Statistical mechanics classical limit

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states tenns in equation (A3.12.15) [3,14]. Thus, to detennine an accurate RRKM lc(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anliannonic corrections, both the molecule and transition state are often assumed to be a collection of hannonic oscillators for calculating the... 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

A method is outlined by which it is possible to calculate exactly the behavior of several hundred interacting classical particles. The study of this many-body problem is carried out by an electronic computer which solves numerically the simultaneous equations of motion. The limitations of this numerical scheme are enumerated and the important steps in making the program efficient on the computer are indicated. The applicability of this method to the solution of many problems in both equilibrium and nonequilibrium statistical mechanics is discussed. 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

Giauque, whose name has already been mentioned in connection with the discovery of the oxygen isotopes, calculated Third Law entropies with the use of the low temperature heat capacities that he measured he also applied statistical mechanics to calculate entropies for comparison with Third Law entropies. Very soon after the discovery of deuterium Urey made statistical mechanical calculations of isotope effects on equilibrium constants, in principle quite similar to the calculations described in Chapter IV. J. Kirkwood s development showing that quantum mechanical statistical mechanics goes over into classical statistical mechanics in the limit of high temperature dates to the 1930s. Kirkwood also developed the quantum corrections to the classical mechanical approximation. 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

Equation (1.24) is very similar to that of the single-particle distribution function of classical statistical mechanics. In the limit h—>0 we get the first equation of the BBGKY hierarchy. 

It is often impossible to obtain the quantized energies of a complicated system and therefore the partition function. Fortunately, a classical mechanical description will often suffice. Classical statistical mechanics is valid at sufficiently high temperatures. The classical treatment can be derived as a limiting case of the quantum version for cases where energy differences become small compared with ksT. 

Nonsequential double and multiple ionization are to a large part classical phenomena. Indeed, the S-matrix approach suggests a pertinent classical limit. We have summarized evidence that the latter reproduces the fully quantum-mechanical results very well in parameter regions where this can be expected. Finally, we have extended such classical avenues to a statistical description of nonsequential triple and quadruple ionization. For neon, such a classical statistical model yields a fair description of the available data. While a more microscopic description of these extremely involved phenomena lies in the future, we believe that the simple models summarized in this paper will remain valuable as benchmark results. 

We have seen that decoherence theory, according to its advocates [128], makes the wave-function collapse assumption obsolete The environmental fluctuations are enough to destroy quantum mechanical coherence and generate statistical properties indistinguishable from those produced by genuine wave-function collapses. All this is unquestionable, and if a disagreement exists, it rests more on philosophy than on physical facts. Thus, there is apparently no need for a new theory. However, we have seen that all this implies the assumption that the environment produces white noise and that the system of interest, in the classical limit, produces ordinary diffusion. As we move from... 

The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. In principle, the density matrix for a spin system includes all the spins, as we have seen, and all the spatial coordinates as well. However, as we discuss subsequently we limit our treatment to spins. For simplicity we deal only with application to systems of spin % nuclei, but the formalism also applies to nuclei of higher spin. 

The most obvious difficulty is that the numbers of oscillators in the classical RRK theory required to fit both the low pressure limit and the fall-off are smaller than the actual number of oscillators in the molecule, typically by a factor of about 2, although there can be wide variations [14]. Thus 14 oscillators have been used in the calculation presented in Fig. 5, but cyclobutane actually contains 30 oscillators. The problem is that the use of classical statistical mechanics overestimates the populations of excited states relative to the ground state. For example, in classical theory the mean energy of an oscillator is kT,... 

In order to make the theory useful it is necessary to know the constant of proportionality, which is calculated in such a way as to give the classical limit of the number of quantum states. This matter is dealt with in standard books on statistical mechanics [26]. The result is that for a system with n degrees of freedom, i.e. n position coordinates q and n momentum coordinates p, the number of states in the infinitesimal volume element rfq rfp is equal to rfq rfp//i", where n is Planck s constant. The association of a phase space volume /i with each quantum state can be thought of as a consequence of the uncertainty principle, which limits the precision with which a phase point can be specified in a quantum mechanical system. 

Quantum mechanical, classical and statistical probabilities agree, on average, reasonably well with the experimental results [133] shown in Fig. 37 (vibrational distributions of NO were also measured by Harrison et al. [310]). In the experiment a high population of the state n o = 1 is found already 100 cm above its threshold. Moreover, the measured probabilities show some indications of fluctuations. Because of the limited number of data points, the inevitable incoherent averaging over several overall rotational states of NO2 and the averaging over the various possible electronic states of the 0 and NO products, these fluctuations are less pronounced than in the quantum mechanical calculations on a single adiabatic PES and for J = 0. 

Quantum effects can be recovered by quantum simulations. Currently there are two main types of quantum simulation methods used. One is based on the time-dependent Schrodinger equation. The other is based on Feynman s path integral (PI) quantum statistical mechanics. [7,8] The former is usually complicated in mathematical treatment and needs also large computational resources. Currently, it can only be used to simulate some very limited systems. [77] MD simulations based on the latter have been used more than a decade and are gaining more and more popularity. The main reason is that in PIMD simulations, the quantum systems are mapped onto corresponding classical systems. In other words the quantum effects can be recovered by making a series of classical simulations with different effective potentials. 

In parallel there exist some attempts trying to introduce a field theory (FT) starting from the standard description in terms of phase space [4—6], Of course, the best way to derive a FT for classical systems should consist in taking the classical limit of a QFT in the same way as the so called classical statistical mechanics is in fact the classical limit of a quantum approach. This limit is not so trivial and the Planck constant as well as the symmetry of wave functions survive in the classical domain (see for instance [7]). Here, we adopt a more pragmatic approach, assuming the existence of a FT we work in the spirit of QFT. 

Phase transitions in statistical mechanical calculations arise only in the thermodynamic limit, in which the volume of the system and the number of particles go to infinity with fixed density. Only in this limit the free energy, or any thermodynamic quantity, is a singular function of the temperature or external fields. However, real experimental systems are finite and certainly exhibit phase transitions marked by apparently singular thermodynamic quantities. Finite-size scaling (FSS), which was formulated by Fisher [22] in 1971 and further developed by a number of authors (see Refs. 23-25 and references therein), has been used in order to extrapolate the information available from a finite system to the thermodynamic limit. Finite-size scaling in classical statistical mechanics has been reviewed in a number of excellent review chapters [22-24] and is not the subject of this review chapter. 

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