynamics
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
Principles of statistical thermodynamics
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
Principles of statistical thermodynamics
Microcanonical ensemble
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?
Principles of statistical thermodynamics
Microcanonical ensemble
Example – counting the microstates
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
Principles of statistical thermodynamics
Microcanonical ensemble
Example – counting the microstates
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
Principles of statistical thermodynamics
Microcanonical ensemble
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
Principles of statistical thermodynamics
Microcanonical ensemble
BTW: James Stirling
Scottish mathematician
Principles of statistical thermodynamics
Microcanonical ensemble
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
Principles of statistical thermodynamics
Microcanonical ensemble
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
Principles of statistical thermodynamics
Microcanonical ensemble
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
Principles of statistical thermodynamics
Microcanonical ensemble
Most probable configuration
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
Principles of statistical thermodynamics
Microcanonical ensemble
Most probable configuration
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 ]
Principles of statistical thermodynamics
Microcanonical ensemble
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
Principles of statistical thermodynamics
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
Principles of statistical thermodynamics
Entropy
BTW: Ludwig Boltzmann
Austrian physicist
Principles of statistical thermodynamics
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
Principles of statistical thermodynamics
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. . .
Principles of statistical thermodynamics
Canonical ensemble
Closed system – 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
Principles of statistical thermodynamics
Canonical ensemble
Closed system – 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
Principles of statistical thermodynamics
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
Principles of statistical thermodynamics
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