In the molecular dynamics simulation method for fluids, the equations of motion for a collection of particles in a fixed volume are solved numerically. The energy, volume, and number of particles are constant for a particular simulation, and it is assumed that time averages of properties of the simulated fluid are equal to microcanonical ensemble averages of the same properties. In some situations, it is desirable to perform simulations of a fluid for particular values of temperature and/or pressure or under conditions in which the energy and volume of the fluid can fluctuate. This paper proposes and discusses three methods for performing molecular dynamics simulations under conditions of constant temperature and/or pressure, rather than constant energy and volume. For these three methods, it is shown that time averages of properties of the simulated fluid are equal to averages over the isoenthalpic–isobaric, canonical, and isothermal–isobaric ensembles. Each method is a way of describing the dynamics of a certain number of particles in a volume element of a fluid while taking into account the influence of surrounding particles in changing the energy and/or density of the simulated volume element. The influence of the surroundings is taken into account without introducing unwanted surface effects. Examples of situations where these methods may be useful are discussed.

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In the dynamics calculation, the momentum of the system is also conserved. Thus, it would be more accurate to say that the trajectory average is equal to an ensemble average in which the total momentum is specified in addition to N, V, and E.
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Strictly speaking, it generates a Markov process rather than a Markov chain, since the time is a continuous rather than discrete variable. The dynamical equations describing the process will in practice be solved by using a discrete grid of times, and we assume that it is correct to regard the process as a chain and apply theorems that have been proven for chains.
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15.
Any distribution function that expresses the probability density at a point in phase space as a function of only the value of the Hamiltonian at that point is invariant with respect to the motion generated by that Hamiltonian.
16.
An aperiodic chain is a chain all of whose states are aperiodic. See Ref. 10, p. 387, for the definition of an aperiodic state.
17.
If such a state exists, it is aperiodic (Ref. 10, p. 387), and hence the chain is aperiodic if it is irreducible (Ref. 10, p. 391).
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