Computer simulation theorem

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

M. Menon and R. E. AUen, New technique for molecular-dynamics computer simulations Hellmann- Feynman theorem and subspace Hamiltonian approach , Phys. Rev. B33 7099 (1986) Simulations of atomic processes at semiconductor surfaces General method and chemisorption on GaAs(llO) , ibid B38 6196 (1988). 

Apart from their pedagogical value, reversible rules may be used to explore possible relationships between discrete dynamical systems and the dynamics of real mechanical systems, for which the microscopic laws are known to be time-reversal invariant. What sets such systems apart from continuous idealizations is their exact reversibility, discreteness assures us that computer simulations run for arbitrarily long times will never suffer from roundoff or truncation errors. As Toffoli points out, ...the results that one obtains have thus the force of theorems [toff84a]. ... 

Our point of view is that the evaluation of the partition function (9.5) can be done by using any available tool, specifically including computer simulation. If that computer simulation evaluated the mechanical pressure, or if it simulated a system under conditions of specified pressure, then /C,x would have been determined at a known value of p. With temperature, composition, and volume also known, (9.2) and (9.1) permit the construction of the full thermodynamic potential. This establishes our first assertion that the potential distribution theorem provides a basis for the general theory of solutions. 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

Answers to the following questions are sought. (1) Can a closure, or several closures, be found to satisfy theorems for simple liquids (2) If such closures exist, will they be improvements over conventional ones Do they give better thermodynamic and structural information (3) Will such closure relations render the IE method more competitive with respect of computer simulations and with other methods of investigation ... 

Better agreement between the values calculated using the above equations and the computer simulations is obtained by introducing the correction proposed by Verlet and Weis [32] which forces g(l,r/) to satisfy the consequence of the virial theorem given by Eqn (A3). For values of R > 4, the value of hs(-R) is set equal to unity. 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

The transition path sampling techniques developed for the study of rare events can also be used to improve the calculation of free energies from non-equilibrium transformations on the basis of Jarzynski s theorem. In this approach equilibrium free energies are linked to the statistics of work performed during irreversible transformations. When applying these ideas in computer simulations, one has to deal with rare events (of a different kind, however) and transition path sampling can help to solve this problem. Due to space limitations we cannot comprehensively discuss all aspects of transition path sampling in this chapter. For further information the reader is referred to several recent review articles [11-15]. Of these articles, [12] is the most detailed and comprehensive. 

According to the fluctuation-dissipation theorem [1], the electrical polarizability of polyelectrolytes is related to the fluctuations of the dipole moment generated in the counterion atmosphere around the polyions in the absence of an applied electric field [2-4], Here we calculate the fluctuations by computer simulation to determine anisotropy of the electrical polarizability Aa of model DNA fragments in salt-free aqueous solutions [5-7]. The Metropolis Monte Carlo (MC) Brownian dynamics method [8-12] is applied to calculate counterion distributions, electric potentials, and fluctuations of counterion polarization. 

In computer simulations, the most convenient way of evaluating G(t) is by using the fluauation-dissipation theorem ... 

Call, J.I., Taylor, D.J.R. and Stepto, R.F.T. (2000) Computer simulation studies of molecular orientation in polyethylene networks orientation functions and the legendre addition theorem. Macromolecules, 33,4966. 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

In an ab initio simulation, the electronic structure problem is solved for each nuclear configuration, and forces are computed using the Hellmann-Feynman theorem. 

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