1549380323-Statistical Mechanics Theory and Molecular Simulation

604 Langevin and generalized Langevin equations reasonable approximation toζ(t)onlyin the high-frequency limit. In fact, Berneet ...

Problems 605 15.2. Consider a single harmonic degree of freedomq, having a massm, that obeys the generalized Langevin equation q ...

606 Langevin and generalized Langevin equations b. Compute the autocorrelation function〈ˆσy(0)ˆσy(t)〉assuming an initial canonic ...

Problems 607 c. Now consider a simple continuous model for the friction kernelγ(t) = λAe−λt. In what time range would you expect ...

608 Langevin and generalized Langevin equations 15.9. a. Derive eqns. (15.7.27), (15.7.30), and (15.7.31). ∗b. Derive eqn. (15.7 ...

16 Critical phenomena 16.1 Phase transitions and critical points In Section 4.7, we studied the liquid–gas phase transition asso ...

610 Critical phenomena P T Solid Gas Liquid Critical point Triple point L L L 1 2 3 Fig. 16.1Representative phase diagram of a s ...

Critical exponents 611 . P V Pc Vc Liquid region Va por region Ideal gas region Critical isotherm C Fig. 16.2Equation of state o ...

612 Critical phenomena universality class by studying its physically simplest members. We start by introducing a set of critical ...

Ising model 613 system). We are interested in the transition from a disordered to an ordered state that occurs either by the app ...

614 Critical phenomena a scalar variableσzithat can take values of±1 only. At this point, since there is only one relevant spati ...

Ising model 615 h T Tc M > 0 M < 0 (Spin up) (Spin down) M = 0 Fig. 16.4Phase diagram of the Ising model. m h T = T T > ...

616 Critical phenomena Table 16.1Comparison of thermodynamic variables and relations between gas–liquid and magnetic systems. Qu ...

Universality classes 617 which is simply a spin-flip transformation, the magnetization in a perfectly ordered state changes sign ...

618 Critical phenomena universality class. By contrast, in the Heisenberg model of eqn. (16.4.2), the spins can point in any spa ...

Mean-field theory 619 β. The location of the thin solid line is determined using a procedure known as the Maxwell construction, ...

620 Critical phenomena ∆(N,h,T)≈ ∑ σ 1 =± 1 ∑ σ 2 =± 1 ··· ∑ σN=± 1 exp { −β [ 1 2 NJm^2 −(Jm+h) ∑ i σi ]} = e−βNJm (^2) / 2 ∑ σ ...

Mean-field theory 621 tanh( )bJm f(m)=m m J < 1 J = 1 bJ > 1 b b m 0 m 0 Fig. 16.6 Graphical solution of the transcendenta ...

622 Critical phenomena m m g(0,T) g(0,T) m 0 m 0 (a) (b) Fig. 16.7Free energy of eqn. (16.5.12),g(0,T). (a)T < Tc. (b)T > ...

One-dimensional Ising model 623 We are interested in the behavior of eqn. (16.5.15) along the criticalisotherm, where m 0 →0 nea ...