1549380323-Statistical Mechanics Theory and Molecular Simulation

264 Isobaric ensembles I 0 (x) = 1 π ∫π 0 dθe±xcosθ. b. Calculate the equation of state by determining the one-dimensional “pres ...

6 The grand canonical ensemble 6.1 Introduction: The need for yet another ensemble The ensembles discussed thus far all have the ...

266 Grand canonical ensemble for an arbitrary parameter,λ. We call such a function ahomogeneous function of degree nin the varia ...

Thermodynamics 267 so that A(N,V,T) =−PV+μN, (6.2.9) which agrees with Euler’s theorem. Similarly, the Gibbs free energyG(N,P,T) ...

268 Grand canonical ensemble A ̃depends on a single extensive variable,V, it is a homogeneous function of degree 1 inV, i.e.A ̃( ...

Phase space and partition function 269 N=N 1 +N 2 , V=V 1 +V 2. (6.4.1) In order to carry out the derivation of the ensemble dis ...

270 Grand canonical ensemble WhenN 1 = 2, we need to place two particles in system 1. The first particle can be chosen inNways, ...

Phase space and partition function 271 f 1 (x 1 ,N 1 ) = ( e−βH^1 (x^1 ,N^1 ) Q(N,V,T)N 1 !h^3 N^1 ) 1 (N−N 1 )!h3(N−N^1 ) ∫ dx ...

272 Grand canonical ensemble 1 eβPV ∑∞ N=0 eβμN 1 N!h^3 N ∫ dx e−βH(x,N)= 1. (6.4.17) Taking the exp(βPV) factor to the right si ...

Ideal gas 273 For other thermodynamic quantities, it is convenient to introduce anew variable ζ= eβμ (6.4.24) known as thefugaci ...

274 Grand canonical ensemble 6.5 Illustration of the grand canonical ensemble: The ideal gas In Chapter 11, the grand canonical ...

Particle number fluctuations 275 which contains the average particle number〈N〉instead ofNas would appear in the canonical ensemb ...

276 Grand canonical ensemble ζ ∂ ∂ζ ζ ∂ ∂ζ lnZ(ζ,V,T). (6.6.2) Using eqn. (6.4.25), this becomes ζ ∂ ∂ζ ζ ∂ ∂ζ lnZ(ζ,V,T) =ζ ∂ ∂ ...

Particle number fluctuations 277 From eqn. (6.6.9), it follows that ∂P ∂μ = ∂P ∂v ∂v ∂μ =− ∂^2 a ∂v^2 ∂v ∂μ . (6.6.10) We can ob ...

278 Grand canonical ensemble ∆N 〈N〉 = 1 〈N〉 √ 〈N〉kTκT v = √ kTκT 〈N〉v ∼ 1 √ 〈N〉 . (6.6.18) Thus, as〈N〉 −→0 in the thermodynamic ...

Problems 279 whereνiare the stoichiometric coefficients in the reaction. Using this nota- tion, the coefficients of the products ...

7 Monte Carlo 7.1 Introduction to the Monte Carlo method In our treatment of the equilibrium ensembles, we have, thus far, exclu ...

Central Limit theorem 281 practice, it would take about 10^6 such dart throws to achieve a reasonable estimate of π/4, which wou ...

282 Monte Carlo For simplicity, we introduce the notation ∫ dxφ(x)f(x) =〈φ〉f, (7.2.5) where〈···〉findicates an average ofφ(x) wit ...

Central Limit theorem 283 with g(σ) = ln ∫ dxf(x)eiσφ(x). (7.2.11) Although we cannot evaluate the integral overσin eqn. (7.2.9) ...