Ensemble probability density

For particles interacting through the potential l/(r ), the canonical-ensemble probability density in phase space is... 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

The nature of the intemuclear distance, r, is the object of interest in this chapter. In Eq. (5.1) it has the meaning of an instantaneous distance i.e., at the instant when a single electron is scattered by a particular molecule, r is the value that is evoked by the measurement in accordance with the probability density of the molecular state. Thus, when electrons are scattered by an ensemble of molecules in a given vibrational state v, characterized by the wave function r /v(r), the molecular intensities, Iv(s), are obtained by averaging the electron diffraction operator over the vibrational probability density. 

As we have seen in Sect. 8.4.2, in the Tsallis generalization of the canonical ensemble [31], the probability density that the system is at position x is... 

I function which carries maximum information about that system. Definition of the -function itself, depends on a probability aggregate or quantum-mechanical ensemble. The mechanical state of the systems of this ensemble cannot be defined more precisely than by stating the -function. It follows that the same -function and hence the same mechanical state must be assumed for all systems of the quantum-mechanical ensemble. A second major difference between classical and quantum states is that the -function that describes the quantum-mechanical ensemble is not a probability density, but a probability amplitude. By comparison the probability density for coordinates q is... 

The isothermal-isobaric ensemble (N,P,T) allows the simulation of chemical systems at a constant temperature and pressure, such as 298 K and 1 at. which are the most usual experimental conditions. At a given set of (N,P,T) the probability density is not only dependent of q N) but also of the volume V ... 

For low density electron ensembles such as electrons in semiconductors, where electrons are usually allowed to occupy energy bands much higher and much lower than the Fermi level, the probability density of electron energy distribution may be approximated by the Boltzmann fimction of Eqn. 1-3, as shown in Fig. 1-3. The total concentration, n.,of electrons that occupy the allowed electron... 

Here, (A) represents the ensemble average of A, and p is the probability density of the state represented by p, r ). Under conditions of constant number of particles, volume and temperature, the probability density is the Boltzmann distribution... 

This expression suggests that the canonical ensemble can be considered to be an incoherent mixture of the QDO s, each with different position and momentum centroids, and the latter having a probability density given by pc (xc, po) / Z. Each QDO can then be interpreted as a representation of a thermally mixed state localized aroimd (xcPc), with its width being defined by the temperature and the system Hamiltonian. 

The conditional probability Pi i(y2, t21 Ti M is the probability density for Y to take the value y2 at t2 given that its value at tx is yx. To put it differently From all sample functions Yx(t) of the ensemble select those that obey the condition that they pass through the point yt at tt the fraction of this subensemble that goes through the gate y2> T2 + dy2 at t2 is denoted by Pili(y2 t2 yi> h) dy2 Clearly P1(1 is nonnegative and normalized ... 

The statistical behavior of interest is encapsulated in the equilibrium probability density function P )( q c). This PDF is determined by an appropriate ensemble-dependent, dimensionless [6] configurational energy 6( q, c). The relationship takes the generic form... 

The connection between the covariant cyclic and cartesian coordinates of the vector J yields Eq. (A.6), whilst (A.5) makes it possible to form the vector itself out of the components (J)q. As follows from (2.18), the components of the multipole moment pq characterize the preferred orientation of the angular momentum J in the molecular ensemble. Fig. 2.3(a, b) shows the probability density p(0, [Pg.30]

For a rather diluted system, the interaction potential U can be represented as a combination of the one-body and pair potentials given by Eqs. (8) and (9). Further, we introduce a phase-space probability density /( r, p v 0- This function describes the probability, with which the system acquires an ensemble configuration, where each ion occupies a point given by the unique combination (r, p,- in the phase-space. Knowing the ensemble configuration, the average value of any function of coordinates and impulses can be calculated [13,14]. 

When one studies random objects, the first question is always what is the probability of occurrence of individual objects in the ensemble In our context the question is what is the probability of occurrence of a specific Hamiltonian matrix H So, clearly, we need a probability density p H) which assigns a probability to a given matrix H. Apparently the matrix H is specified by stating the three matrix elements H, H 2 and H22-Accordingly, the density p depends on three arguments. 

An avenue that has received exploration is the development of equations for evolution of probability-density functions. If, for example, attention is restricted entirely to particular, fixed values of x and t, then the variable whose value may be represented by v becomes a random variable instead of a random function, and its statistics are described by a probability-density function. The probability-density function for v may be denoted by P(v where P(v) dv is the probability that the random variable lies in the range dv about the value v. By definition P(v) > 0, and P(v) dv = 1, One approach to obtaining an equation of evolution for P(v) is to introduce the ensemble average of a fine-grained density, as described by O Brien in [27], for example another is formally to perform suitable integrations in... 

Figure 2.2 already shows that the number of nodes increases with n, corresponding to a decrease of A and an increase of both p and E. The probability density varies according to Fig. 2.4. It also depicts some of the positions where the particle can be instantaneously found in many determinations of an ensemble of identical systems equally prepared in such a way that we know them to all be in a given state n. For very high n (high energy), the probability distribution is practically uniform, thus approaching the classical description according to which the averaged residential time of the particle in each position is the same. This is another example of the tendency of quantum-mechanical predictions to approach classical predictions w hen...

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