1549380323-Statistical Mechanics Theory and Molecular Simulation

84 Microcanonical ensemble As an example of the use of the virial theorem, consider the choicexi=pi, a momentum component, andi= ...

Thermal equilibrium 85 andV =V 1 +V 2 , respectively. The entropy of each system is given in terms of the partition function for ...

86 Microcanonical ensemble Consider a sum of the form σ= ∑P i=1 ai, (3.4.7) whereai>0 for allai. Letamaxbe the largest of all ...

Ideal gas 87 S(N,V,E) =kln Ω 1 (N 1 ,V 1 ,E ̄ 1 ) +kln Ω 2 (N 2 ,V 2 ,E ̄ 2 ) =S 1 (N 1 ,V 1 ,E ̄ 1 ) +S 2 (N 2 ,V 2 ,E ̄ 2 ) +O ...

88 Microcanonical ensemble Ω(L,E) = E 0 L √ 2 m h ∫∞ −∞ dy δ ( y^2 −E ) . (3.5.4) Then, using the properties of the Diracδ-funct ...

Ideal gas 89 ∑N i=1 y^2 i=E. (3.5.11) Eqn. (3.5.11) is the equation of a (3N−1)-dimensional spherical surface of radius √ E. Thu ...

90 Microcanonical ensemble Ω(N,V,E) = E 0 (2m)^3 N/^2 VN N!h^3 N 2 π^3 N/^2 Γ(3N/2) 1 2 √ E E(3N−1)/^2 = E 0 E 1 N! 1 Γ(3N/2) [ ...

Ideal gas 91 P kT = ( ∂ln Ω ∂V ) N,E . (3.5.25) Since Ω∼VN, ln Ω∼NlnV so that the derivative yields P kT = N V , (3.5.26) or P= ...

92 Microcanonical ensemble 3.5.1 The Gibbs Paradox According to eqn. (3.5.21), the entropy of an ideal gas is S(N,V,E) =kln Ω(N, ...

Harmonic systems 93 Let us now repeat the analysis using eqn. (3.5.29). Introducing Stirling’s approx- imation as a logarithm of ...

94 Microcanonical ensemble The partition function is Ω(E) = E 0 h ∫∞ −∞ dp ∫∞ −∞ dx δ ( p^2 2 m + 1 2 kx^2 −E ) . (3.6.3) In ord ...

Harmonic systems 95 where ̄h=h/ 2 π. The integration overI′now proceeds using eqn. (A.2) and yields unity, so that Ω(E) = E 0 ̄h ...

96 Microcanonical ensemble Ω(N,E) = ( 2 πE h ) 3 N 1 Γ(3N) ∏N i=1 1 ω^3 i , (3.6.18) or, using Stirling’s approximation Γ(3N)≈(3 ...

Introduction to molecular dynamics 97 the control parameters. Moreover, extreme conditions, such as high temperature and pressur ...

98 Microcanonical ensemble This fact suggests an intimate connection between the microcanonical ensemble and classical Hamiltoni ...

Integrating the equations of motion 99 prove the ergodicity or lack thereof in a system with many degrees of freedom. The ergodi ...

100 Microcanonical ensemble ri(t+ ∆t) +ri(t−∆t) = 2ri(t) + ∆t^2 mi Fi(t), (3.8.4) which, after rearrangement, becomes ri(t+ ∆t) ...

Integrating the equations of motion 101 is a fundamental symmetry of Hamilton’s equations that should be preserved by a nu- meri ...

102 Microcanonical ensemble x f(x) s Fig. 3.3Gaussian distribution given in eqn. (3.8.11). erally, ifxis a Gaussian random varia ...

Integrating the equations of motion 103 Gaussian random number. If the equation is solved forMvaluesξ 1 ,...,ξMto yield valuesX ...