1、2.3.4 Potential truncation and its corrections The chemical potential may also be related to g(r) Parameter coupling the two atomThermal de Broglie wavelenthIn the MD or MC, in order to save time, we introduce a spherical cutoff. The cutoff distance should be sufficient large to ensure that it is a
2、small perturbation, but it must be no greater than BOXL/2 for the consistency with minimum image convention. Long-range corrections Therefore simulation calculates only the part of integration in energy, pressure and chemical potential equations (from 0 to rC). They should be corrected.Assuming the
3、radial distribution function as unit in the distance r rC, the energy, pressure and chemical potential can be corrected bySubscript C refers to the quantities calculated from simulation.2.3.5 Time correlation functions and transport coefficientsCorrelation between two different quantities A and B ar
4、e measured in the usual statistical sense, via the correlation coefficient cABThe absolute value of cAB lies between 0 and 1, with values close to 1 indicating a high degree of correlation.By considering A and B to be evaluated at two different times, the resulting quantity is a function of the time
5、 difference and called time correlation function cAB(t).Time correlation functions and transport coefficients-continueFor identical phase functions, cAA(t) is called an autocorrelation function and its time integral (from t =0 to t =) is a correlation time tA. These functions are of great interest i
6、n computer simulation because1) They give a clear picture of the dynamics in a fluid2) tA is related to macroscopic transport coefficients3) Fourier transform is related to experimental spectra.The non-normalized correlation function is defined asTime correlation functions and transport coefficients
7、-continueSo that Transport coefficients are defined in terms of the response of a system to a perturbation.Transport coefficientA variable appearing in the perturbation term in the Hamiltonian. Einstein relationDiffusion coefficientWe can use time correlation function or Einstein relation to calcula
8、te transport coefficients in computer simulations. Or go back to first principles and conducting a suitable non-equilibrium simulation.For using equilibrium MD, we give here the equations in the microcanonical ensemble, for a fluid composed of N identical molecules. The diffusion coefficient is give
9、n by Green-Kubo formula or Einstein relationship The center-of-mass velocity of a single moleculesMolecular position at time t.3directionsVelocity autocorrelation function and mean squared displacement Velocity autocorrelation functions for liquid argonVariation in mean squared displacement during a
10、 MD simulation of argonShear viscosityGreen-Kubo Formula:An off-diagonal element of the pressure tensorEinstein relationshipBulk viscosityGreen-Kubo formulaIn calculation, we use andEinstein relationshipThermal conductivityA component of the energy current, i.e., the time derivative of .Make no contribution if ri=0, as in the case in a normal one-component MD simulation.Green-Kubo FormulaEinstein relationshipHow to get i ?
