Simulations in other ensembles

1
Simulations in other ensembles

• Nathan Baker
• (baker_at_biochem.wustl.edu)
• BME 540

2
What are some other ensembles?

• Vary the independent variables
• Useful for describing open and closed physical
  systems
• Variants
• NVE microcanonical
• NVT canonical
• NPT isobaric-isothermal
• NsT constant stress
• µVT grand canonical
• NPH isenthalpic-isothermal

NVE
NVT
µVE
NPT
µVT
NsT
NPH
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Extensive vs. intensive variables

• Extensive variables
• Measure the extent of the system
• Think change when the size of the system
  changes
• Examples total energy, particle number, volume,
  mass
• Intensive variables
• Intrinsic to the system
• Think does not change with the size of the
  system
• Examples temperature, chemical potential,
  pressure, heat capacity, density
• Most ensembles have at least one extensive
  variable
• Thermodynamic potential defines ability to do
  work
• No potential without extensive variable ( 0)

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Where do other ensembles come from?

• Physical situations
• Necessity
• Some ensembles arent well suited to certain
  phenomena
• Choice based on fluctuations
• Density fluctuations in closed ensembles
• Volume fluctuations in constant-stress ensembles
• Transformation of variables
• Usually intensive for extensive
• Conceptually
• Coupling system to bath with desired extensive
  properties
• Sampling extensive variable (with appropriate
  weight)

5
Transforming ensembles macroscopic view

• How do we change between sets of independent
  variables in a function?
• Legendre transformation
• Identify conjugate variables
• as slope of function at each point

6
Transforming ensembles macroscopic view
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Transforming ensembles microscopic view

• Consider the conceptual model of a bath
• Exchange states between systems
• Sample all states of a variable with given
  probability
• Start with the microcanonical ensemble
• Identify the probability function for coupling
  to the bath (e.g., what variable can fluctuate?)
• Integrate out the variable of interest
• Identify undetermined multiplier via
  thermodynamics

Variable to be averaged out
Conjugate variable
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Transforming ensembles

• Thermodynamic potential transformed by Legendre
  method
• Microscopic probability transformed by entropy
  maximization
• Conjugate variables determined from thermodynamics

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Isobaric-isothermal ensemble

• Start from the canonical ensemble
• Transform out the volume
• Identify the multiplier by thermodynamic
  relationships
• The conjugate variable to the volume is the
  pressure
• Not too surprising
• A piston to compensate for volume fluctuations

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Coupling pressure and volume

• Domains are not necessarily symmetric
• Pressures are not necessarily isotropic
• How do we calculate (pV)?
• Use a stress tensor diagonals are pressure
  terms, off-diagonals are shear forces
• Shear forces (simple) do not change volume
• Changes in V are made through qi

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NpT simulations

• What do we expect?
• Energy fluctuations variance increases with the
  heat capacity
• Volume fluctuations

Variance increases with the isothermal
compressibility
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NpT Monte Carlo

• Start with NVT type of framework
• Scale coordinates
• Include changes in box dimensions

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NpT Metropolis Monte Carlo

• Choose a move
• Move particles
• Scale box
• Evaluate the energy (one can often cheat for
  rigid bodies)
• Calculate transition probability
• If accepted, evaluate observable
• Accumulate integral
• The temperature and pressure are prescribed and
  fixed via the Boltzmann factor

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NpT Monte Carlo pros and cons

• Pros
• Your temperature is set (exactly) through the
  distribution
• Your pressure is set (exactly) through the
  distribution
• Fancy move sets
• Biased sampling
• Cons
• Convergence
• Random number coverage issues
• Ergodicity
• Lack of dynamic information
• Software

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Constant pressure molecular dynamics

• Same problems as NVT molecular dynamics
• Exact ODE integration is an NVE method
• How do we sample various p and T energy surfaces
  with our method?
• We need a barostat
• Similar approaches
• Stochastic (Langevin)
• Extended Lagrangian (effective potential)

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What is pressure related to?

• For NVT, we related instantaneous temperature to
  the kinetic energy
• This was the same for ideal and non-ideal systems
• For NpT, we need an instantaneous pressure the
  following ingredients
• Ideal gas pressure
• Excess pressure
• Ideal and excess pressures have different
  fluctuations

Intermolecular pair virial function. Can be
tensor-valued. Long-range contributions
essential.
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Andersen how to simulate NpH

• Easy extended Lagrangian to understand add a
  piston to the system
• Unfortunately, this simulates an isenthalpic
  ensemble

Piston mass
Piston PE
Instantaneous pressure
Piston KE
Coordinate acceleration
Piston acceleration
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Berendsen close but no cigar

• Pressure coupling obtained via bath concept
• Strength of coupling related to isothermal
  compressibility and frequency of coupling
• Similar to methods used for temperature
  regulation
• Neither reproduces a standard ensemble (close,
  though)

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Other extended Lagrangian methods

• All similar in spirit to Nosé-Hoover (as
  discussed last time)
• Add additional variables
• Rescale positions, velocities based on equations
  of motion
• Not all methods are stable and/or give the
  correct ensemble
• Martyna, Tobias, Klein
• Reproduces NpT
• Extended variable is ln(V/V0) includes piston
  momentum, position (volume), mass
• Use Nosé-Hoover chain type of method
• Rahman-Parrinello
• Reproduces NpT
• Good stability
• Allows shear motion of box
• Not so good for liquids
• Useful for solids, liquid crystals, etc.
• Permits investigation of phase changes

20
Grand canonical ensemble

• Start from the canonical ensemble
• Transform out the number
• Identify the multiplier by thermodynamic
  relationships
• The conjugate variable is the chemical potential
  (activity)
• Not too surprising
• A reservoir in contact through a permeable
  membrane

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Coupling chemical potential and number

• Particles are allowed to enter and leave the
  system at any point
• Allows for number fluctuations
• Important for
• Open systems
• Concentration hard to predict
• Particle number coupled to internal degrees of
  freedom
• Phase transitions

Isothermal compressibility
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µVT Monte Carlo

• Start with NVT type of framework
• Scale coordinates
• Include changes in particle numbers

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µVT Metropolis Monte Carlo

• Choose a move
• Move particles
• Add particle(s)
• Delete particle(s)
• Evaluate the energy (one can often cheat for
  rigid bodies)
• Calculate transition probability
• If accepted, evaluate observable
• Accumulate integral
• The temperature and activity are prescribed and
  fixed via the Boltzmann factor

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µVT Metropolis Monte Carlo

• Choose move
• Displace
• Create
• Destroy

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µVT MC in practice

• Works great for dilute systems
• Gases
• Implicit solvent models
• Requires excess chemical potential
• Needs to be consistent with parameters
• Obtained via simulations to achieve target
  concentration
• Obtained via particle insertion/deletion
• Used in several applications
• Electrolyte modeling
• Ion binding
• Phase transitions

26
µVT molecular dynamics

• Not widely used
• Hard to find cavities in most condensed phase
  simulations
• Difficult to determine dynamics of discrete
  particle insertion
• This would be an interesting class project,
  though

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Summary

• Transforming ensembles
• Conjugate variables
• Legendre transformation
• Entropy maximization
• NpT simulations
• Pressure coupling via piston
• MC implementation simple
• MD implementation more complication
• µVT simulations
• Coupling via excess chemical potential
• Particles added and deleted
• Activity set by extra simulations
• No straightforward MD implementation