Generation of the NVT Ensemble

In a series of papers,Nose showed that a Hamiltonian mechanics could be written down that would generate the distribution function for the NVT or canonical ensemble. The basic approach involves extending the phase space of the system, in a manner similar to that originally laid out by Andersen and by Parrinello and Rahman. Namely, in addition to the dN coordinates and dN momenta, where d is the number of spatial dimensions, an additional variable s, representing a heat bath, and its conjugate momentum are included. The Hamiltonian for the extended system is given by 

The integration over s can now be performed using the 8 function and the identity 

The interpretation of this result is that to obtain the canonical energy of the system, one computes the thermodynamic derivative with respect to P of the extended-system microcanonical partition function and adds back k T/2 for the thermostat kinetic energy and a constant E, which merely changes the energy scale. Similarly, the heat capacity at constant volume is given by 

The difficulty inherent in Eqs. [63] is the presence of p,/s. The analysis carried out above shows that a canonical distribution in pj = p,/s results. However, if we introduce such a noncanonical transformation into the equations of motion, the result is 

Although H is conserved by the equations of motion, it clearly is not a Hamiltonian for Eqs. [65]. Since non-Hamiltonian systems tend to be more difficult than Hamiltonian systems to integrate stably numerically, the existence of a conserved energy for a non-Hamiltonian system is of vital importance as a check on the stability of the numerical integration scheme employed. 

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