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Introduction to Statistical Thermodynamics (Recall)

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2 Basic assumption Each 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

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3 Consistent with classical thermodynamics? Systems 1 and 2 are weakly coupled such that they can exchange energy. What will be E 1 ?

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4 Summary: micro-canonical ensemble (N,V,E) Partition function: Probability to find a particular configuration Free energy

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5 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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6 Canonical ensemble Consider a small system that can exchange heat with a big reservoir 1/k B T Hence, the probability to find E i : Boltzmann distribution

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7 Thermodynamics What is the average energy of the system? Compare: Hence: Thermo recall (2) First law of thermodynamics Helmholtz Free energy:

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8 Summary: Canonical ensemble (N,V,T) Partition function: Probability to find a particular configuration Free energy

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9 Constant pressure simulations: N,P,T ensemble Consider a small system that can exchange volume and energy with a big reservoir 1/k B T Hence, the probability to find E i,V i : p/k B T Thermo recall (4) First law of thermodynamics Hence and

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10 N,P,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration:

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11 Grand-canonical simulations: μ,V,T ensemble Consider a small system that can exchange particles and energy with a big reservoir 1/k B T Hence, the probability to find E i,N i : -μ/k B T Thermo recall (5) First law of thermodynamics Hence and

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12 μ,V,T ensemble (2) In the classical limit, the partition function becomes The probability to find a particular configuration:

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Monte Carlo in different ensembles Chapter 5 NVT ensemble NPT ensemble Grand-canonical ensemble Exotic ensembles

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14 Statistical Thermodynamics Partition function Ensemble average Free energy Probability to find a particular configuration

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15 Detailed balance o n

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16 NVT-ensemble

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18 NPT Ensemble Partition function: Probability to find a particular configuration: Sample a particular configuration: Change of volume Change of reduced coordinates Acceptance rules ?? Detailed balance

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19 Detailed balance o n

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20 NPT-ensemble Suppose we change the position of a randomly selected particle

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21 NPT-ensemble Suppose we change the volume of the system

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22 Algorithm: NPT Randomly change the position of a particle Randomly change the volume

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26 NPT simulations

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27 Grand-canonical ensemble What are the equilibrium conditions?

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28 Grand-canonical ensemble We impose: –Temperature –Chemical potential –Volume –But NOT pressure

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29 MuVT Ensemble Partition function: Probability to find a particular configuration: Sample a particular configuration: Change of the number of particles Change of reduced coordinates Acceptance rules ?? Detailed balance

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30 Detailed balance o n

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31 VT-ensemble Suppose we change the position of a randomly selected particle

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32 VT-ensemble Suppose we change the number of particles of the system

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35 Application: equation of state of Lennard-Jones

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36 Application: adsorption in zeolites

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37 Exotic ensembles What to do with a biological membrane?

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38 Model membrane: Lipid bilayer hydrophilic head group two hydrophobic tails water

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40 Questions What is the surface tension of this system? What is the surface tension of a biological membrane? What to do about this?

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41 Phase diagram: alcohol

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42 Simulations at imposed surface tension Simulation to a constant surface tension –Simulation box: allow the area of the bilayer to change in such a way that the volume is constant.

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43 Constant surface tension simulation A A’ LL’ A L = A’ L’ = V

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44 (A o ) = -0.3 +/- 0.6 (A o ) = 2.5 +/- 0.3 (A o ) = 2.9 +/- 0.3 Tensionless state: = 0

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