1549380323-Statistical Mechanics Theory and Molecular Simulation

104 Microcanonical ensemble X=σ √ −2 lnξ 2 ′cos 2πξ 1 Y=σ √ −2 lnξ 2 ′sin 2πξ 1. (3.8.20) Thus, we obtain two Gaussian random nu ...

Constraints 105 3.9.1 The SHAKE and RATTLE algorithms For time-independent holonomic constraints, the Euler–Lagrangeequations of ...

106 Microcanonical ensemble Substituting in forri(∆t), we obtain a set ofNcnonlinear equations for theNcun- known multipliersλ ̃ ...

Constraints 107 Eqn. (3.9.14) has a simple solution δλ ̃(1)l =− σl(r(1) 1 ,...,r(1)N) ∑N i=1(1/mi)∇iσl(r (1) 1 ,...,r (1) N)·∇iσ ...

108 Microcanonical ensemble then cycling through the constraints again to compute a new increment to the multi- plier until conv ...

Time evolution operator 109 for the ... in the Poisson bracket expression.” It can also be written as a differential operator iL ...

110 Microcanonical ensemble ThatiL 1 andiL 2 donotgenerally commute can be seen in a simple one-dimensional example. Consider th ...

Time evolution operator 111 theorem is somewhat involved and is, therefore, presented in Appendix C for interested readers. Appl ...

112 Microcanonical ensemble ( x(∆t) p(∆t) ) ≈exp ( ∆t 2 F(x(0)) ∂ ∂p(0) ) ×exp ( ∆t p(0) m ∂ ∂x(0) ) ×exp ( ∆t 2 F(x(0)) ∂ ∂p(0) ...

Time evolution operator 113 In the same way, the third operator, which involves another derivative with respect to momentum, yie ...

114 Microcanonical ensemble iL= ∑N i=1 pi mi · ∂ ∂ri + ∑N i=1 Fi·pi. (3.10.34) If we writeiL=iL 1 +iL 2 withiL 1 andiL 2 defined ...

Multiple time-scale integration 115 immensely powerful technique that we term thedirect translationmethod (Martyna et al., 1996) ...

116 Microcanonical ensemble which the fast forces vary naturally, the slow forces change verylittle. In the simple ve- locity Ve ...

Multiple time-scale integration 117 p ̇=Ffast(x) and has the associated single-time-step propagator exp(iLfast∆t). The full prop ...

118 Microcanonical ensemble exp(iL∆t) = exp ( ∆t 2 Fslow ∂ ∂p ) × [ exp ( δt 2 Ffast ∂ ∂p ) exp ( δt p m ∂ ∂x ) exp ( δt 2 Ffast ...

Symplectic quaternions 119 3.12 Symplectic integration for quaternions In Section 1.11, we showed that the rigid body equations ...

120 Microcanonical ensemble Eqn. (3.12.4) leads to a slightly modified set of equations for the angular velocity components. Ins ...

Time step dependent Hamiltonians 121 eiL∆t≈eiL^5 ∆t/^2 × [ eiL^4 δt/^2 eiL^3 δt/^2 eiL^2 δteiL^3 δt/^2 eiL^4 δt/^2 ]n ×eiL^5 ∆t/ ...

122 Microcanonical ensemble et al., 2005). The one example for which the shadow Hamiltonian is known is,not sur- prisingly, the ...

Time step dependent Hamiltonians 123 p x Fig. 3.4Phase space plot of the shadow Hamiltonian in eqn. (3.13.3) for different time ...