Canonical ensemble average

In principle, these fomuilae may be used to convert results obtained at one state point into averages appropriate to a neighbouring state point. For any canonical ensemble average... 

A consideration of the transition probabilities allows us to prove that microscopic reversibility holds, and that canonical ensemble averages are generated. This approach has greatly extended the range of simulations that can be perfonned. An early example was the preferential sampling of molecules near solutes [77], but more recently, as we shall see, polymer simulations have been greatly accelerated by tiiis method. 

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average... 

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

Another element in this whirlwind notational tour of statistical mechanics is a more explicit notation for averages. A canonical ensemble average of a phase function 0 will be denoted by... 

Then a grand canonical ensemble average of the same property is... 

After the system has reached equilibrium, in order to compute the canonical ensemble average of any phase space function, one may continue integrating the trajectories using the constant temperature algorithm. Details about technical issues involved in this calculation can be found in the standard texts mentioned above. Here we briefly discuss the computation methodology of properties that are relevant to simulating interfacial systems. 

Computer simulations of isothermal systems, in general, have fallen into two main categories, constant-temperature MD calculations and MC calculations. Many algorithms have been introduced for the former, but only a few have been shown capable of producing canonical distributions. The MC method applied to an ensemble under constant-temperature conditions directly yields canonical ensemble averages. The MC method is straight-... 

A particular example of Eq. (2.4) would be the estimation of the canonical ensemble average of any mechanical quantity M(qN) in a particular state ( , V), according to... 

Here the inner product of two row vectors Ai and A2 is defined as the canonical ensemble average,... 

Bulgac, A., Kusnezov, D. Canonical ensemble averages from pseudomicrocanonical dynamics. Phys. Rev. A 42, R5045-R5048 (1990). doi 10.1103/PhysRevA.42.5045... 

The discussion so far has focused on the use of canonical ensemble averages like (2), in which the variables (N, V, T) are held fixed. That is the most common application of MC work to fluids. However, averages in any ensemble can be put into form (1) and evaluated using similar techniques. In practice computations have been carried out in the grand canonical ensemble [with (fi, V, T) fixed] and in the isothermal isobaric [with (N, P, 7) fixed]. In the former the number of particles N is a fluctuating quantity that must be sampled during the MC experiment, while in the latter this is true of the volume V. 

The thermodynamic functions, as defined above, can be shown to be given by ensemble averages. For a dynamical variable A = A(, Pj ), the canonical-ensemble average is given by... 

Express thermodynamic functions as canonical ensemble averages. 

Relate the canonical ensemble averages to MD ensemble averages. 

We now turn to the problem of how to relate MD-ensemble averages to canonical-ensemble averages. The MD ensemble has two constants of the motion (besides N and V), namely H and J. The ordinary canonical ensemble has only one parameter, namely p which is conjugate to H, and it has zero for the average of J. 

Equation (41) ensures that the time averages resulting from (40) are equivalent to the canonical ensemble averages. The algorithm is rather robust, and very useful if one is just interested in static averages of the model. However, when one considers the dynamics of polymer solutions, one must be aware that the additional terms in (40) seriously disturb the hydrodynamic interactions, for instance. This latter problem can be avoided by using a more complicated form of friction plus random force, the so-caUed dissipative particle dynamics (DPD) thermostat [225-227]. 

Exjuations (5.229) (5.233) provide the basis for the thermod)niamic integration scheme employed in Section 5.8.4. However, it is worthwhile noting at this point that key quantities such as p and 14 in Eqs. (5.229)-(5.233) can be calculated readily as thermal averages of s< and H, respectively. Th(W( grand canonical ensemble averages will be ( alcnlatcxl via GCEMC simulations described in Section 5.8.2. 

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