Computational Physics

7.2 Examples of statistical models; phase transitions 183 factor, and a small number of ordered states, with a large Boltzmann f ...

184 Classical equilibrium statistical mechanics the calculation has been completed, the limitH→0 is taken. It is to be noted tha ...

7.3 Phase transitions 185 T<Tt T=T T<T 1st order 2nd order c c T>Tc t t T=T T>T Figure 7.4. Typical behaviour of the ...

186 Classical equilibrium statistical mechanics The phase transition is characterised by the liquid phase going from stable to m ...

7.3 Phase transitions 187 above is an ideal model for visualising what is going on close to a second order phase transition. An ...

188 Classical equilibrium statistical mechanics as a whole: the differences only show up at the smallest scales, i.e. comparable ...

7.3 Phase transitions 189 exhibits a singularity near the phase transition: χm(T)∝|T−Tc|−γ (7.53) whereγis also called the ‘crit ...

190 Classical equilibrium statistical mechanics A T L 2 L 3 L ∞ Figure 7.5. Typical behaviour of a physical quantityAvs temperat ...

7.3 Phase transitions 191 in terms of additional exponents, the so-calledfinite-size scaling exponents: The shift in the positi ...

192 Classical equilibrium statistical mechanics These are the finite-size scaling laws for any thermodynamic quantity which dive ...

7.4 Determination of averages in simulations 193 We can estimate the standard deviation as a time average: σ^2 =A^2 −A 2 . (7.71 ...

194 Classical equilibrium statistical mechanics If we definel=n−m, then this can be rewritten as ε^2 = 1 M^2 ∑M n= 1 n∑−M l=n− 1 ...

References 195 (b) We assume that the incoming rays have intensityI 0. Show that the average total intensity of waves with wave ...

196 Classical equilibrium statistical mechanics [11] A. Rahman, ‘Correlations in the motion of atoms in liquid argon,’Phys. Rev. ...

8 Molecular dynamics simulations 8.1 Introduction In the previous chapter we saw that the experimental values of physical quanti ...

198 Molecular dynamics simulations Consider a collection ofNclassical particles in a rectangular volumeL 1 ×L 2 ×L 3. The partic ...

8.1 Introduction 199 Figure 8.1. Periodic boundary conditions for molecular dynamics. Each particle interacts not only with ever ...

200 Molecular dynamics simulations whereLμare vectors along the edges of the rectangular system volume and the first sum on the ...

8.2 Molecular dynamics at constant energy 201 conserved too, so the time averages of physical quantities obtained by this type o ...

202 Molecular dynamics simulations wherer(t)is the position of the particle at timet=nh(his the time step;nis an integer). From ...