Fluctuations in volume

The average value and root mean square fluctuations in volume Vof the T-P ensemble system can be computed from the partition fiinction Y(T, P, N) ... 

The density dependence of the entropy can also be studied by introducing fluctuations in volume rather than particle number. Typically the particle number approach is favored the computational demands of volume scaling moves scale faster with system size than do addition and deletion moves. Nevertheless, the Wang-Landau approach provides a means for studying volume fluctuations as well. In this case, the excess entropy is determined as a function of volume and potential energy for fixed particle number one, therefore, calculates (V, U). Here the microstate probabilities follow ... 

We have SS = 0, as P ITX = P2 T2 = PIT at equilibrium, and the second derivative becomes negative. Equation (12.8) shows that the state of equilibrium is stable to fluctuations in volume because the isothermal compressibility is positive. 

The magnitude of the fluctuations in volume (dilatation) and density (condensation) associated with US wave is controlled by the properties of the medium and the applied forces. The velocity of sound in mixtures and suspensions will therefore be controlled by the mean density and mean compressibility as expressed by the Urick equation (see Eq.9.7-9.10). The equation can be formulated in terms of partial molar volumes by forming an identity between the volume fraction of the solute, its partial volume Vm2), the mean molar volume of the solution Vm) and the mole fraction Cm) as follows ... 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

The last formula shows that isothermal compressibility is related to the fluctuation in the total volume (AV) of the system. Again, the amplitude of this fluctuation in volume, measured by the second moment, is a measure of the volumes accessible to the system. Nevertheless, Eq. (2.A.2) is really an elegant expression. 

Variability takes two basic and interrelated forms, as summarized in Figure 28.1. The first form arises from fluctuations in volume. Volume variability can be externally or internally caused. The principal external cause (top box in Figure 28.1) is variability in demand as actual sales or sales forecasts increase or decline. Variability can also occur when customers cancel orders or move orders forward or backward in time. 

We again consider the director no oriented along the z-axis. Its fluctuating part has components (n riy, 0). The total Frank elastic energy, related to the fluctuations in volume V, is given by the integral of free energy density (8.15b) ... 

Disruption of hydrodynamical stability may take place either as a result of local fluctuations in volume properties of a continuous medium (viscosity or density) or due to disturbances introduced into the boundary conditions. A general equation system for the differential balance of forces can be written as ... 

To show how to use the above reformulation, consider fluctuations in volume dVoi the system that arise when the rigid partition between system and surroundings is replaced by a flexible one. This in turn engenders fluctuations in T and P. With dE specified explicitly as usual at constant n, we then find from Eq. (1.17.28b) that (for the molar values)... 

This result is identical to that obtained from a canonical ensemble approach in the thennodynamic limit, where the fluctuations in N vanish and (N) = N. The single-particle expression for the canonical partition fiinction = (-" can be evaluated using ih r rV i f<,2M) or a particle in a cubical box of volume V. 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

As noted at the end of the last section, it is fluctuations in concentration 5c2 rather than density which act as the scattering centers of interest for solutions of small molecules. There is nothing in the forgoing theory that prevents us from placing 6p by 6c2, the solute concentration in mass volume" units. Therefore we write for a solution of small molecules... 

The phase Doppler method utilizes the wavelength of light as the basis of measurement. Hence, performance is not vulnerable to fluctuations in light intensity. The technique has been successfully appHed to dense sprays, highly turbulent flows, and combustion systems. It is capable of making simultaneous measurements of droplet size, velocity, number density, and volume flux. 

Liquids can also exert pressure due to thermal expansion. Table 4.15 provides an indication of pressure increases due to temperature increases for selected common liquids in full containers or pipes. Serious accidents can arise unless the design of rigid plant items such as pipework takes into account the changes in volume of liquids with temperature fluctuation by the following or combinations thereof ... 

The large size of now is responsible for mean-field theory being reliable for large N Invoking the Ginzburg criterion one says mean field theory is self-consistent if the order parameter fluctuation in a correlation volume is much smaller than the order parameter itself. 

As mentioned earlier, there is an inverse relationship between water volumes and oxygen concentration in soil. As soils dry, conditions become more aerobic and oxygen diffusion rates become higher. The wet-dry or anaerobic-aerobic alternation, either temporal or spatial, leads to higher corrosion rates than would be obtained within a constant environment. Oxygen-concentration-cell formation is enhanced. This same fluctuation in water and air relations also leads to greater variation in biological activity within the soil. 

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