Quasiergodic hypothesis

It may be necessary to amend the underlying theory before the reduction process. In order to derive thermodynamics from classical statistical mechanics, Boltzmann was, for example, obliged to introduce the quasiergodic hypothesis. There are vague speculations that quantum theory may have to be amended for systems with a very large number of particles [17]. 

The next step in preparing to calculate thermodynamic functions from MD Is to invoke the quasiergodic hypothesis. This is well described by Reif, and for a general ensemble composed of systems in contact with a reservoir, it may be stated as follows. 

Speaking pictorially, the quasiergodic hypothesis implies that the trajectory of a single system, while it does not cover phase space, uniformly samples phase space. Questions regarding ergodlc properties of Hamiltonian systems are extremely complicated, and the subject is under active study today. But in practice an MD system is not quite Hamiltonian, because solving the equations of motion as finite difference equations Introduces a certain randomness Into the phase space trajectory. This property of the trajectory, which will be discussed In detail In Section III, presumably justifies the quasiergodic hypothesis for an MD system. 

Then how is P(X) related to the actual MD data, as represented by a graph of the function X(t)2 The answer to this is found in the quasiergodic hypothesis, which may be restated in the following form. 

Quasiergodic hypothesis restated. In the course of a sufficiently long time, the variable X for a single equilibrium MD system is distributed according the the probability distribution P(X). 

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