1549380323-Statistical Mechanics Theory and Molecular Simulation

204 Canonical ensemble α= ∑N ∑i=1Fi·pi/mi N i=1p 2 i/mi . (4.12.8) When eqn. (4.12.8) is substituted into eqn. (4.12.5), the equ ...

Isokinetic ensemble 205 a single kinetic-energy constraint based on the virial theorem, since the number of degrees of freedom i ...

206 Canonical ensemble s ̈(t) = ̇s(t)h ̇(t) = [∑N j=1(Fj·pj(t))/mj 2 K ] s ̇(t) = [∑N j=1(Fj·pj(0))/mj 2 K ] + [∑N j=1(Fj·Fj)/mj ...

Liquid structure 207 Note,{s(∆t/2),s ̇(∆t/2)}are evaluated by substituting the present momenta and forces into eqns. (4.12.20) w ...

208 Canonical ensemble the box edge. Let these intervals be indexed by an integeri= 0,...,Nr−1 with radial valuesr 1 ,...,rNr. ...

Problems 209 p ̇η 1 = p^2 m −kT p ̇η 2 = p^4 3 m^2 −(kT)^2. a. Show that these equations of motion are non-Hamiltonian. b. Show ...

210 Canonical ensemble Show that these equations obey the following two conservation laws: C= p^2 2 m + p^2 η 2 Q +kTη≡H′ K=peη. ...

Problems 211 i. Finally, show the conservation ∑N i=1Fi= 0 isnotobeyed by eqns. (4.10.7) and, therefore, that they also generate ...

212 Canonical ensemble c. Show that the second virial coefficient in the low density limit becomes B 2 (T) =− 2 π ∫∞ 0 dr r^2 f( ...

Problems 213 4.7. A classical system ofNnoninteracting diatomic molecules enclosed in a cubic box of lengthLand volumeV =L^3 is ...

214 Canonical ensemble 4.10. The canonical ensemble version of the classical virial theoremis credited to Richard C. Tolman (191 ...

Problems 215 d. Denote the single-molecule partition function asq(n,V,T) =V f(n,T). Now suppose that the system is composed ofdi ...

216 Canonical ensemble u(r) = A rn , wherenis an integer andA >0. In the low density limit, compute the pressure of such a sy ...

Problems 217 c. Express the average energy in terms of the radial distribution functiong(r) and the new function ̃g(3)(r,s). d. ...

5 The isobaric ensembles 5.1 Why constant pressure? Standard handbooks of thermodynamic data report numerical values of physical ...

Thermodynamics 219 for molecular dynamics calculations in these ensembles. We will show how the volume and density distributions ...

220 Isobaric ensembles Eqns. (5.2.5) constitute the basic thermodynamic relations in the isoenthalpic-isobaric ensemble. The rea ...

Phase space and partition functions 221 the volume of the container adjusts to keep the internal pressure equal to the external ...

222 Isobaric ensembles Noting that the thermodynamic relations in eqn. (5.2.5) can also be written in terms of the entropyS=S(N, ...

Phase space and partition functions 223 N , V , E 2 2 2 H 2 ( x 2 ) N , V , E H ( x ) 1 1 1 1 1 Fig. 5.1Two systems in contact w ...