1549380323-Statistical Mechanics Theory and Molecular Simulation

324 Free energy calculations 0 0.2 0.4 0.6 0.8 1 l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A (l) 0 0.2 0.4 0.6 0.8 1 l 0 0.2 0.4 0.6 0.8 1 f ...

Jarzynski’s equality 325 generate a distributionZ(λ,β) as a function ofλin which the physical coordinates sample essentially all ...

326 Free energy calculations which states that if an amount of workWABtakes a system from stateAto stateB, thenWAB≥∆AAB, where e ...

Jarzynski’s equality 327 H(r,p,t) = ∑N i=1 p^2 i 2 mi +U(r 1 ,...,rN,t). (8.4.3) For time-dependent Hamiltonians, the usual cons ...

328 Free energy calculations = CN QA(N,V,T) ∫ dx 0 e−βH(xτ(x^0 ),τ). (8.4.8) Now let us change variables from x 0 to xτ(x 0 ) in ...

Jarzynski’s equality 329 equations have an associated phase space metric √ g = e^3 Nη. Note that the heat term in eqn. (8.4.4) d ...

330 Free energy calculations 〈 e−βWAB 〉 A= 1 ZT(0) ∫ dxτe^3 Nητe−βWτ(x^0 (xτ)) ×δ(H′(xτ,τ)−Wτ(x 0 (xτ))−C). (8.4.18) The integra ...

Jarzynski’s equality 331 fixed v Fig. 8.3Extension of deca-alanine by pulling via eqn. (8.4.22) (reprinted with permission from ...

332 Free energy calculations Wt P(Wt )e Wt P(Wt )e P(Wt ) -3Wt Fig. 8.4Shift in a Gaussian work distribution as a result of th ...

Rare events 333 distribution by performing a short run with a thermostat coupling, even if such a relaxation process has been sh ...

334 Free energy calculations two dihedral angles contains four minima, then in a protein of modestsize containing 50 such pairs, ...

Reaction coordinates 335 the relative coordinate between A and B,r=rB−rA, and a third relative coordinate sbetween H and the cen ...

336 Free energy calculations constantKw(Trout and Parrinello, 1998). A clearer picture of this reaction waspro- vided by Chandle ...

Blue moon ensemble 337 A(s) =−kTlnP(s). (8.7.1) In the second line, the integration over momenta has been performed giving the t ...

338 Free energy calculations ∂ ∂s δ(q 1 −s) =− ∂ ∂q 1 δ(q 1 −s). (8.7.6) Finally, we integrate by parts to obtain 1 P(s) dP ds = ...

Blue moon ensemble 339 of the transformation is not unity. LetJ(q) =|∂(r 1 ,...,rN)/∂(q 1 ,...,q 3 N)|denote this Jacobian. Eqn. ...

340 Free energy calculations p ̇i=Fi+λ ∂σ ∂ri , (8.7.15) whereλis the Lagrange multiplier needed to impose the constraint. In or ...

Blue moon ensemble 341 √ g= e−w= ∑ i 1 mi ( ∂σ ∂ri ) 2 ≡z(r). (8.7.20) This metric factor arises frequently when holonomic const ...

342 Free energy calculations compute the average ofz−^1 /^2 (r)O(r) and normalize by the average ofz−^1 /^2 (r). That is, 〈O(r)〉 ...

Umbrella sampling 343 2.0 2.5 3.0 3.5 0.0 10.0 20.0 30.0 ξ (Α) ∆ A (kcal/ mol ) Butadiene 2,F-Butadiene Fig. 8.6Free energy prof ...