The ensembles

One of the ways that statistical thermodynamics tries to understand the thermodynamic state of a large macroscopic system is by separating it into tiny, or microscopic, parts. These parts are called microsystems. The state of each microsystem is called 

FIGURE 17.3 For a smooth distribution, an integral can be substituted for a summation. This allows us to use calculus in our derivation of ejcpressions in statistical thermodynamics. 

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A canonical ensemble is an ensemble separated intoj individual microstates such that the numbers of particles in each microstate Nj, the volumes of the microstates Vp and the temperatures of the microstates Tj are the same. As extensive variables, particle numbers and the volumes are additive over the microstates, whereas the temperature, an intensive variable, is not additive over the ensemble. Another way of saying this is by defining the total number of particles N, the systems total volume V, and the system s overall temperature T as 

An ensemble of microsystems, whose overall thermodynamic properties are determined from the combined states of the constituent microsystems 

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We are interested in < E (0[,(t)i)E3(62,, where <> means the average over the ensemble of surfaces, the subindexes 1 and 2 refer to two different points of observation and the subindexes A and B belong to two different conditions of illumination, which for example arise from two different wavelengths, two different incident angles, etc.. If A = B and 1 = 2, the above expression gives the angular distribution of the mean scattered intensity, otherwise it turns to be the intensity correlation coefficient y from < E Eb >, assuming that we deal with fully developed speckle. 

Let us consider the consequence of mechanics for the ensemble density. As in subsection A2.2.2.1. let D/Dt represent differentiation along the trajectory in F space. By definition,... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

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... 

The ensemble density p g(p d ) of a mixing system does not approach its equilibrium limit in die pointwise sense. It is only in a coarse-grained sense that the average of p g(p,. d ) over a region i in. S approaches a limit to the equilibrium ensemble density as t —> oo for each fixed i . 

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... 

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. 

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... 

The macroscopic rate coefficient k (cm s for elastic collisions between the ensembles A and B is... 

Here =MkT. In a real system the thennal coupling with surroundings would happen at the surface in simulations we avoid surface effects by allowing this to occur homogeneously. The state of the surroundings defines the temperature T of the ensemble. 

The ensemble average in the Widom fomuila, ((exp -p is sometimes loosely referred to as the... 

Here we have adopted a Dirac bracket notation which should be distmguished from the ensemble... 

For these sequences the value of Gj, is less than a certain small value g. For such sequences the folding occurs directly from the ensemble of unfolded states to the NBA. The free energy surface is dominated by the NBA (or a funnel) and the volume associated with NBA is very large. The partition factor <6 is near unify so that these sequences reach the native state by two-state kinetics. The amplitudes in (C2.5.7) are nearly zero. There are no intennediates in the pathways from the denatured state to the native state. Fast folders reach the native state by a nucleation-collapse mechanism which means that once a certain number of contacts (folding nuclei) are fonned then the native state is reached very rapidly [25, 26]. The time scale for reaching the native state for fast folders (which are nonnally associated with those sequences for which topological fmstration is minimal) is found to be... 

One way to overcome this problem is to start by setting up the ensemble of trajectories (or wavepacket) at the transition state. If these bajectories are then run back in time into the reactants region, they can be used to set up the distribution of initial conditions that reach the barrier. These can then be run forward to completion, that is, into the products, and by using transition state theory a reaction rate obtained [145]. These ideas have also been recently extended to non-adiabatic systems [146]. 

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... 

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