Thermodynamic fluctuations

However, nonequilibrium steady states may be unstable even if the system is stable with respect to diffusion. For a nonequilibrium state, the stability condition for a chemical reaction in terms of excess entropy production is 

Example 12.2 Stability of chemical systems Consider the following chemical reaction system  

Assuming that the chemical system is far from equilibrium, and for = 1 and kb = 0 (disregarding the backward reaction), the reaction velocity and the affinity become 

The excess entropy production due to a fluctuation in the concentration of X around a steady-state value is 

Such a fluctuation would not violate the stability condition, and the inequality above would have a positive sign. 

---

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

Let us analyse the effect caused by the thermodynamic fluctuations of order parameter on the ground state of a dipole honeycomb lattice. Two factors determine the situation just as they do in the case of a square lattice.54,56,81 These are the... 

The thermodynamic fluctuation is obtained by multiplying the fluctuation in the nth state by the weighting factor f(En), and summing... 

G. Ruppeiner. Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605-59 (1995). 

We have already established that proteins are highly dynamic structures, which are constantly sampling different conformations due to local thermodynamic fluctuations. Some such transient conformations can be stabilized in new micro-environments, such as at a solid-liquid interface. 

Let us estimate the increase of the disturbance amplitude at the channel wall with respect to the initial value taken from the thermodynamic fluctuation theory. This theory predicts the initial disturbance of the order of Ao/iN)1/2, where N is the number of gas molecules in the volume Aq. Therefore, the condition for reaching a noticeable value of the disturbance amplitude leads to the relation... 

Since it is important to judge the accuracy of the description that will be given to the fluid, a third important quantity, the isothermal compressibility %T of the system, can be defined via two independent routes. On the one hand, by making use of the thermodynamic fluctuation theory, one obtains... 

The minimum of Landau free energy functional with respect to the order parameter determines the equilibrium state of the system. The deviations of the order parameter from the equilibrium value describe equilibrium thermodynamic fluctuations. 

H. Qian. Open-system nonequilibrium steady state statistical thermodynamics, fluctuations and chemical oscillations. J. Phys. Chem. B., 110 15063-15074, 2006. 

Equation (12.24) does not describe the dynamics of thermodynamic fluctuations, although the nonequilibrium thermodynamics theory and the thermodynamic fluctuations are connected in deriving the Onsager rules. However, the introduction of the notion of internal degrees of freedom into nonequilibrium thermodynamics implicitly accounts for the fluctuations. 

In addition, cure time is increased five minutes for every 0.25 inches of thickness of a molding [6, 7]. In general, these rules do not apply to most polymeric systems because the phenomena of heat transfer and cure kinetics have been over-simplified. The cure rate depends on the basic polymers, curatives, cure temperature, and filler loading. The prediction of cure rate will be discussed from a new model of cure kinetics which is developed from the concept of a non-equilibrium thermodynamic fluctuation theory of chemical relaxation. 

As has been discussed by Hsich [12, 13], any chemical relaxation (reaction) or structural relaxation can be explained from irreversible thermodynamic fluctuation theory in which changes of physical and mechanical properties during the relaxation process can be interpreted and predicted from the mean square fluctuations of thermodynamic ordering parameters. Then the physical or mechanical properties at any given cure time can be written as a relaxation function ... 

D. Magde, E. Elson, W.W. Webb, Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy. Phys. Rev. Lett. 29(11), 705-708 (1972)... 

We find that the nucleation proceeds via "the local structuring mechanism/ " i.e., a thermal fluctuation causing the local ordering of CO2 molecules leads to the nucleation of the clathrate, and not by the labile cluster hypothesis, one current conceptual picture. The local ordering of the guest molecules induces ordering of the host molecules at the nearest- and next-to-nearest-neighbor shells, which are captured by a three-body host-host order parameter, f these thermodynamic fluctuations lead to the formation of the critical nucleus. 

The dependence Blue light scatters more than red. (This is often associated with the blue of the sky and the red of the sunset," but other factors are involved here including the fact that in perfectly clean air (no particles) scattering occurs from small, thermodynamic fluctuations in the air density.)... 

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

---