1549380323-Statistical Mechanics Theory and Molecular Simulation

284 Monte Carlo Eqn. (7.2.18) is a function ofyalone, and we can now analyze itsydependence in greater detail. First, the extrem ...

Sampling 285 algorithms. Otherwise, a very large sample will be needed beforeI ̃Mbecomes a good estimator for the integral. 7.3 ...

286 Monte Carlo exists a variable transformationr=g(x) whereg(x) is a nondecreasing function of x. This function maps the interv ...

Sampling 287 7.3.2 Importance sampling Let us return to the problem of calculating the multidimensional integral I= ∫ dxφ(x)f(x) ...

288 Monte Carlo F[h] =σ^2 [h]−λ ∫ dxh(x). (7.3.17) Computing the functional derivativeδF/δh(x), we obtain the condition φ^2 (x)f ...

Sampling 289 0 0.5 1 x 0 0.2 0.4 0.6 0.8 1 f (x ), h( x) (^05) ́ 105 1 ́ 106 MC Steps 0 0.2 0.4 0.6 0.8 1 Estimator (^05) ́ 105 ...

290 Monte Carlo sampling of the state space. It has been argued that the detailed balance condition is a sufficient but not stri ...

Sampling 291 r(xk+1|xk)< ξ, the move is rejected, and xk+1is set equal to xk. Remember, however, that even when a move is rej ...

292 Monte Carlo A simple yet fairly standard way to apply the M(RT)^2 algorithm is based on choosing the trial probabilityT(x|y) ...

Sampling 293 A simple remedy for this problem lies at the other extreme, in which we attempt to only move one particle at a time ...

294 Monte Carlo As a final observation, it is interesting to compare Monte Carlo with molecular dynamics as methods for sampling ...

Sampling 295 of a random vector and normalizing the vector by the length l= √ ζ^2 x+ζy^2 +ζz^2 (7.3.39) to given= (ζx/l,ζy/l,ζz/ ...

296 Monte Carlo ∆(N,P,T) = 1 V 0 N!λ^3 N ∫∞ 0 dVe−βPVVN × ∫ ds 1 ···dsNe−βU(V 1 / (^3) s 1 ,...,V 1 / (^3) sN) . (7.3.43) From e ...

Hybrid Monte Carlo 297 arises from the use of scaled coordinates in the configurational partition function in eqn. (7.3.45). Sim ...

298 Monte Carlo this consistency. We can achieve this by accompanying each “bad” molecular dynamics move by an acceptance step b ...

Hybrid Monte Carlo 299 uniform moves of Section 7.3. Typically, in a hybrid Monte Carlo calculation, one aims for a higher avera ...

300 Monte Carlo that d(−p) d(−p′) = dpdp′, we obtain eqn. (7.4.6), which emphasizes the importance of using a reversible, measur ...

Replica exchange 301 Clearly, a straightforward molecular dynamics or Monte Carlo calculation car- ried out on a rough potential ...

302 Monte Carlo T 1 T 2 T 3 T 4 TM Fig. 7.3Schematic of the parallel-tempering replica exchange Monte Carlo. (r(K),r(K+1))→( ̃r( ...

Replica exchange 303 0 10 20 30 40 50 Dihedral angle number 0 2 4 6 8 10 Number of barrier crossings 0 10 20 30 40 50 Dihedral a ...