NVT Simulations

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NVT Simulations

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

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Canonical ensemble recap

• Canonical ensemble
• Constant number of particles (N)
• Constant system volume (V)
• Constant system temperature (T)
• Weighting function dependent on temperature
• Result canonical ensemble average

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Maxwell distribution

• Consider the ideal gas Hamiltonian
• and partition function.
• The weight of each momentum value is
• giving a probability distribution of

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Averages from the Maxwell distribution

• Mean
• Variance
• Kinetic variables

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Two approaches

• Do we care about kinetic properties
• We cannot pre-average over momenta
• We need to sample velocities and positions
• We need molecular dynamics
• or just thermodynamic ones?
• We can pre-average over momenta
• We need to sample only positions from stationary
  distribution
• We can use Monte Carlo or molecular dynamics

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Monte Carlo approaches

• Consider an observable that depends only on
  position
• Assume position and momentum are separable in
  Hamiltonian
• Pre-average over the momenta
• Sample observable based on Boltzmann distribution

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Monte Carlo methods

• How do we sample?
• Importance sampling
• Markov chain methods
• Usually Metropolis Monte Carlo (or similar)
• Choose a move
• Evaluate the energy
• Accept/reject with Boltzmann probability
• If accepted, evaluate observable
• Accumulate integral
• The temperature is prescribed and fixed via the
  Boltzmann factor

Temperature!
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Monte Carlo pros and cons

• Pros
• Your temperature 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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Molecular dynamics approaches

• When kinetic information is important
• We need to integrate equations of motion
• If energy is conserved, then we choose our
  initial conditions such that

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NVE molecular dynamics

• If energy is conserved, then we choose our
  initial conditions such that
• If the observable evolves according to this
  constant-E simulation, then (ergodic theorem)
• Problem the title of this lecture is NVT
  Simulations not NVE simulations

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NVE to NVT

• Conceptual solution (per NVT lecture)
• Not very practical
• Simulations are expensive
• Wed need a lot of E values
• We cant conserve energy very well anyway
• How do we run a simulation at constant
  temperature?

We could run lots of NVE simulations with
different energies (via initial conditions). The
results could be used in a Boltzmann-weighted
average.
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Velocity rescaling

• How do we get our momenta to satisfy Maxwell
  distribution?
• Some possibilities
• Constrain instantaneous velocities by rescaling
• Occasionally randomize from Maxwell distribution
• Problem the instantaneous kinetic energy in an
  NVT ensemble fluctuates
• Upshot not sampling NVT
• Good for thermal equilibration of systems
• Kinetic observables improperly sampled
• Thermodynamic observables less sensitive some
  (related to fluctuations) can be incorrect
• Is there a better way?

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Thermostats

• Regulate the momenta to obey the NVT
  distribution
• Correct mean
• Correct higher-order moments
• Consider interaction of system with thermal
  bath
• Additional degrees of freedom in system
• Interact with particles momenta
• Bath properties recover desired distribution
• Methodology
• Describe bath dynamics
• Describe interaction of system with bath

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Andersen thermostat concept

• What if there was a Markov process to move our
  NVE simulation from E to E with the correct
  distribution?
• What are the important properties of this
  process?
• Frequency
• Magnitude of energy change
• There is a physical process with these
  properties random collisions
• Basic ingredients
• NVE dynamics simulation
• Stochastic collisions with heavy particles

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Andersen thermostat algorithm

• Standard NVE dynamics are supplemented with the
  following step
• Result
• A subset of the particle velocities are
  reassigned each step
• The frequency of reassignment is
  Poisson-distributed

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Andersen thermostat properties

• Pros
• Reproduces NVT distribution
• Simple implementation
• Few parameters
• Cons
• Artificial introduction of noise into system
• Enhanced decay of kinetic correlations
• Potentially poor prediction of kinetic variables
• Worse for larger frequencies
• Smaller frequencies allow possible drift in
  temperature and skewed statistics

Relative diffusion constant of LJ fluid. Adapted
from Fig. 6.3 of Frenkel Smit.
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Langevin methods

• Fluctuation-dissipation the way a system
  fluctuates is related to the way it dissipates
  energy
• If we couple our system to a heat bath, then
• We should include random collisions with the
  bath
• And we should include a mechanism for dissipating
  the energy
• These two conditions are combined in the Langevin
  equation

Potential gradient
Random term
Collision frequency
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Langevin thermostats fluctuation-dissipation

• The properties of the random force are related to
  the collisional damping
• The time dependence of the random force can be
  chosen to satisfy various (non-Markovian)
  relaxation schemes
• or instantaneous relaxation.

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Langevin thermostats explicit solvent
applications

• Thermostat coupled to each particle (think
  Andersen)
• Regulates temperature via collisions
• Can influence kinetic variables but can also be
  corrected!
• Gives correct equilibrium distribution (if
  fluctuation-dissipation satisfied)

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Langevin thermostats implicit solvent
applications

• Implicit solvent bath coupled to simulation
• Regulates temperature and mimics solvent
  frictional effects
• Friction coefficient related to particle radii
• The price of implicit solvent?
• Coupling between particles (hydrodynamic
  interactions)
• Complicated dynamic effects (memory kernels)

Coupling between collisions
Frictional coupling
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This is not a Nosé-Hoover thermostat

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Nosé-Hoover thermostat

• Example of extended Lagrangian dynamics methods
• Introduce a new degree of freedom in the system
  get approximate Hamiltonian
• Simulations carried out at NVE in expanded
  ensemble
• Result
• Fluctuations in s model kinetic energy
  fluctuations
• Recover NVT ensemble statistics

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Nosé-Hoover thermostat equations of motion

• Extra variable new equations
• From inspection this looks like a
  time-dependent viscosity!
• In other words, this method scales momenta in
  response to the conjugate variable

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Nosé-Hoover thermostat application

• There are several caveats in the implementation
• System constraints (center of mass, etc.)
  required to achieve NVT statistics
• Couple the Nosé-Hoover thermostat to a chain of
  thermostats
• With the correct implementation, the end result
  is excellent
• Good accuracy in kinetic variables (less
  sensitive to effective mass)
• Convergence to NVT distribution

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Problems with NVT simulations

• Rapid changes in potential energy are problematic
• Rapid changes in kinetic energy
• Tougher temperature control
• Limits energy landscapes and perturbations to
  system
• Viscosity limits sampling
• Dynamics has intrinsic viscosity
• Viscosity controls rate of change in
  configurations
• Potential decrease in sampling rate
• Did you really want MC instead?

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Summary

• Monte Carlo
• Metropolis (and other) algorithms ensure correct
  distribution
• No kinetic information momenta pre-averaged
• Flexible implementation
• Molecular dynamics
• Thermostats to control temperature
• Stochastic
• Deterministic
• Kinetic information available
• Watch out for thermostat artifacts
• Watch out for timescale limitations
• Less flexible application
• Energy surfaces
• Timescale constraints