1549380323-Statistical Mechanics Theory and Molecular Simulation

124 Microcanonical ensemble From the Jacobi identity, it follows that {{F(x),H 2 (x)},H 1 (x)}= −{{H 1 (x),F(x)},H 2 (x)}−{{H 2 ...

Examples 125 3.14.1 The harmonic oscillator The phase space of a harmonic oscillator with frequencyωand massmwas shown in Fig. 1 ...

126 Microcanonical ensemble -3 -2.5 -2 -1.5 -1 log 10 (Dt) -8 -7 -6 -5 -4 -3 -2 log 10 (D E ) Velocity Verlet RESPA RESPA (l=0.9 ...

Examples 127 the face directly opposite. The correct handling of periodic boundary conditions within the force calculation is de ...

128 Microcanonical ensemble The implication of this exercise is that a single dynamical trajectory conveys very little informati ...

Examples 129 Fig. 3.8 (Top) Ball-and-stick model of the isolated alanine dipeptide. (Bottom) Schematic representation of the ala ...

130 Microcanonical ensemble can be seen that the temperature exhibits regular fluctuations, leading to a well-defined thermodyna ...

Problems 131 Ω(N,V,E) =MN ∫ dE′ ∫ dNpδ ( ∑ i=1 p^2 i 2 m −E′ ) × ∫ D(V) dNrδ(U(r 1 ,...,rN)−E+E′), which provides a way to separ ...

132 Microcanonical ensemble as completely rigid, with internal bond lengthsdOHanddHH, so that the constraints are: |rO−rH 1 |^2 ...

Problems 133 c. IfU(x) =mω^2 x^2 /2, find the exactly conserved Hamiltonian. Hint: Assume the exactly conserved Hamiltonian take ...

134 Microcanonical ensemble 3.8. Use the direct translation technique to produce a pseudocodefor the algo- rithm in eqn. (3.11.1 ...

4 The canonical ensemble 4.1 Introduction: A different set of experimental conditions The microcanonical ensemble is composed of ...

136 Canonical ensemble ploying the canonical ensemble. In addition, we will examine how physical observables of experimental int ...

Phase space and partition function 137 A process in whichN,V, andTchange by small amounts dN, dV, and dTleads to a change dAin t ...

138 Canonical ensemble N , V , E 2 2 2 H 2 ( x 2 ) N , V , E 1 1 1 H 1 ( x 1 ) Fig. 4.1A system (system 1) in contact with a the ...

Phase space and partition function 139 lnf(x 1 )≈ln ∫ dx 2 δ(H 2 (x 2 )−E) + ∂ ∂H(x 1 ) ln ∫ dx 2 δ(H 1 (x 1 ) +H 2 (x 2 )−E) ∣ ...

140 Canonical ensemble f(x)∝e−H(x)/kT. (4.3.12) The overall normalization of eqn. (4.3.12) must be proportional to ∫ dx e−H(x)/k ...

Phase space and partition function 141 The link between the macroscopic thermodynamic properties in eqn.(4.2.6) and the microsco ...

142 Canonical ensemble μ=−kT ( ∂lnQ ∂N ) N,V P=kT ( ∂lnQ ∂V ) N,T S=klnQ+kT ( ∂lnQ ∂T ) N,V E=− ( ∂ ∂β lnQ ) N,V . (4.3.23) Noti ...

Energy fluctuations 143 which measures the width of the energy distribution, i.e. the root-mean-square devia- tion ofH(x) from i ...