Time and ensemble average

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r a and M(r) = c for r a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential 

Here cr is the collision diameter and e is the depth of the potential well at the minimum of zz(r). For molecules we often use combinations of atomic pair potentials, adding several body potentials that describe bending or torsion when needed. For dipolar fluids we have to add dipole-dipole interactions (or, in a more sophisticated description. Coulomb interactions between partial charges on the atoms) and for ionic solutions also Coulomb interactions between the ionic charges. 

Consider an equilibrium thennodynamic ensemble, say a set of atomic systems characterized by the macroscopic variables T (temperature), Q (volume), andTV (number of particles). Each system in this ensemble contains N atoms whose positions and momenta are assigned according to the distribution function (5.2) subjected to the volume restriction. At some given time each system in this ensemble is in a particular microscopic state that coiTesponds to a point (r, p- ) in phase space. As the system evolves in time such a point moves according to the Newton equations of motion, defining what we call a phase space trajectory (see Section 1.2.2). The ensemble coiTesponds to a set of such trajectories, defined by their starting point and by the Newton equations. Due to the uniqueness of solutions of the Newton s equations, these trajectories do not intersect with themselves or with each other. 

In this microscopic picture, any dynamical property of the system is represented by a dynamical variable—a fimction of the positions and momenta of all particles. 

Equation (5.10) defines an ensemble average. Alternatively we could consider another definition of the thermodynamic quantity, using a time average 

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X, ..., X. Such a transformation induces a trivial tensor transformation for the instantaneous force Tip(t). We show in the Appendix, Section H, by evaluating the time and ensemble average of the instantaneous force over a short time interval, that, in the case of a nonsingular mobility matrix, such a transformation creates a transformed force bias... 

Practically all NMR observables bear information on the dynamics of the investigated molecules. This is a quite inherent feature of solution-state NMR measurements as there are 1016-1017 solute molecules in the sample tube, and all of them are interconverting from one conformer to another on a range of time scales. Thus, all parameters obtained are essentially time- and ensemble averages of a high number of conformers and their proper interpretation should take this into account. In other words, in a general case no single conformer can be expected to fulfill all measured parameters simultaneously.4 Apparently, this is in contrast with the considerations above for "conventional" structural calculations where multiple structures were compatible with the data used. This problem can be referred to as... 

The stationary nature of our system and the ergodic theorem (see Section 1.4.2) imply that time and ensemble averaging are equivalent. This by no means implies that the statistical information in a row of the table above is equivalent to that in a column. As defined, the different systems j = 1,..., 7 7 are statistically independent, so, for example, statistically independent so that (ni(Zi)ni(Z2)) 7 (ni(Zi))time series provide information about time correlations that is absent from a single time ensemble data. The stationary nature of our system does imply, as discussed in Section 6.1, that ( (Zl) (Z2)> depends only on the time difference Z2 — Zi. 

We also note the equality of time and ensemble averages which is a fundamental tenet of statistical mechanics (ergodic theorem)... 

F. 6 Time- and ensemble-averaged mean square displacements of hydrogen molecules in the Co stmcture at different temperatures and 100 % cage occupancy... 

Small correction of a variable in FVM discretization Double prime, fluctuation component of an instantaneous quantity around it s mean weighted value. Used in the time-and ensemble averaging approaches Fractional step index, or intermediate value... 

With the molecular dynamics method the differential equations describing the classical equations of motion for interacting atoms are solved by step-by-step numerical integration. These differential equations involve derivatives of the energy and, hence, force fields. The solution to the differential equations gives the position and momenta of each atom at each time step (typically a femtosecond) over a total time interval (typically 10 ps-10 ns) limited by the computational speed of the computer, the complexity of the force field, and the size of the molecular system. Thermodynamic properties are obtained from the time and ensemble average of appropriate molecular properties. 

In MQ-NMR motional dynamics are parameterized by one or, more commonly, a distribution of residual dipolar coupling values, which represent the time- and ensemble-averaged interaction strength between two protons along the chain length. It is important to note that while the majority of the alternative techniques outlined above also rely on this phenomenon, the data resulting from the MQ-NMR experiments pioneered by Saalwachter and coworkers [63] can be analyzed in a model-free way with no assumptions about various motional limits, functional forms of dynamical correlation functions, etc. 

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