1549380323-Statistical Mechanics Theory and Molecular Simulation

164 Canonical ensemble Proceeding as we did in derivingg(r) (Section 4.6), we introduce the change of variables in eqn. (4.6.14) ...

Spatial distribution functions 165 that the volume differentiation cannot be easily performed. The task would be made considerab ...

166 Canonical ensemble The quantity in the angle brackets in eqn. (4.6.56) is an instantaneous estimatorP(r,p) for the pressure ...

Spatial distribution functions 167 β 3 V 〈N ∑ i=1 ri·Fi 〉 = β 6 V 〈 ∑ i,j,i 6 =j rij·fij 〉 = β 6 V Z ∫ dr 1 ···drN   ∑ i,j,i 6 ...

168 Canonical ensemble P kT =ρ− 2 πρ^2 3 kT ∫∞ 0 dr r^3 u′(r)g(r), (4.6.69) which is a simple expression for the pressure in ter ...

van der Waals equation 169 4.7 Perturbation theory and the van der Waals equation Up to this point, the example systems we have ...

170 Canonical ensemble 〈a〉 0 = 1 Z(0)(N,V,T) ∫ dr 1 ···drNa(r 1 ,...,rN) e−βU^0 (r^1 ,...,rN). (4.7.5) IfU 1 is a small perturba ...

van der Waals equation 171 We can equate the two expressions forA(1)by further expanding the natural log in eqn. (4.7.11) using ...

172 Canonical ensemble The expressions forω 1 ,ω 2 , andω 3 are known as the first, second, and thirdcumulants ofU 1 (r 1 ,...,r ...

van der Waals equation 173 Suppose next thatU 0 andU 1 are both pair-wise additive potentials of the form U 0 (r 1 ,...,rN) = ∑N ...

174 Canonical ensemble 5 10 r -500 0 500 1000 u^0 (r )+u ( 1 r) u 0 (r) u 1 (r) 0 Fig. 4.6Plot of the potentialu 0 (r) +u 1 (r). ...

van der Waals equation 175 σ σ Fig. 4.7Two hard spheres of diameterσat closest contact. The distance between their cen- ters is ...

176 Canonical ensemble model, we cannot expect it to be applicable over a wide range ofP,V, andTvalues. Nevertheless, if we plot ...

van der Waals equation 177 at the critical point. The first and second derivatives of eqn. (4.7.35) with respect to Vyield two e ...

178 Canonical ensemble For example, a simple molecular system that can exist as a solid, liquid, or gas has a critical point on ...

Extended phase space 179 ∂P ∂ρ = ∂P ∂V ∂V ∂ρ ∂^2 P ∂ρ^2 = [ ∂^2 P ∂V^2 ( ∂V ∂ρ ) 2 + ∂P ∂V ∂^2 V ∂ρ^2 ] . (4.7.49) Both derivati ...

180 Canonical ensemble which equilibrium observables can be computed. The problem of generating dynamical properties consistent ...

Extended phase space 181 gappearing in eqn. (4.8.1) will be determined by the condition that amicrocanonical distribution of 2dN ...

182 Canonical ensemble s 0 = e(E−H(r,p)−p (^2) s/ 2 Q)/gkT 1 |f′(s 0 )| = 1 gkT e(E−H(r,p)−p (^2) s/ 2 Q)/gkT . (4.8.7) Substitu ...

Extended phase space 183 dri dt′ = p′i mi dp′i dt′ =Fi− sp′s Q p′i ds dt′ = s^2 p′s Q dp′s dt′ = 1 s [N ∑ i=1 (p′i)^2 mi −gkT ] ...