1549380323-Statistical Mechanics Theory and Molecular Simulation

64 Theoretical foundations physical and chemical processes for which the underlying atomic and molecular me- chanics are of sign ...

Liouville’s theorem 65 point that results from the evolution of x 0. As we noted in Section 1.6, xtis a unique function of x 0 t ...

66 Theoretical foundations d dt J(xt; x 0 ) =J(xt; x 0 ) ∑ k,l [ ∂ ̇xkt ∂xl 0 ∂xl 0 ∂xkt ] . (2.4.9) The summation overlof the t ...

Ensemble distribution 67 p q p q dx 0 dxt Fig. 2.3 Illustration of phase space volume conservation prescribed by Liouville’s the ...

68 Theoretical foundations nˆ dS Fig. 2.4An arbitrary volume in phase space. dSis a hypersurface element andnˆis the unit vector ...

Ensemble distribution 69 or ∫ Ω dxt [ ∂ ∂t f(xt,t) +∇xt·( ̇xtf(xt,t)) ] = 0. (2.5.6) Since the choice of Ω is arbitrary, eqn. (2 ...

70 Theoretical foundations forf(x,t) require input of further information; we will return to this point again as specific ensemb ...

Problems 71 number of microscopic states in the phase space accessible within a given ensemble. Each ensemble has a particular p ...

72 Theoretical foundations d. Finally, suppose that the volume changes fromV 1 toV 2 in an adiabatic process (∆Q= 0). The pressu ...

Problems 73 T S Fig. 2.5Thermodynamic cycle. a. Find a solution of the Liouville equation that also satisfies this initial distr ...

74 Theoretical foundations Here,f(x,t) satisfies the Liouville equation eqn. (2.5.13). The notationS(t) expresses the fact that ...

3 The microcanonical ensemble and introduction to molecular dynamics 3.1 Brief overview In the previous chapter, it was shown th ...

76 Microcanonical ensemble 3.2 Basic thermodynamics, Boltzmann’s relation, and the partition function of the microcanonical ense ...

Basic thermodynamics 77 andE). If the number of particles is increased fromN 1 toN 2 > N 1 , then chemical work W 12 (chem)= ...

78 Microcanonical ensemble That logarithmic dependence of the entropy on Ω(N,V,E) will be explained shortly. Assuming we can det ...

Basic thermodynamics 79 entire phase space is an integration over the momentumpiand positionriof each particle in the system and ...

80 Microcanonical ensemble small compared to all of phase space, eqn. (3.2.17) can be very wellapproximated by an integral Ω(N,V ...

Classical virial theorem 81 The form of the microcanonical partition function shows that the phase space vari- ables are not all ...

82 Microcanonical ensemble 〈 xi ∂H ∂xj 〉 =kTδij, (3.3.1) where the average is taken with respect to a microcanonical ensemble. T ...

Classical virial theorem 83 〈 xi ∂H ∂xj 〉 = MNδij Ω(N,V,E) ∫ H(x)<E dx = δij Ω(N,V,E) MN ∫ dxθ(E−H(x)). = E 0 δij Ω(N,V,E) CN ...