Statistical ensemble averaging

Classical molecular dynamics is a computer simulation method to study the equilibrium and transport properties of a classical many-body system by solving Newton s equations of motion for each component. The hypothesis of this methodology is that the properties of the matter or the transport phenomena can be understood through the observation of statistical properties of a small molecular system under certain microscopic interactions among its constituents. The main justification of the classical molecular dynamics simulation method comes from statistical mechanics in that the statistical ensemble averages are equal to the time averages of a system. 

Both Equation 7.31 and Equation 7.32 contain the same information, but they lead to two different ways of handling the correlations. Indeed, one can define two types of pair correlation fnnctions through the same statistical ensemble average either the full angular distribution fnnction. 

It should be noted that the separation of W into W and is dynamical in character, and that there appears to be no analogous statistical ensemble average equivalent to this form of the equation of state. 

As mentioned previously, the definition of an empirical potential establishes its physical accuracy those most commonly used in chemistry embody a classical treatment of pairwise particle-particle and n-body bonded interactions that can reproduce structural and conformational changes. Potentials are useful for studying the molecular mechanics (MM), e.g., structure optimization, or dynamics (MD) of systems whereby, from the ergodic hypothesis from statistical mechanics, the statistical ensemble averages (or expectation values) are taken to be equal to time averages of the system being integrated via (7). 

According to the theory of ergodicity, averaging a quantity over an infinitely long time series is identical to performing the statistical ensemble average ... 

For the study of the folding transition, the introduction of an effective parameter that uniquely describes the macrostate of the ensemble of heteropolymer conformations is useful. A typical measure is the contact number q( X), which for a given conformation X is simply defined as the fraction of the already formed native contacts n(X) in conformation X and the total number of native contacts tot in the final fold, i.e., (X) = (X)/ tot- Then, the statistical ensemble average of this quantity q X)) at a given temperature characterizes its macrostate. Roughly, for a two-state folder, if q X)) > 0.5, native-like conformations are dominating the statistical ensemble. If less than half the total number of contacts is formed, the heteropolymer tends to reside in the pseudophase of denatured conformations. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

Thus, unlike molecular dynamics or Langevin dynamics, which calculate ensemble averages by calculating averages over time, Monte Carlo calculations evaluate ensemble averages directly by sampling configurations from the statistical ensemble. 

A key problem in the equilibrium statistical-physical description of condensed matter concerns the computation of macroscopic properties O acro like, for example, internal energy, pressure, or magnetization in terms of an ensemble average (O) of a suitably defined microscopic representation 0 r ) (see Sec. IVA 1 and VAl for relevant examples). To perform the ensemble average one has to realize that configurations = i, 5... 

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). 

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