http:dbeveridge.web.wesleyan.eduwescourses2004schem38801mdcourse

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http//dbeveridge.web.wesleyan.edu/wescourses/2004
s/chem388/01/mdcourse/
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Introduction Chapter 1 Haile

• System
• Modeling vs. Simulation
• Models for molecular simulation
• Steps in Simulation
• Simulation Theory or experiment?
• Simulations Stochastic vs. Deterministic

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What is a system?

• A system is
• A portion of the world on which to focus
  attention
• A subset of the universe (System Surrounding)
• Composed of any number of similar or dissimilar
  parts (i.e. be homogeneous or heterogeneous)

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• What is a state of the system?
• A specific conditions of all the parts
• What are the Observables of a system?
• Numerical values referring to a state or function
  which are, in principle, measurable
  experimentally
• Eg. PV NkBT

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State Variables
Extensive variable Scales linearly with system
size Intensive variable Conjugate field,
size-independent
Ensembles
EVN Microcanonical TPN Isothermal-isobaric TVN
Canonical TV? Grand-canonical
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Modeling vs. Simulation
Studying a system involves
Manipulate and control certain observables
(inputs)
Measure or compute other observables (outputs)
System
Problem To define the state in such a way that
complicated interactions among state variables
are decoupled so that observable outputs can be
computed.
A model is simpler than the system it mimics.
A simulation is more complex than the system it
simulates.
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Modeling vs. Simulation

• A model is a representation of the system,
  conceptual or mathematical, which
• Behaves like the system
• But involves fewer states
• i.e. some interactions of lesser importance are
  neglected
• A simulation is a numerical calculation of
  properties of a model based on a trajectory.
• Decompose interactions
• More constraints

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Modeling vs. Simulation
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Models for Molecular Simulation
Molecular Interactions

Simulated Model
Boundary Conditions
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Steps in Simulation

• Model building
• Calculation of a trajectory
• Analysis of the trajectory

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Hierarchy of Scientific modes of Investigation
System of Interest
Experiment
Theory
Simulation
Reductionism
Computer simulation
Mathematical models
Analytically Solvable
Computer simulation
Experiment
Theory

• Development of Models occurs as a result of
  Reductionistic approach to a system.

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Simulation Deterministic vs. Stochastic
Deterministic
Stochastic
Metropolis Monte Carlo
Force-Biased Monte Carlo
Brownian Dynamics
Langevin Dynamics
Molecular Dynamics
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Assessment of a Simulation

• What are the possible sources of error or
  uncertainty?
• Are sampled populations representative?
• Are the results reproducible?
• Are the results internally consistent?
• Do the results confirm results from elsewhere?

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Fundamentals Chapter 2 Haile

• Newtonian Dynamics
• Hamiltonian Dynamics
• Phase space
• Classification of Dynamical Systems
• Phase space Problem
• Stability of Trajectories
• Determination of Properties
• Sampling Theory
• Periodic Boundary Conditions

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Newtonian Dynamics

• First Law
• A system will be at rest or moving at a specific
  velocity until a force acts on it.
• Second Law
• F ma
• Third Law
• Any force exerted by molecule 1 on molecule 2
  must be balanced by a force exerted by 2 on 1

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Hamiltonian Dynamics
Hamiltonian
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Difference Between Newtonian and Hamiltonian
Dynamics

• Newtonian Dynamics considers forces explicitly
• Hamiltonian Dynamics conserves Hamiltonian

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Phase-Space of 1-DHO
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Phase Space

• A 6N dimensional hyper-surface composed of 3N
  configuration space terms and 3N momentum space
  terms

R. Penrose, The Emperors New Mind, OUP, NY 1989.
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Classification of Dynamical Systems
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Classification of Dynamical Systems

• Recurrent Trajectories
• For a phase space of finite volume, the phase
  point will pass, arbitrarily closely and
  indefinitely often, almost every accessible
  configuration on the trajectory.
• Non-Recurrent Trajectories
• Motion of most comets

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Classification of Dynamical Systems

• Hamiltonian
• The trajectory is constrained in the phase space
  to the hyper-surface of constant H
• Non-Hamiltonian
• Eg Dissipative Systems

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Classification of Dynamical Systems

• Integrable
• Number of constants of the motion equals the
  number of degrees of freedom
• E.g. I-DHO
• Non-Integrable
• Equation of motion has nonlinear terms
• E.g. Classic 3-body problem

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Classification of Dynamical Systems

• Periodic
• Within a finite time, the phase trajectory will
  intersect itself
• Quasi-Periodic
• The phase-space trajectory is confined to the
  surface of a torus. The torus is a subspace of
  the constant-Hamiltonian surface.

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Quasi-Periodic Motion
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Classification of Dynamical Systems
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Classification of Dynamical Systems

• Non-Ergodic
• Ergodic
• Over a sufficiently long time the phase point
  passes through all configurations on the constant
  Hamiltonian surface
• Ergodicity does not guarantee that an isolated
  system started from an arbitrary non-equilibrium
  state will irreversibly evolve to equilibrium
  states.

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Stability of a Trajectory
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Classification of Dynamical Systems Ability to
Evolve from Non-Equilibrium to Equilibrium States

• Mixing
• A perturbed trajectory must drift away from its
  unperturbed parent trajectory, may be slow or
  fast
• K-Flow
• In response to a small perturbation, a K-flow
  trajectory diverges exponentially
• B-Flow
• The state observed at an instant on a perturbed
  trajectory is completely uncorrelated with the
  state on the parent trajectory

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Classification of Dynamical Systems
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Classification Summary

• Seven classes of Hamiltonian Dynamics are not
  mutually exclusive
• High levels of instability encompass the
  desirable properties of lower levels
• In MD simulations, the equations of motion are
  non-integrable and the computed trajectories are
  recurrent, Hamiltonian, at least mixing and
  preferably K-flows.

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Meta-Stable States