Introduction to (Statistical) Thermodynamics

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Introduction to (Statistical) Thermodynamics
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Molecular Simulations
MD

• Molecular dynamics solve equations of motion
• Monte Carlo importance sampling

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MC
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Questions
What is the desired distribution?

• 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?

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Summary

• Thermodynamics
• 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

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Entropy

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

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Helmholtz free energy
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Statistical Thermodynamics

• System of N molecules

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

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Statistical Thermodynamicsthe 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

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Basic assumption
, but there are not many microstates that give
these extreme results
Each individual microstate is equally probable
If the number of particles is large (gt10) these
functions are sharply peaked
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Does the basis assumption lead to something that
is consistent with classical thermodynamics?
Systems 1 and 2 are weakly coupled such that
they can exchange energy. What will be E1?
BA each configuration is equally probable but
the number of states that give an energy E1 is
not know.
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Energy is conserved! dE1-dE2
This can be seen as an equilibrium condition
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Entropy and number of configurations
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)

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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)
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And this gives the statistical definition of
temperature
In short Entropy and temperature are both
related to the fact that we can COUNT states.

• Basic assumption
• leads to an equilibrium condition equal
  temperatures
• leads to a maximum of entropy
• leads to the third law of thermodynamics

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Number of configurations

• How large is ??
• 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.

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Canonical ensemble
1/kBT
Consider a small system that can exchange heat
with a big reservoir
Hence, the probability to find Ei
Boltzmann distribution
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Thermodynamics
What is the average energy of the system?
Compare
Hence
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Remarks (1)
We have assume quantum mechanics (discrete
states) but we are interested in the classical
limit
Integration over the momenta can be carried out
for most systems
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Remarks (2)
Define de Broglie wave length
Partition function
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Example ideal gas
Free energy
Pressure
Energy
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Ideal gas (2)
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SummaryCanonical ensemble (N,V,T)
Partition function
Probability to find a particular configuration
Free energy
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Summarymicro-canonical ensemble (N,V,E)
Partition function
Probability to find a particular configuration
Free energy
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Thermodynamic properties
Ensemble average
For many properties the momenta do not matter
only the configurational part
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Other ensembles?
COURSE MD and MC different ensembles
In the thermodynamic limit the thermodynamic
properties are independent of the ensemble so
buy a bigger computer
However, it is most of the times much better to
think and to carefully select an appropriate
ensemble.
For this it is important to know how to simulate
in the various ensembles.
But for doing this wee need to know the
Statistical Thermodynamics of the various
ensembles.
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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?

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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?

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Ensembles

• Micro-canonical ensemble E,V,N
• Canonical ensemble T,V,N
• Constant pressure ensemble T,P,N
• Grand-canonical ensemble T,V,µ

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Constant pressure simulations N,P,T ensemble
1/kBT
p/kBT
Consider a small system that can exchange volume
and energy with a big reservoir
Hence, the probability to find Ei,Vi
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N,P,T ensemble (2)
In the classical limit, the partition function
becomes
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Grand-canonical simulations µ,V,T ensemble
1/kBT
-µ/kBT
Consider a small system that can exchange
particles and energy with a big reservoir
Hence, the probability to find Ei,Ni
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µ,V,T ensemble (2)
In the classical limit, the partition function
becomes