1549380323-Statistical Mechanics Theory and Molecular Simulation

644 Critical phenomena Note that the matrix T is not required to be symmetric. Consequently, we define a left eigenvalue equatio ...

Universality and linearized RG 645 RG eigenvaluesytandyh. An analysis of the scaling properties of the singular part ̃g(h,T) of ...

646 Critical phenomena .P K K K 1 2 3 Fig. 16.18A renormalization group trajectory. defines the critical surface, which in this ...

Problems 647 K K 1 2 .P Fig. 16.19 Example curves defined by u 1 (K 1 ,K 2 ). The critical curve defined by u 1 (K 1 ,K 2 ) = 0 ...

648 Critical phenomena ∗16.3. The Maxwell construction in the van der Waals theory attempts to fix the unphysical behavior of th ...

Problems 649 Hint: Try redefining the coupling constant byu= eKand show that P′can be put in the same form as P, i.e., P′(u) =c( ...

650 Critical phenomena d. What is the probability, for largeN, that the angles will alternate trans, gauche, trans, gauche, ...? ...

Problems 651 b. Consider trying to write one of the terms in brackets in the aboveexpres- sion as eK(σ^1 +σ^2 +σ^3 +σ^4 )+ e−K(σ ...

652 Critical phenomena g. Show thatK′can be estimated by K′≈K 1 +K 2. h. Derive the RG equation that results, and show that it p ...

Appendix A Properties of the Dirac delta-function The Dirac delta-function is informally defined as having infinite height,zero ...

654 Diracδ-function Let us employ the normalized Gaussian sequence in eqn. (A.5) to prove eqns. (A.2) and (A.3). Eqn. (A.2) foll ...

Diracδ-function 655 Consider, next, aδ-function of the formδ(ax). This can be simplified according to δ(ax) = 1 |a| δ(x), (A.14) ...

Appendix B Evaluation of energies and forces In any Monte Carlo or molecular dynamics calculation of a many-body system, the mos ...

Energies and forces 657 in real space and one that is short ranged in reciprocal space. Given two rapidly convergent terms, the ...

658 Energies and forces Experience has shown that a good balance between the short- andlong-range com- ponents is achieved ifα= ...

Energies and forces 659 = 16 πρ^2 ǫσ^3 3 [ 2 3 ( σ rc ) 9 − ( σ rc ) 3 ] . (B.8) Eqns. (B.7) and (B.8) are called thestandard lo ...

660 Energies and forces i rc r + δ c Fig. B.1Building a Verlet list of the neighbors of particlei. be correct, which means that ...

Energies and forces 661 be evaluated efficiently in real space. However, because a long-ranged function in real space is short r ...

662 Energies and forces As written, the sum over particles in eqn. (B.12) scales asO(N^2 ). However, the compu- tational overhea ...

Energies and forces 663 Ulong(r 1 ,...,rN) = 1 V ∑ g∈S 4 π |g|^2 e−|g| (^2) / 4 α 2 |S(g)|^2 − α √ π ∑ i qi^2. (B.19) Eqn. (B.19 ...