Number density, definition

We begin with the definition of the grand potential f] as a functional of the number density of a fluid [49], p(r)... 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

In astrophysics, particle sizes in protoplanetary disks are usually described in the context of a distribution, dn = f(a)da, where a is the particle radius and (in is the number density of particles with radii between a and a + da. Note that in cosmochemistry, grain size usually refers to the largest diameter of a monomer. The description is often associated with two assumptions. First, it is often assumed that the distribution function, /(a), is close to being a power law in the ISM, except at very large or very small sizes (Weingartner Draine 2001), and power laws are therefore often used in protoplanetary disk studies. Second, the definition of a particle radius assumes that the particles are spherical - a convenience for converting the size distribution to an opacity law. 

The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equilibrium, the local states may be in local thermodynamic equilibrium all intensive thermodynamic variables become functions of position and time. The definition of energy and entropy in nonequilibrium systems can be expressed in terms of energy and entropy densities u(T,Nk) and s(T,Nk), which are the functions of the temperature field T(x) and the mole number density Y(x) these densities can be measured. The total energy and entropy of the system is obtained by the following integrations... 

We noted earlier that the detection limit is directly related to SNR and is often defined as an analyte concentration yielding a signal that is some factor, k, larger than the standard deviation of the blank, (Jbk- It is useful to define the detection limit for Raman spectroscopy as the minimum detectable value of the cross section-number density product, or ( SD)min- Of course, the concentration detection limit in terms of D or molarity will depend on the magnitude of p, but (PD)min is a more general definition that directly indicates spectrometer performance. In the vast majority of analytical Raman measurements at low values of PD, the SNR is background noise limited, so abk In... 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

This average property of the distribution is defined through the number density itself and it represents the mean particle size with respect to the number of particles in the system. Of course, other definitions of the mean particle size are possible, as will become clear below. If we define the th moment of the length-based NDF as... 

Here, pa is the average number density of molecules of species a, i.e., pa = Na)/V, with V the volume of the system. We also recall the definition of the spatial pair correlation function... 

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