1549380323-Statistical Mechanics Theory and Molecular Simulation

144 Canonical ensemble Ebecomes negligible. The implication of this result is that in the thermodynamic limit, the canonical ens ...

Examples 145 We now extend this derivation to the case ofNparticles in three dimensions, i.e., an ideal gas ofNparticles in a cu ...

146 Canonical ensemble which is just the cube of eqn. (4.5.1). Using eqn. (4.5.4), we obtain the partition function as Q(N,V,T) ...

Examples 147 the integration ofxover all space with no significant loss of accuracy. Therefore, the partition function becomes Q ...

148 Canonical ensemble Q(N,V,T,r,r′) = 1 h^3 N ∫ dNpdNrexp { −β [N ∑ i=1 p^2 i 2 m + 1 2 mω^2 ∑N i=0 (ri−ri+1)^2 ]} . (4.5.23) W ...

Examples 149 I 2 = (π 2 α ) 3 / 2 e−α(r (^2) +2r (^23) )/ 2 eα(r+2r^3 ) (^2) / 6 ∫ all space dr 2 e−^3 α[r^2 −(r+2r^3 )] (^2) / ...

150 Canonical ensemble Q(N,V,T) = 1 h^6 ( 2 π βhω ) 3 N( 2 πm β ) 3 1 (N+ 1)^3 /^2 × ∫ dr 0 drN+1e−βmω (^2) (r 0 −rN+1) (^2) /(N ...

Examples 151 In order to illustrate how the recursive inverse works, consider the special case of N= 3. If we setk= 3 in eqn. (4 ...

152 Canonical ensemble Thus, substituting the transformation into eqn. (4.5.24), we obtain Q(N,V,T,r,r′) = 1 h^3 N ( 2 πm β ) 3 ...

Spatial distribution functions 153 〈|r−r′|^2 〉=λ^3 ( βhω 2 π ) 3 N+3 1 h^6 ( 2 π βhω ) 3 N( 2 πm β ) 3 × 4 π (N+ 1)^3 /^2 ∫∞ 0 d ...

154 Canonical ensemble expect peaks at particular values characteristic of the average structural motifs present in the system, ...

Spatial distribution functions 155 of the locations of the remainingn+1,...,Nparticles. This probability can be obtained by simp ...

156 Canonical ensemble g(n)(r 1 ,...,rn) = N! (N−n)!ρn 〈n ∏ i=1 δ(ri−r′i) 〉 r′ 1 ,...,r′N . (4.6.11) Of course, the most importa ...

Spatial distribution functions 157 around a given pair does not depend on where the pair is in the system. Thus, we integrate ov ...

158 Canonical ensemble (^051015) r (Å) -200 -100 0 100 200 V (r ) (K) (^051015) r (Å) 0 0.5 1 1.5 2 2.5 3 g( r) 100 K 200 K 300 ...

Spatial distribution functions 159 0 1 2 3 4 5 6 r (Å) 0 1 2 3 4 g( r) O O O H 0 1 2 3 4 5 6 r (Å) 0 1 2 3 4 5 6 N (r ) (a) (b) ...

160 Canonical ensemble ......... ......... ......... d dsinψ ψ ψ ki ks θ θ r 1 r 2 Fig. 4.4Illustration of Bragg scattering from ...

Spatial distribution functions 161 This simple analysis suggests that a similar scattering experiment performed in a liquid coul ...

162 Canonical ensemble 0 2 4 6 8 10 q (Å -1 ) -1 0 1 2 S (q ) 100 K 200 K 300 K 400 K 0 2 4 6 8 10 q (Å -1 ) -1 0 1 2 S (q ) 213 ...

Spatial distribution functions 163 E= 3 2 NkT+〈U〉. (4.6.37) Moreover, since 3NkT/2 =〈 ∑N i=1p 2 i/^2 mi〉, we can write eqn. (4.6 ...