Computational Physics

10.6 Further applications and Monte Carlo methods 323 1 10 100 1000 10 000 10 100 N (^2) R Figure 10.3. Scaling behaviour of the ...

324 The Monte Carlo method of their ‘parent’. This causes the total weight of this conformation in the population to be constant ...

10.6 Further applications and Monte Carlo methods 325 In this example that constant should be near 1/(0.75Nθ), whereNθis the num ...

326 The Monte Carlo method This can also be written as Z= ∑ Xm;m=1,...,M exp ( − ∑M m= 1 βmEXm ) , (10.55) whereXmdenotes a syst ...

10.6 Further applications and Monte Carlo methods 327 exponent −(βm−βn)(EXn−EXm) (10.59) should be of order 1. Note that before ...

328 The Monte Carlo method is formulated in a continuum phase space. An example is finding the minimum- energy conformation of a ...

10.6 Further applications and Monte Carlo methods 329 done by Grassberger for polymer chains consisting of two different types o ...

330 The Monte Carlo method following steps. WHILE No acceptable solution found DO Calculate fitness (merit function) of all indi ...

10.7 The temperature of a finite system 331 temperature. This procedure thus leads to the expression kBT= 2 〈K〉 3 N− 3 (10.64) f ...

332 The Monte Carlo method Taking the derivative ofS =kBlnwith respect to energy then leads to the following expression for the ...

10.7 The temperature of a finite system 333 This is a Taylor expansion of the exponent around its maximumE∗which satisfies β= 1 ...

334 The Monte Carlo method and 1/K^3 are determined in a molecular dynamics simulation at constant energy in order to calculateα ...

References 335 Metropolis method, in which creations are tried much more often than annihilations. Suppose the trial probabiliti ...

336 The Monte Carlo method [12] J. A. Barker, ‘Monte Carlo calculations of the radial distribution functions for a proton-electr ...

References 337 [35] G. M. Torrie and J. P. Valleau, ‘Monte Carlo study of a phase separating liquid mixture by umbrella sampling ...

11 Transfer matrix and diagonalisation of spin chains 11.1 Introduction In Chapters 8 and 10 we studied methods for simulating c ...

11.2 The one-dimensional Ising model and the transfer matrix 339 itself, even without reference to a transfer matrix relating th ...

340 Transfer matrix and diagonalisation of spin chains It is therefore convenient to define the transfer matrix which contains t ...

11.2 The one-dimensional Ising model and the transfer matrix 341 necessary, but for matrix sizes up to 10 000×10 000, diagonalis ...

342 Transfer matrix and diagonalisation of spin chains We see that the correlation drops off exponentially, unlessλ 0 =λ 1 , tha ...