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33) D and κ(dp) = β 2π 3N/2 N −3/2 mi i=1 β exp − pT M −1 p 2 dp. 34) Under µ, the position a and the momentum p are independent random variables. Therefore, sampling configurations (q, p) according to the canonical measure µ(dq dp) can be performed by independently sampling positions according to ν(dq) and momenta according to κ(dp). It is straightforward to sample from κ since the momenta are Gaussian random variables. The actual issue is therefore to sample from ν. 2. Some elements on the derivation of the canonical measure.
Let us stress that the above derivation is performed under the assumption that the system is at equilibrium. In particular, no notion of dynamics is required. For nonequilibrium systems in a steady state, the dynamics has to be made precise. It is not always clear whether a stationary probability measure exists, and, when it exists, whether it is unique and whether the distribution of the microscopic configurations converges to it. There are some positive results, see [Rey-Bellet (2006)] in the case of heat transport in one-dimensional atom chains.
32) can be seen as the phase space probability measure maximizing the statistical entropy among the set of phase space probability measures compatible with the observed macroscopic data (in this case, average energy given). The derivation performed for an average energy fixed may be performed for any average thermodynamic quantity, leading to other thermodynamic ensembles. The choice of the ensemble amounts to choosing which quantities are fixed exactly or in average. We present in this section a general derivation of thermodynamic ensembles associated with a given set of constraints, and next focus on two examples, the isobaric-isothermal ensemble (NPT) where the number of particles, the pressure and the temperature are fixed, and the grand-canonical ensemble (µVT) where the chemical potential, the volume and the temperature are fixed.