分子动力学模拟基础知识
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Molecular Dynamics Simulation
o MD: Theoretical Background
Newtonian Mechanics and Numerical Integration
The Liouville Operator Formalism to Generating MD Integration Schemes
o Case Study 1: An MD Code for the Lennard-Jones Fluid
Introduction The Code, mdlj.c
o Case Study 2: Static Properties of the Lennard-Jones Fluid (Case Study 4 in F&S) o Case Study 3: Dynamical Properties: The Self-Diffusion Coefficient •
Ensembles
o Molecular Dynamics at Constant Temperature
Velocity Scaling: Isokinetics and the Berendsen Thermostat
Stochastic NVT Thermostats: Andersen, Langevin, and Dissipative Particle Dynamics The Nosé-Hoover Chain
Molecular Dynamics at Constant Pressure: The Berendsen Barostat
Molecular Dynamics Simulation
We saw that the Metropolis Monte Carlo simulation technique generates a sequence of states with appropriate probabilities for computing ensemble averages (Eq. 1). Generating states probabilitistically is not the only way to explore phase space. The idea behind the Molecular Dynamics (MD) technique is that we can observe our dynamical system explore phase space by solving all particle equations of motion . We treat the particles as classical objects that, at least at this stage of the course, obey Newtonian mechanics. Not only does this in principle provide us with a properly weighted sequence of states over which we can compute ensemble averages, it additionally gives us time-resolved information, something that Metropolis Monte Carlo cannot provide. The ``ensemble averages'' computed in traditional MD simulations are in practice time averages
:
(99)
The ergodic hypothesis partially requires that the measurement time, , is greater than the longest
relaxation time
,
, in the system. The price we pay for this extra information is that we must at least access if not store
particle velocities in addition to positions, and we must compute interparticle forces in addition to potential energy. We will introduce and explore MD in this section.
Newtonian Mechanics and Numerical Integration
The Newtonian equations of motion can be expressed as
(100)
where is the acceleration of particle , and the force acting on particle is given by the negative gradient of the total
potential, , with respect to its position:
(101)
Whereas in a typical MC simulation, in which all we really need is the ability to evaluate the potential energy of a configuration, in MD we actually need to evaluate all interparticle forces for a configuration.
We first encountered interparticle forces in Sec. 4.6 in a discussion of the virial in computing pressure in a standard Metropolis Monte Carlo simulation of the Lennard-Jones liquid. At this point, it suffices to consider a system with generic pairwise interactions, for which the total potential is given by:
(102)
where is the scalar distance between particles and , and is the pair potential specific to pair . For a system
of identical particles, Eq. 102 is a summation of terms. So, the force on any particular particle, , selects
terms from the above summation; that is, those terms involving particle :
(103)
where we can define the quantity is the force exerted on particle by virtue of the fact that it interacts with particle . Because is a function of a scalar quantity, we can break the derivative up:
