1549380323-Statistical Mechanics Theory and Molecular Simulation

484 The Feynman path integral 12.6.2 Path-integral Monte Carlo We saw in Chapter 7 that Monte Carlo methods are very effective f ...

Numerical evaluation 485 we obtain ∑j k=0 (xl+k−xl+k+1)^2 = ∑j k=1 k+ 1 k u^2 l+k+ 1 j+ 1 (xl+j+1−xl)^2 (12.6.28) (see, also, eq ...

486 The Feynman path integral acceptance probability is 40%. For a system ofNparticles, the algorithm extends straightforwardly. ...

Numerical evaluation 487 (^01) ́ 105 2 ́ 105 Steps -30 -15 0 15 30 eprim (^01) ́ 105 2 ́ 105 Steps -30 -15 0 15 30 eprim (^01) ́ ...

488 The Feynman path integral 〈α〉f= CP 2 QP ∫ dx 1 ···dxP ∑P k=1 xk ∂α ∂xk e−βα(x^1 ,...,xP)e−βγ(x^1 ,...,xP) =− CP 2 βQP ∫ dx 1 ...

Numerical evaluation 489 εvir εvir εvir <ε > vir <ε > vir <ε > vir σ σ σ Steps Steps Steps Steps Steps Steps B ...

490 The Feynman path integral Pvir({r(1),...,r(P)}) = dNkT V − 1 V ∑N i=1 1 P r(ic)· ∑P k=1 ∂ ∂r(ic) U(r( 1 k),...,r(Nk),V) − 1 ...

Problems 491 O O H O O H O O H -0.8 -0.4 0 0.4 0.8 d c (Å) 0 1 2 3 4 D A (kcal/mol) Classical Quantum Quantum H Fig. 12.14Top: S ...

492 The Feynman path integral 12.7 Problems 12.1. Derive primitive and virial estimators for the full pressure tensorPαβdefined ...

Problems 493 ∗12.6. Consider two distinguishable particles in one dimension with respective coor- dinatesxandyand conjugate mome ...

494 The Feynman path integral whereεn(X) are the eigenvalues that result from the solution of the Schr ̈odinger equation [ − ̄h^ ...

13 Classical time-dependent statistical mechanics 13.1 Ensembles of driven systems Our discussion of both classical and quantum ...

496 Classical time-dependent statistical mechanics transport properties. The coefficient of shear viscosity is an example of atr ...

Linear response 497 ing feature of Fig. 13.2(a) is the fact that many more phase space points are visited, even on the short tim ...

498 Classical time-dependent statistical mechanics q ̇i= ∂H ∂pi +Ci(q,p)Fe(t) p ̇i=− ∂H ∂qi +Di(q,p)Fe(t), (13.2.1) whereFe(t) i ...

Linear response 499 can use a perturbative approach to solve the Liouville equation. When the external perturbation is small, we ...

500 Classical time-dependent statistical mechanics this approximation. Interestingly, however, the approximation oflinear respon ...

Linear response 501 iL(s)f 0 (H(x)) =− ∂f 0 ∂H j(x)Fe(s). (13.2.18) Now, suppose thatf 0 (H(x)) is given by a canonical distribu ...

502 Classical time-dependent statistical mechanics a∗(xt) =a∗(x 0 )e−iL^0 t a(xt) =a(x 0 )e−iL^0 t, (13.2.24) where the last lin ...

Linear response 503 shorthand notation for a time correlation function that stands in for the left side of the equation. Using t ...