Relation to Microcanonical Ensemble

The microcanonical ensemble is characterized by fixed values of the thermodynamic variables N, the total particle number, V, the volume of the system, and E, the total energy. The phase space distribution function for the micro-canonical ensemble is 

The same dynamical system can also be used to generate an equilibrium time correlation function C b( ) = ( 4(0)B(t)). Since any physical observable can be shown to evolve according to 

In terms of a trajectory average, the time correlation function can be written as 

The ability to generate an equilibrium ensemble from a dynamical trajectory has a number of useful features. One can obtain not only ordinary static equilibrium properties from Eq. [25], but also dynamical information. In fact, dynamical information is available on two levels. On the one hand, equilibrium time correlation functions can be calculated, leading to the prediction of vibrational spectra, transport coefficients and so on. On the other, the trajectory allows access to the microscopic detailed motion of individual atoms. Therefore, one can, in a sense, visualize at an atomistic level the dynamical behavior of the system as a function of time, which can lead to valuable insights about chemical reaction mechanics, structural rearrangements, and other details of the system that can be captured only by visualization at this level of detail. 

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