Class Theory in HYPE

A critical test of the classical nucleation theory applied to condensed phases has been carried out for the liquid miscibility gap system, methylcyclohexane-perfluoromethylcyclohexane (C7H14–C7F14). The densities of, and the interfacial free energies between the coexisting liquid phases have been measured as a function of temperature. The interfacial free energy is found to vary as the 32 power of the critical temperature Tc, less the experimental temperature, and these data are used to calculate the gradient energy parameter κ in the Cahn-Hilliard theory of nucleation. The magnitudes of the relevant parameters are such that the Cahn-Hilliard theory reduces to the classical theory for the present system. These data, together with literature data on the thermodynamic properties of the bulk solutions, enable the degree of undercooling for a sensible nucleation rate to be calculated from classical theory. The experimentally determined undercoolings necessary to nucleate the C7H14-rich phase from the solution are found to be much greater than those predicted by the classical theory by factors ranging from 8.5 at 10 deg below Tc to 340 at 0.3 deg below Tc. These very large discrepancies are attributable to the breakdown of the traditional approximation in the classical nucleation theory that the number of molecules involved in the embryo population is a negligible fraction of the number of molecules in the system. When this approximation is avoided, the very complicated expression that results can be evaluated for the present case, leading to reasonable agreement between the predicted and the observed undercoolings. This is believed to constitute a satisfactory verification of the ``corrected'' classical theory of nucleation for condensed systems. The ``embryo population'' factor can be neglected in homogeneous nucleation in solidification and condensation (except near the critical temperature) and in most of the corresponding cases of heterogeneous nucleation. However, it is extremely important for nucleation in miscibility gap systems, or in any system in which the properties of the parent and daughter phases become identical at a critical temperature.

The classical impact theory of Gordon is used to calculate half-widths and shifts of spectral lines of the pure rotational band of 12C16O isotopologue broadened by He. Two rotational transitions are examined: J=0-> J=1 and J=1-> J=2 in the wide temperature range from 1.3 to 600 K. The main purpose of this work is the study of the validity limits of classical impact theory at low temperatures. Dynamical calculations were performed on the accurate CO-He ab initio potential energy surface. The results of calculations are in good agreement with experimental data with the exception of very low temperatures. The contributions of collisions of different types (elastic, inelastic, quasibound complexes) are clearly examined in the classical picture frame. It is shown that the mismatches between classical theory and measurements are caused by the too high contribution of elastic collisions into broadening and shift in the present variant of theoretical model. The idea in the spirit of the Weisskopf theory is applied to try to diminish this contribution. The classical results are also compared with the results of fully quantum close coupling calculations made with using four CO-He interaction potentials. The roots of discrepancies at low temperatures as well as the virtues and the shortcomings of a classical approach are discussed. Keywords: collisional line broadening and shift, intermolecular interactions, classical impact theory, classical trajectory method, quasibound complexes.

What is the utility of classical political theory and modern writing about it for current-day positive (that is, explanatory) political science? By “classical,” I mean works from Plato through at least Weber. In modern American political science, the conventional answer to this question has been that classical theory offers normative illumination. It helps out with certain “should” questions—that is, recommendations about how political systems should be constructed or how individuals acting as political beings should behave. Important as this line of thinking may be in its own terms, I believe that it has worked to marginalize or trivialize classical theory for many writers and teachers in the positive sectors. It has allowed the kind of dismissal that Ayer (1948), writing at the high tide of logical positivism, gave to normative concerns. Classical theory scholars made a tactical mistake a few generations back, if they had any choice in the matter, to let themselves become labeled “normative.” It is much better to see classical theory as a source of ontological illumination—that is, as a window to the nature of political reality. What is the nature of the political reality that political scientists should be studying? If we members of the profession possessed a clear, singular answer to this question, we might not feel the need to keep classical theory alive. But we do not possess a clear, singular answer, or at least we do not agree on any.

Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature.

Modeling of nucleation and growth in glass-forming alloys using a combination of classical and phase-field theory

It is understood that in free bosonic theories, the classical field theory accurately describes the full quantum theory when the occupancy numbers of systems are very large. However, the situation is less understood in interacting theories, especially on time scales longer than the dynamical relaxation time. Recently there have been claims that the quantum theory deviates spectacularly from the classical theory on this time scale, even if the occupancy numbers are extremely large. Furthermore, it is claimed that the quantum theory quickly thermalizes while the classical theory does not. The evidence for these claims comes from noticing a spectacular difference in the time evolution of expectation values of quantum operators compared to the classical micro-state evolution. If true, this would have dramatic consequences for many important phenomena, including laboratory studies of interacting BECs, dark matter axions, preheating after inflation, etc. In this work we critically examine these claims. We show that in fact the classical theory can describe the quantum behavior in the high occupancy regime, even when interactions are large. The connection is that the expectation values of quantum operators in a single quantum micro-state are approximated by a corresponding classical ensemble average over many classical micro-states. Furthermore, by the ergodic theorem, a classical ensemble average of local fields with statistical translation invariance is the spatial average of a single micro-state. So the correlation functions of the quantum and classical field theories of a single micro-state approximately agree at high occupancy, even in interacting systems. Furthermore, both quantum and classical field theories can thermalize, when appropriate coarse graining is introduced, with the classical case requiring a cutoff on low occupancy UV modes. We discuss applications of our results.

In these notes we shall be mainly concerned with some global aspects of classical continuum Yang-Mills theory which may be of relevance to non-perturbative considerations in the quantum theory of the continuum Yang-Mills field. In doing so we make some assumptions about which one should be clear from the outset. In classical field theory it is necessary to introduce boundary conditions in order to solve the dynamics. In order to construct a (Euclidean) quantum theory of interacting fields it is customary to put in at the outset some cutoffs: a space (space-time) volume cutoff, as well as an ultraviolet cutoff. The latter is to be removed after renormalization, and then the burden is to take the infinite volume limit. We shall adhere to this philosophy [1]. A volume cutoff is introduced in these notes by compactifying space (space-time) in some way, right from the start in the classical theory. This is because in the functional integral approach to the quantum theory one integrates over classical field configurations. Ultraviolet regularization is not mentioned because we deal mostly with the classical theory, but it should be kept in mind. The other major hypothesis is with respect to regularity of classical field configurations.

From design to digital model: A quantitative analysis approach to Garden Cities theory

The methodology of “theory construction” developed in the U.S. around 1970 established an approach for the development of sociological theories. However, this methodology underestimates the value of studies conducted on classical sociological theories. According to Laudan's philosophy of science, the studies on the classical theories can develop current sociological theories by solving conceptual problems, identifying anomalies, and comparing the strengths and weaknesses of several sociological research traditions. By studying these classical sociological theories, we can find problems that require a solution and clues to solve these problems. However, it is very difficult to develop a theory only through studies on these classical theories because conceptual and empirical problems are closely connected to each other. In order to solve an empirical problem, we must gather and analyze data. Therefore, we must solve both conceptual and empirical problems simultaneously in order to develop sociological theories. This can be achieved through derivation and by joining a “good” research group.

In this paper, based on the modi ed couple stress theory, the size-dependent dynamic behavior of circular rings on elastic foundation is investigated. The ring is modeled by Euler-Bernoulli and Timoshenko beam theories, and Hamilton's principle is utilized to derive the equations of motion and boundary conditions. The formulation derived is a general form of the equation of motion of circular rings and can be reduced to the classical form by eliminating the size-dependent terms. On this basis, the size-dependent natural frequencies of a circular ring are calculated based on the non-classical Euler-Bernoulli and Timoshenko beam theories. The ndings are compared with classical beam theories. Response of the micro-ring under application of static and dynamic loads is investigatedand compared with the classical theories. Results show that when the thickness of the ring is in the order of the length scale of the ring material, the natural frequencies evaluated using the modi ed couple stress are considerably more than those predicted based on the classical beam theories, while the defection and natural frequencies of the classical and non-classical beam theories approach one another for the rings with thickness much larger than the material length scale.

The classical theory of mesons only contains the fundamental constant ξ. The rest mass μ of the meson is introduced only when the theory is quantized by the relation μ=ℏχ. As a result, although the quantum theory goes over strictly into the classical theory when ℏ → 0, the classical theory corresponds not only to the limit in which the momentum properties of mesons can be neglected, but also to the limit μ=0 (but ξ a finite constant). Due however to the fortunate circumstance that in nature M≫μ (M being the neutron mass) the classical theory can be used extensively to calculate processes involving mesons and heavy particles, but is entirely inadequate for calculating processes involving mesons and electrons or neutrinos, just because here μ≫m (m being the electron mass).

Three different derivations of the classical binary nucleation theory are considered in detail. It is shown that the derivation originally presented by Wilemski [J. Chem. Phys. 80, 1370 (1984)] is consistent with more extensive derivations [Oxtoby and Kashchiev, J. Chem. Phys. 100, 7665 (1994)]; Debenedetti, Metastable Liquids: Concepts and Principles (Princeton University Press, Princeton, 1996) if and only if the assumption is made that the surface of tension of the binary nucleus coincides with the dividing surface specified by the surface condition ∑nsivli=0, where the nsi denote surface excess numbers of molecules of species i, and the v’s are partial molecular volumes. From this condition, it follows that (1) the surface tension is curvature independent and (2) that the nucleus volume is V=∑nlivli=∑givli, where the nli are the numbers of molecules in the uniform liquid phase of the droplet model encompassed by the surface of tension, and the gi are the total molecular occupation numbers contained by the nucleus. We show, furthermore, that the above surface condition leads to explicit formulas for the surface excess numbers nsi in the nucleus. Computations for the ethanol–water system show that the surface number for water molecules (ns,H2O) causes the negative occupation numbers (gH2O) obtained earlier using the classical nucleation theory. The unphysical behavior produced by the classical theory for surface active systems is thus a direct consequence of the assumption of curvature independence of surface tension. Based on the explicit formulas for nsi, we calculate the full free-energy surfaces for binary nucleation in the revised classical theory and compare these with the free-energy surfaces in the Doyle (unrevised classical) theory. Significant differences in nucleus size and composition are found between these models and they are related to surface excess density. It is shown that only for the revised classical theory is the nucleus composition consistent with the Gibbs dividing surface model.

The study of the classical theory of optical fluctuations and coherence is an important preliminary to the development of the corresponding quantum-mechanical theory. The classical theory is useful for a physical understanding of the various effects and many of the classical concepts carry over into the quantum theory. In addition, a knowledge of the classical theory is useful for identification of the nature of the specifically quantum properties that some light beams may exhibit. Thus it is shown in Chapter 5 that the classical and quantum theories yield the ame predictions for chaotic light, but that there exist other kinds of light that elude any satisfactory classical description. The nature of such ‘nonclassical’ light i5, of course, more sharply appreciated by contrast with the limitations on the possible kinds of light inherent in the classical theory.

Starting from the pole-dipole approximation to the equations of motion for an extended body in an electromagnetic field, we develop a classical special-relativistic theory of a charged point particle with spin in an electromagnetic field. This theory is seen to take an especially simple form for a gyromagnetic ratiog=e/m, and this case is treated in detail. We then study the classical limit of the Dirac equation for an electron in a new way, and we obtain for the Hamiltonian and equations of motion expressions which agree with the energy and equations of motion in the classical theory to first order in the charge. In the process, we obtain Poisson brackets between the dynamical variables which enable us to recover the equations of motion from our classical expression for the energy, which may thus be regarded as the Hamiltonian in the classical theory. The assumptions of the classical theory are then discussed in more detail.