1549380323-Statistical Mechanics Theory and Molecular Simulation

464 The Feynman path integral = ∫x(t)=x′ x(0)=x Dx(s) exp { i ̄h ∫t 0 dsL(x(s),x ̇(s)) } = ∫x′ x DxeiA[x]/ ̄h. (12.4.7) At this ...

Functional integrals 465 small, then the complex exponential oscillates very slowly in moving from path to path. Without frequen ...

466 The Feynman path integral ∫t 0 ds [ m 2 ( dx ds ) 2 −U(x(s)) ] = ∫−iβh ̄ 0 d(−iτ) [ m 2 ( dx d(−iτ) ) 2 −U(x(−iτ)) ] =i ∫β ̄ ...

Functional integrals 467 to imaginary time usings=−iτ, then the equation of motion becomesmd^2 x/ds^2 −→ md^2 x/d(−iτ)^2 =−md^2 ...

468 The Feynman path integral ρ(x,x′;β) = ∫x(β ̄h)=x′ x(0)=x Dx(τ) exp [ − 1 ̄h ∫β ̄h 0 dτ ( 1 2 mx ̇^2 + 1 2 mω^2 x^2 )] . (12. ...

Functional integrals 469 The first term in the penultimate line of eqn. (12.4.21) is the classical Euclidean action integral. Th ...

470 The Feynman path integral y(τ) to an integral over the coefficientscn. Using eqn. (12.4.27), let us first determine the argu ...

Many-body path integrals 471 g 0 = [ m 2 πβ ̄h^2 ] 1 / 2 , gn= [ mβω^2 n 2 π ] 1 / 2 . (12.4.35) Whenω= 0, the product is exactl ...

472 The Feynman path integral arex 1 andx 2 , then, as we saw in Section 9.4, the coordinate eigenvectors for bosons and fermion ...

Many-body path integrals 473 x(τ) (^0) ℏ x x τ 1 2 x(τ) (^0) ℏ x x τ 1 2 β β Fig. 12.9Representative paths in the direct (left) ...

474 The Feynman path integral ×e−βφ ( x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P) )[ perm ( A ̃ )] (12.5.7) for bosons. The matrixA ...

Numerical evaluation 475 Q(N,V,T) = ∮ Dr 1 (τ)···DrN(τ) ×exp { − 1 ̄h ∫β ̄h 0 dτ ∑N i=1 1 2 mir ̇^2 i(τ) +U(r 1 (τ),...,rN(τ)) } ...

476 The Feynman path integral 12.6.1 Path-integral molecular dynamics We begin our discussion with the molecular dynamics approa ...

Numerical evaluation 477 1 2 3 5 4 6 7 8 k=mωP^2 Fig. 12.10Classical isomorphism: The figure shows a cyclic polymer chain having ...

478 The Feynman path integral coordinate transformation capable of uncoupling the harmonic term in eqn. (12.6.3), then we can re ...

Numerical evaluation 479 ×exp { −β ∑P k=1 [ p^2 k 2 m′k + 1 2 mkω^2 Pu^2 k+ 1 P U(xk(u)) ]} . (12.6.9) In eqn. (12.6.9), the par ...

480 The Feynman path integral u ̇k= pk m′k p ̇k=−mkω^2 Puk− 1 P ∂U ∂uk − pηk, 1 Q 1 pk η ̇k,γ= pηk,γ Qk p ̇ηk, 1 = p^2 k m′k −kT ...

Numerical evaluation 481 and eigenvectors; 3) From the eigenvectors, construct the orthogonal matrixOijthat diagonalizes A. The ...

482 The Feynman path integral (1/P) ∑P k=1∂U/∂xk. It can be shown that, for the staging transformation, the force on the modeu 1 ...

Numerical evaluation 483 with the conditionr(iP+1)=r(1)i. An important point to make about eqn. (12.6.24) is that the potentialU ...