1549380323-Statistical Mechanics Theory and Molecular Simulation

224 Isobaric ensembles Q 2 (N 2 ,V−V 1 ,T) = e−βA(N^2 ,V−V^1 ,T) Q(N,V,T) = e−βA(N,V,T) Q 2 (N 2 ,V−V 1 ,T) Q(N,V,T) = e−β[A(N−N ...

Phase space and partition functions 225 function depends on extensive variables such asNandV, it can be reexpressed as a sum of ...

226 Isobaric ensembles to eqn. (5.3.26). other thermodynamic quantities follow in a manner similar to the canonical ensemble. Th ...

Virial theorems 227 =P 1 ∆ ∫∞ 0 dVe−βPVQ(N,V,T) =P. (5.4.3) The boundary term in the first line of eqn. (5.4.3) vanishes at both ...

228 Isobaric ensembles The quantity〈P(int)V〉can be defined for any ensemble. However, because the volume can fluctuate in an iso ...

Anisotropic cells 229 H=− ∂ ∂β ln ∆ = (N+ 1)kT+ 3 2 NkT≈ 5 2 NkT, (5.5.8) from which the constant pressure heat capacity is give ...

230 Isobaric ensembles a b c α β γ Fig. 5.2A general parallelepiped showing the convention for the cell vectors and angles. In t ...

Anisotropic cells 231 asdh 0 =V−^3 dh. In addition, det(h 0 ) = det(h)/V. Thus, substituting the cell-matrix transformation into ...

232 Isobaric ensembles = 1 ∆(N,P,T) ∫ dh[det(h)]−^2 e−βPdet(h) kT det(h) ∑^3 γ=1 hβγ ( ∂Q ∂hαγ ) N,T .(5.6.10) An integration by ...

Pressure tensor estimator 233 5.7 Derivation of the pressure tensor estimator from the canonical partition function Molecular dy ...

234 Isobaric ensembles into eqn. (5.7.5) to yield L(s, ̇s) = 1 2 ∑ i mi ∑ α,β,γ hαβs ̇i,βhαγs ̇i,γ−U(hs 1 ,...,hsN) = 1 2 ∑ α,β, ...

Pressure tensor estimator 235 H= ∑ i ∑ α,β 1 mi πi,αh−αβ^1 pi,β−L= ∑ i ∑ α,β,γ 1 mi πi,αh−αβ^1 πi,γh−γβ^1 −L. (5.7.14) Since the ...

236 Isobaric ensembles ∂H ∂hαγ =− ∑ i ∑ μ,ν,λ πi,μπi,ν 2 mi ∑ ρ,σ ( h−μρ^1 ∂hρσ ∂hαγ h−σλ^1 h−νλ^1 +h−μλ^1 h−νρ^1 ∂hρσ ∂hαγ ) h− ...

Molecular dynamics 237 P(int)αβ (r 1 ,...,rN,p 1 ,...,pN) = 1 det(h) ∑N i=1 [ pi,αpi,β mi +Fi,αri,β ] , (5.7.27) which is equiva ...

238 Isobaric ensembles but also to promote the volume to a dynamical variable. Moreover, itleads to a force that is used to prop ...

Molecular dynamics 239 πi=V^1 /^3 pi π ̇i=V^1 /^3 p ̇i+ 1 3 V−^2 /^3 V ̇pi. (5.8.5) Substituting eqns. (5.8.5) into eqns. (5.8.4 ...

240 Isobaric ensembles In most molecular dynamics calculations, the isoenthalpic-isobaric ensemble is em- ployed only seldom: th ...

Molecular dynamics 241 transformed the incompressible equations (5.8.6) into compressibleones; the compress- ibility of eqns. (5 ...

242 Isobaric ensembles r ̇i= pi mi + pǫ W ri p ̇i=F ̃i− ( 1 + d Nf ) pǫ W pi− pη 1 Q 1 pi V ̇=dV pǫ W p ̇ǫ=dV(P(int)−P) + d Nf ∑ ...

Molecular dynamics 243 be modified to include a thermostat on each particle or on each degree of freedom (“massive” thermostatti ...