Thermal equilibrium description

The evaluation of rnel a in this case requires an a priori description of the phase temperature profiles. In general, investigators have assumed that gas and liquid are in thermal equilibrium. Therefore, Eq. (77) can be simplified to yield... 

The canonical ensemble was developed as the appropriate description of a system in thermal equilibrium with its surroundings by free exchange of energy. Following the discussion of classical systems the density operator of the canonical ensemble is introduced axiomatically as... 

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

The emissivity, 8, is the ratio of the radiant emittance of a body to that of a blackbody at the same temperature. Kirchhoff s law requires that a = e for all bodies at thermal equilibrium. For a blackbody, a = e = 1. Near room temperature, most clean metals have emissivities below 0.1, and most nonmetals have emissivities above 0.9. This description is of the spectrally integrated (or total) absorptivity, reflectivity, transmissivity, and emissivity. These terms can also be defined as spectral properties, functions of wavelength or wavenumber, and the relations hold for the spectral properties as well (71,74—76). 

The exact transmission factor (within classical molecular dynamics) kmd was calculated using the approach described in Section 5.1.2 that is, trajectories were sampled from the thermal equilibrium distribution at a dividing surface. Good agreement between kmd and kgh was found (with a transmission coefficient 0.5), whereas kkr severely underestimates the transmission (with a transmission coefficient < 0.05). The transmission coefficient in the non-adiabatic (frozen solvent) regime gives, on the other hand, a description that is in much better agreement with the numerical value of kmd-... 

Here, let us only remark that for a precise description of the state of the thermal equilibrium and of the conditions of decoupling of the various particles of the early Universe see Signore Puy (1999) and more recently Melchiorri et al. (2003). 

A simple fluid in thermal equilibrium, at its most primitive level of description, is characterized by a particle density n(r) controlled by an applied external potential u(r). In the grand ensemble, it is only the combination... 

Dummy level population. With no laser, the population of the dummy level is set at 11% of the total, the thermal equilibrium fraction in v=l at 2000°K. Because vibrational energy transfer rates are generally slow, the laser excitation causes a sizeable fraction of the total to be pumped into the dummy level. Fig. 3 shows the dummy level population for three laser intensities as a function of assumed a. (In the imensionless notation used in the computer, 1=1 corresponds to 10 erg sec- cm Hz-, or that of the unfocussed output of the fundamental from an efficient dye pumped by a powerful doubled Nd YAG laser). At the nominal 0.4 A, nearly 40% of the population is driven into the dummy level at high I. Clearly the value of C, a poorly known parameter, is important for a quantitative description of fluorescence saturation. 

In Section II, motivated by the fact that in typical experiments an aging system is not isolated, but coupled to an environment which acts as a source of dissipation, we recall the general features of the widely used Caldeira-Leggett model of dissipative classical or quantum systems. In this description, the system of interest is coupled linearly to an environment constituted by an infinite ensemble of harmonic oscillators in thermal equilibrium. The resulting equation of motion of the system can be derived exactly. It can be given, under suitable conditions, the form of a generalized classical or quantal Langevin equation. 

The light-induced and thermal equilibrium defect reactions are aspects of the same general process. Indeed, the structural models proposed are virtually identical (compare Figs. 6.13 and 6.30). In the two-well description of Fig. 6.1, excitation over the barrier in either direction can, in principle, be thermal or by an external excitation. The... 

Several other sources of external excitation result in metastable defect or dopant creation in a-Si H. Most have the characteristic property that a shift in the Fermi (or quasi-Fermi) energy causes a slow increase in the density of states and that annealing to 150-200 °C reverses the effect. The phenomena are therefore similar in origin to the optically-induced states and fall within the same general description of departures from the thermal equilibrium state induced by excess carriers. 

The MCT-ITT approach thus provides a microscopic route to calculate the generalized shear modulus g t, y) and other quantities characteristic of the quiescent and the stationary state under shear flow. While MCT has been reviewed thoroughly, see, e.g., [2, 38, 39], the MCT-ITT approach shall be reviewed here, including its recent tests by experiments in model colloidal dispersions and by computer simulations. The recent developments of microscopy techniques to study the motion of individual particles under flow and the improvements in rheometry and preparation of model systems, provide detailed information to scrutinize the theoretical description, and to discover the molecular origins of viscoelasticity in dense colloidal dispersions even far away from thermal equilibrium. 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

However the description of complex pulse sequences soon becomes cumbersome and less instructive and consequently the main application of the energy level scheme is for describing processes such as the population changes associated with a transition in a SPI experiment. NMR-SIM can be used to construct energy level schemes for a spin system in the state of thermal equilibrium. 

Adopting the two models of laser-induced initial conditions developed above, a nonequilibrium MD description of the photoinduced relaxation processes can be rationalized as follows. As the system is in thermal equilibrium before the excitation, we first perform a standard canonical MD simulation using the GROMACS simulation program package. From this trajectory, a number of (typically some hundred) statistically independent snapshots are chosen in which the positions and... 

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