Deterministic canonical ensembles

In well-known papers [22,23] Nose showed how a deterministic canonical ensemble N, V, T) MD simulation scheme could be constructed. Again the system phase space variables were augmented with an additional degree of freedom s. The physical system variables (qiiP ) were related to virtual variables (q,-,p,-,f) by a non-canonical transformation q, - = q,-, p = p,/i and = J df/i. The Hamiltonian of the extended system is... 

It is important to understand the conceptual difference between the quantities E and S in Eqs (1.161) and (1.15 8), and the corresponding quantities in Eq. (1.149). In the microcanonical case E, S, and the other derived quantities (P, T, /.i are unique numbers. In the canonical case these, except for T which is defined by the external bath, are ensemble averages. Even T as defined by Eq. (1.151) is not the same as T in the canonical ensemble. Equation (1.151) defines a temperature for a closed equilibrium system of a given total energy while as just said, in the canonical ensemble T is determined by the external bath. For macroscopic observations we often disregard the difference between average quantities that characterize a system open to its environment and the deterministic values of these parameters in the equivalent closed system. Note however that fluctuations from the average are themselves often related to physical observables and should be discussed within their proper ensemble. 

A virtue of this simple method is that it deterministically samples phase space in a manner consistent with the canonical ensemble, 4io while providing a quantity, similar to a total energy, which is conserved. The latter prop-... 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

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