D. Frenkel and B. Smit. Understanding Molecular Simulation, Second Edition. Academic Press, San Diego, 2002. 57  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as the time required for a simulated system to achieve equilibrium. The diffusion coefficient D is proportional to the slope of R( over long times via the Einstein relation. More details for computing these quantities can be found in Allen and Tildesley [55] Rapaport [56] and Frenkel and Smit [57]. Table 2. Some Computable quantities Specific Heat at constant volume CV = 3N 4 9 h(T Gamma hT i t ) i t hT i Gamma1 kB Velocity autocorrelation function Z( D v(t) Delta v(t ) E Pair correlation function (radial distribution function) N n(r) 4r ....

....r OH1 = r OH2 = r 0 ; r HH = r 1 : Numerical integration of these constrained equations of motion again provides a simple test problem. 3.2 Lennard Jones Fluids Careful description of argon simulations are given in [52] and [58] we summarize these here. Chapter 4 of Frenkel and Smit [57] includes detailed algorithms for simulating a Lennard Jones system and analyzing the resulting trajectory. The atoms of liquid argon are assumed to interact with a Lennard Jones potential (1.1) In reduced unit computer simulation of the Lennard Jones Table 4. Potential energy parameters for a ....

D. Frenkel and B. Smit, Understanding molecular simulation. From algorithms to applications, Academic Press, 1996

....the time reversal symmetry. In numerical experiments, it is shown that the new method exhibits enhanced stability when the temperature uctuation is large. Extensions are presented for Nos e chains, holonomic constraints, and rigid bodies. 1. INTRODUCTION Molecular dynamics computer simulation [1, 2] has become a standard tool in computational biophysics and chemistry. Traditional molecular dynamics samples con gurations from a constant energy or microcanonical distribution. This is often inappropriate since experiments are usually performed at constant temperature (canonical ensemble) ....

....changes in the ow. Methods using ad hoc non reversible temperature controls [4] and isokinetic constraints [5 7] have also been proposed in the literature. These methods succeed in producing smooth trajectories, but they fail to yield the correct canonical uctuations in the kinetic energy [1]. This paper will focus on the newer dynamical methods derived from the extended Hamiltonian proposed by Nos e [8, 9] HNos e = g k T ln s: 1) Here g = N f 1, where N f is the number of degrees of freedom of the real system. The constants T and k are temperature and ....

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D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic Press, New York, 1996).

....likely to occur leads to a Boltzmann probability distribution P (instate i ) 1 Q(c) e Gamma E i c where the Q(c) is systems partition function and c = k b T . This probability distribution of energy states leads to Metropolis criterion, the proof can be found for example from reference[6] Simulated annealing can then be viewed as a sequence of Monte Carlo simulations where the control parameter c (temperature) is slowly decreased between sequences of Monte Carlo moves. The cost function that is optimized is the analog of the potential energy landscape. In a pseudo code ....

D. Frenkel, B. Smit, Understanding Molecular Simulation

....= 01, where is the set of real numbers (in the program, represented by a double precision variable) and we use a vector notation, r r i . Data strcutures in md.c) int nAtom: N, the number of atoms. NMAX: Maximum number of atoms that can be handled by the program. double r[NMAX][3]: r[i] 0] r[i] 1] and r[i] 2] are the x, y, and z coordinates of the i th atom, where i = 0, N 1. x y 1 2 N . Trajectory: A mapping from time to a point in the 3 dimensional space, trt i a r ( 3 . In fact, a trajectory of an N atom system is regarded as a curve in 3N dimensional ....

....N NNN = 000111 1 1 1 . r i (t=0) v i (t=0) r i (t=t 1 ) v i (t=t 1 ) Velocity: Short time limit of an average speed (how fast and in which direction the particle is moving) r r rrr vt rt dr dt rt rt ii ii ( lim ( 0 . double rv[NMAX][3]: rv[i] 0] rv[i] 1] and rv[i] 2] are the x, y, and z components of the velocity vector, r v i , of the i th atom. Acceleration: Rate at which a velocity changes (whether the particle is accelerating or decelerating) rr r rrr at rt dr dt dv dt vt vt ii iii ( lim ( ....

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D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic Press, New York, 1996).

....going beyond standard Hamiltonian dynamics. Today a large variety of methods exist, which drive the system into the canonic state, e.g. by introduction of artificial degrees of freedom or by coupling the system to a heat bath via stochastic methods. The reader will find more details in Refs. [16, 17]. The choice for the present work is a Langevin thermostat [18] This means that instead of integrating Newton s equations of motion, one solves a set of Langevin equations m i r i = GammarU (fr i g) Gamma Gamma r i i (t) 1) with i (t) being a ffi correlated Gaussian noise source with ....

....the two rods. Even for v = 1 it is only half as large as the distance between rod axis and Wigner Seitz boundary. Note that this statement is independent of density. The main advantage of this approach is that such a system can be treated with the plain cubic Ewald sum or one of its mesh upgrades [17, 27, 28]. This permits a very efficient way for computing the long range electrostatic interactions. Its main disadvantage is the coupling of counterion number and cell volume. 2.3 Interaction potentials A specification of two interaction potentials is necessary two describe the model system: i) an ....

Frenkel D, Smit B. Understanding Molecular Simulation. San Diego: Academic Press, 1996.

....requires only 0.29 kJ mol. Whereas the determination of mechanical properties is now routine for computer simulation, the determination of (relative and absolute) free energies and other thermal properties, which depend on the volume of phase space, remains one of the most challenging problems [15,21]. Over the past few years many new methods have been proposed which greatly aid in the calculation of phase equilibria [21 27] The simulations described herein 5 were carried out using a combination of the Gibbs ensemble Monte Carlo (GEMC) method [28 30] and the con gurational bias Monte Carlo ....

.... of (relative and absolute) free energies and other thermal properties, which depend on the volume of phase space, remains one of the most challenging problems [15,21] Over the past few years many new methods have been proposed which greatly aid in the calculation of phase equilibria [21 27]. The simulations described herein 5 were carried out using a combination of the Gibbs ensemble Monte Carlo (GEMC) method [28 30] and the con gurational bias Monte Carlo (CBMC) algorithm [11,3134 ] GEMC utilizes two separate simulation boxes that are in thermodynamic contact, but do not have an ....

D. Frenkel and B. Smit, Understanding Molecular Simulation, (Academic Press, New York, 1996).

.... the exact Schrodinger equation by some numerically tractable theoretical model, then apply the machinery of control theory to this model, and finally get into the numerics, or should we proceed the other way round, applying the 10 why not use a molecular dynamics model like those of [15, 17, 33], coupled with a quantum model for the reactive part of the system, if necessary 90 ESAIM: Proc. Vol. 8, 2000, 77 94 Schrodinger Equation # Control # approximation # # # # # # # # # Continuous approximation # Control # Discretization Figure 2: Some possible ways to ....

D. Frenkel & B. Smit, Understanding molecular simulation, Academic Press, 1996.

....of parameters, and present some numerical test results. PACS: 02.50. r; 02.70.Ns; 05.20.Gg; 82.20.Fd; 82.20.Wt Keywords: Molecular Dynamics; NPT Ensemble; Symplectic Algorithms; Stochastic Dynamics; Langevin Equation; Fokker Planck Equation I. INTRODUCTION Molecular Dynamics (MD) simulations [1 3] are a very efficient tool to study the statistical properties of thermodynamic systems, especially at high densities where the acceptance rates of standard Monte Carlo (MC) simulations [4] are small. They are also very well suited to study dynamical properties. MD simulations are most naturally ....

.... P is the external pressure, and fi = 1= k B T ) From this one immediately reads off that one has to run a standard Metropolis algorithm on the state space (L; f s i g) using an effective Hamiltonian U eff = U PV Gamma NkBT ln V: 3) The MD approach [1,2] to non microcanonical ensembles [1 3,5 7], pioneered by Andersen [8] Nos e [9] and Hoover [10] is slightly more involved. Like in MC, one defines an additional dynamical variable whose fluctuations allow to keep the thermodynamically conjugate variable fixed. In our example, this variable is L, while P is fixed. However, the dynamics ....

D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic Press, New York, 1996).

....and the absence of defects and nucleation centers. The rapid heating and quenching rates causes hysteresis, making it difficult to specify the precise temperature corresponding to the phase transition. One approach to avoid this problem is to calculate the free energy by thermodynamic integration [26] or by two phase simulation techniques [9] Instead, we choose to find the phase coexistence curve by using the Clausius Clapeyron equation, dP dT = 1 T DeltaH DeltaV ; 10) which relates the derivative of the coexistence curve between two phases to the change in enthalpy and volume ....

D. Frenkel and B. Smit, Understanding Molecular Simulation, from algorithms to applications (Academic Press, San Diego, 1996).

....ff) 3) where z(T ; ff) is the normalizing constant of the system density. Simulation of from p( jT ; ff) q( jT ; ff) z(T ; ff) is typically carried out via MCMC methods. For detailed discussions of this and related topics, see, among others, Ciccotti and Hoover (1986) Ceperley (1995) and Frankel and Smit (1996). A more statistically oriented review is given in Neal s (1993) comprehensive overview on simulation techniques. In applications in both genetics and physics, the real interest is not a single normalizing constant itself, but rather ratios, or equivalently differences of the logarithms, of them ....

....application of (7) in conjunction with (3) using log q( j = ff) GammaH ( ff) kT ) Similarly, we can calculate free energy difference for systems with different temperatures but the same ff; identity (10) also allows for different ff s and different T s simultaneously. See Frenkel (1986) Frankel and Smit (1996), and Neal (1993, Section 6.2) for more discussions of thermodynamic integration so named because identities such as (12) were originally derived from differential equations for describing thermodynamic relationships. Applying (7) Ogata (1989; also see Ogata and Tanemura, 1984) proposed an ....

Frankel, D. and Smit, B. (1996). Understanding Molecular simulation. Academic Press, Boston.

....particle number. Additional molecular dynamics simulations are used to randomize and thermalize the large system (see Figure 1.4) The MD part of the program MCMD is optimized for large particle numbers. The linked cell method is used to reduce the effort for the calculation of pair distances [4]. The calculation of the force matrix which usually comsumes more than 98 percent of the computing time is parallelized for shared memory and distributed memory parallel computing machines (SGI,SUN,Cray) Parallelization was not a primary goal of this project, but a speedup of 10 is easily ....

....velocity distribution (see Figure 3.3) 3.2 Benchmarks The parameters for the benchmark runs are given in Tables 3.1 and 3.2. The radial cutoff for simulations with 1,000,000 particles is reduced to 10 Theta 10 Gamma10 m. The trajectories are calculated using the velocity Verlet al..gorithm [4] with strict temperature scaling is applied in every time step. Computing times per MD time step for systems with one million particles are shown in Table 3.3. These values were obtained from an average interval over 10 time steps. oe ffl =10 Gamma10 m =K Ar Ar 3.405 119.8 Kr Kr 3.633 167.0 ....

D. Frenkel and B. Smit. Understanding molecular simulation. Academic Press, Amsterdam, 1996.

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D. Frenkel and B. Smit. Understanding Molecular Simulation, Second Edition. Academic Press, San Diego, 2002. 57

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D. Frenkel, B. Smit, Understanding Molecular Simulation, Second Edition, Academic Press, San Diego, 2002.

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D. Frenkel and B. Smit; Understanding Molecular Simulation, Academic Press, Boston (1996).

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D. Frenkel and B. Smit; Understanding Molecular Simulation, Academic Press, Boston (1996).

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D. Frenkel and B. Smit, Understanding molecular simulation. From algorithms to applications, Academic Press, San Diego, 1996.

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D. Frenkel and B. Smit, Understanding Molecular Simulation. New York: Academic Press, 1996.

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D. Frenkel and B. Smit, Understanding molecular simulation. From algorithms to applications. Academic Press, San Diego (1996).

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D. Frenkel and B. Smit, Understanding molecular simulation. From algorithms to applications. Academic Press, San Diego (1996).

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D. Frenkel and B. Smit, Understanding Molecular Simulations, (Academic Press, New York, 1996).

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D. Frenkel, B. Smit, Understanding Molecular Simulation (Academic Press, San Diego, 1996).

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