1549380323-Statistical Mechanics Theory and Molecular Simulation

304 Monte Carlo one is forced to use a finer temperature “grid” in order to have a reasonable average acceptance probability, th ...

Transition path sampling 305 is determined entirely by the potential energyU. For any trial move, the acceptance probability is ...

306 Monte Carlo aim hasonlybeen to sample, there is no dynamical information in the sequence of configurations we have produced. ...

Transition path sampling 307 in Fig. 7.5, which shows a thermostatted trajectory of a particle in aone-dimensional double-well p ...

308 Monte Carlo P[X(T)] =f(x 0 ) n∏− 1 k=0 T(x(k+1)∆t|xk∆t), (7.7.3) wheref(x 0 ) is the equilibrium distribution of initial con ...

Transition path sampling 309 generate a trajectory X(T) starting from Y(T). Both X(T) and Y(T) must be transi- tion paths fromAt ...

310 Monte Carlo A B .yj∆t . xj∆t . xj∆t Accepted Rejected Fig. 7.7The shooting algorithm. The original path Y(T) is shown as the ...

Transition path sampling 311 where the phase space displacement ∆ is chosen randomly from a symmetric distribu- tionπ(∆) satisfy ...

312 Monte Carlo methods can become trapped in local regions of configuration space, so transition path sampling can become trapp ...

Problems 313 a. Show that the sequence of distributionsπn(x) satisfies the recursion πn+1(x) = ∫ 1 x x y πn(y)dy+ ∫x 0 πn(y)dy+π ...

314 Monte Carlo Can this algorithm be used in conjunction with hybrid Monte Carlo? Whyor why not? 7.6. Write a replica exchange ...

8 Free energy calculations Our treatment of the classical equilibrium ensembles makes clear that the free energy is a quantity o ...

316 Free energy calculations acterized by potential energy functionsUA(r 1 ,...,rN) andUB(r 1 ,...,rN). For example, in a drug-b ...

Free energy perturbation theory 317 ZB= ∫ dNre−βUB(r^1 ,...,rN) = ∫ dNre−βUB(r^1 ,...,rN)e−βUA(r^1 ,...,rN)eβUA(r^1 ,...,rN) = ∫ ...

318 Free energy calculations compute the average of ∆Uα,α+1in each stateα. The free energy difference ∆AAB is the sum of contrib ...

Adiabatic switching and thermodynamic integration 319 In Section 4.2, we showed that derivatives ofAwith respect toN,V, orTgive ...

320 Free energy calculations f(λ) andg(λ), which means that eqn. (8.2.7) always yields the reversible work via the free energy d ...

Adiabatic switching and thermodynamic integration 321 ∆Ab ∆A 1 ∆A 2 ∆A 3 E I H O 2 Fig. 8.1Two thermodynamic pathways for the ca ...

322 Free energy calculations 8.3 Adiabatic free energy dynamics Although we cannot entirely eliminate the need to visit the unph ...

Adiabatic free energy dynamics 323 Thus, if the region betweenλ= 0 andλ= 1 is transformed into a low-probability region,λwill sp ...