An ensemble

We are going to specify the meaning of the term ensemble in the field of statistical thermodynamics, whose definition will be a bit more precise in this context than in everyday language. 

An ensemble is an identical replica of a system. This replica is imaginary the ensemble does not physically exist, which allows us to readily choose the number of elements Ne, as many as one wishes, or if needed, to make it tend towards infinity. The ensemble has the energy Eg and contains a certain number of elementswhose energy is Ej. The ensemble could be considered a collection of non-quantum objects, or elements, and we can describe the configurations or complexions of the ensemble. 

Phase Modeling Tools. Applications to Gases, First Edition. Michel Soustelle. ISTE Ltd 2015. Published by ISTE Ltd and John Wiley Sons, Inc. 

There are several types of ensembles used in statistical thermodynamics (see section 5.6). The ensemble that we will mainly focus on in this chapter is called a canonical ensemble . 

Canonical ensembles are widely used they are particularly well-adapted to the microscopic description of solid or fluid phases, which must be studied in detail. 

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The VIGRAL approach represents the reflecting surface of a defect as an ensemble of virtual point sources. At every measurement point of the B-Scan, the detecting transducer responds... 

It is customary in statistical mechanics to obtain the average properties of members of an ensemble, an essentially infinite set of systems subject to the same constraints. Of course each of the systems contains the... 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

Such an ensemble of systems can be geometrically represented by a distribution of representative points m the F space (classically a continuous distribution). It is described by an ensemble density fiinction p(p, q, t) such that pip, q, t)S Q is the number of representative points which at time t are within the infinitesimal phase volume element df p df q (denoted by d - D) around the point (p, q) in the F space. 

In the last subsection, the microcanonical ensemble was fomuilated as an ensemble from which the equilibrium, properties of a dynamical system can be detennined by its energy alone. We used the postulate of... 

The statement of the mixing condition is equivalent to the followhig if Q and R are arbitrary regions in. S, and an ensemble is initially distributed imifomily over Q, then the fraction of members of the ensemble with phase points in R at time t will approach a limit as t —> co, and this limit equals the fraction of area of. S occupied by... 

Statistical mechanics and kinetic theory, as we have seen, are typically concerned with the average behaviour of an ensemble of similarly prepared systems. One usually hopes, and occasionally can demonstrate, that the variations of these properties from one system to another in the ensemble, or that the variation with time of the properties of any... 

Consider an ensemble of Brownian particles. The approach of P2 to as 00 represents a kmd of diflfiision process in velocity space. The description of Brownian movement in these temis is known as the Fo/c/cer-PIanc/c method [16]- For the present example, this equation can be shown to be... 

In an ensemble of collisions, the impact parameters are distributed randomly on a disc with a probability distribution P(b) that is defined by P(b) db = 2nb db. The cross section da is then defined by... 

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Wliatever the deteetion teehnique, the window stage of the 4WM event must eonvert these evolved vibrational wavepaekets into the third order polarization field that oseillates at an ensemble distribution of optieal frequeneies. One must be alert to the possibility that the window event after doorway ehaimel B may involve resonanees from eleetronie state manifold e to some higher manifold, say r. Thus ehaimel B followed by an e (ket) or a (bra) event might be enlianeed by an e-to-r resonanee. However, it is nonnal to eonfine the... 

The nonlinear response of an individual molecule depends on die orientation of the molecule with respect to the polarization of the applied and detected electric fields. The same situation prevails for an ensemble of molecules at an interface. It follows that we may gamer infonnation about molecular orientation at surfaces and interfaces by appropriate measurements of the polarization dependence of the nonlinear response, taken together with a model for the nonlinear response of the relevant molecule in a standard orientation. 

Write (r) = Qy treating it as a component of a (very large) coliinm vector. Consider an ensemble of systems... 

Fluse D A and Leibler S 1988 Phase behavior of an ensemble of noninterseoting random fluid films J. Physique 49 605... 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

For an ensemble of tire systems described above, we can represent tire time rate of change of tire population density as... 

Figure C2.17.7. Selected area electron diffraction pattern from TiC nanocrystals. Electron diffraction from fields of nanocrystals is used to detennine tire crystal stmcture of an ensemble of nanocrystals [119]. In tliis case, tliis infonnation was used to evaluate the phase of titanium carbide nanocrystals [217].

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

Figure C3.6.3 The spreading of an ensemble of four points on the WR chaotic attractor, (a) The initial tight, four-point ensemble of open circles (o) at = 5.287. .., = 24.065. .. and variable = 2.884. ..,2.984. ..,3.084. .., and...

In a classical limit of the Schiodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

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