Thermodynamic theory of fluctuations

From this expression it is clear that the leading contribution to the entropy change due to a fluctuation in the equilibrium state is of second order and we may make this explicit by using 6 5/2 in place of Aj5. In terms of 5 5/2, equations (14.1.14) and (14.1.15) can be written as 

In the previous sections we have discussed stability of a thermodynamic state in the face of fluctuations. But the theory that we presented does not give us the probability for a fluctuation of a given magnitude. To be sure, our experience tells us that fluctuations in thermodynamic quantities are extremely small in macroscopic systems except near critical points still we would like to have a theory that relates these fluctuations to thermodynamic quantities and gives us the conditions under which they become important. 

In an effort to understand the relation between microscopic behavior of matter, which was in the realm of mechanics, and macroscopic laws of thermodynamics, Ludwig Boltzmann (1844-1906) introduced his famous relation that related entropy and probability (Box 3.1)  

Albert Einstein (1879-1955) proposed a formula for the probability of a fluctuation in thermod5mamic quantities by using Boltzmann s idea in reverse whereas Boltzmann used microscopic probability to derive thermod5mamic entropy, Einstein used thermodynamic entropy to obtain the probability of a fluctuation through the following relation  

To obtain the probability of a fluctuation, we must obtain the entropy change associated with it (Fig. 14.2). The basic problem then is to obtain Ae A5 in terms of the fluctuations 5T, 5p, etc. But this has already been done in the previous sections. Expression (14.1.13) gives us the entropy associated with a fluctuation  

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According to the thermodynamic theory of fluctuations, the mean-square concentration fluctuation is given - by... 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

If we look at a small portion of a macroscopic system or study a mesoscopic system, we must study fluctuations. The probability of fluctuations is phenomenologically described by the thermodynamic theory of fluctuations (5). From the ensemble theory point of view, the fluctuation theory is the study of large deviations from the expectation value. This is the reason why large deviation theory is becoming increasingly important in statistical thermodynamics. Standard works on large deviation theory are References 16 and 17 perhaps as accessible introduction to the topic may be found in Reference 18. 

Glansdorff, G., and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Chapter 3, Wiley, London, 1971. 

P. Glansdorff and I. Progogine, Thermodynamic theory of Structure, Stability and Fluctuations, 1971, Wiley, New York. 

This principle is very general, relating neither to the linearity nor to the symmetry of the transport laws. On the other hand, it is difficult to attribute a physical meaning to dxP- The authors later attempted to derive a local potential from this property, and they applied this concept to the study of the chemical and hydrodynamical stability (e.g., the Benard convection). The results of this approach were published in Glansdorff and Prigogine s book Thermodynamic Theory of Structure, Stability and Fluctuations (LS.IO, 10a), published in 1971. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Glansdorff, P. Prigogine, I. "Thermodynamic Theory of Structure, Stability and Fluctuations" Interscience-Wiley, New York, 1971. 

Refs. [i] Prigogine I-Autobiography http //nobelprize.org/chemistry/ laureates/1977/prigogine [ii] Prigogine I (1967) Introduction to the thermodynamics of irreversible processes. Wiley Interscience, New York [iii] Prigogine I (1962) Non-equilibrium statistical mechanics. Wiley Interscience, New York [iv] Glansdorff P, Prigogine I (1971) Thermodynamic theory of structure stability and fluctuations. Wiley, London... 

From the perspective of the fluctuation-dissipation approach, Dewey (1996) proposed that the time evolution of a protein depends on the shared information entropy. S between sequence and structure, which can be described with a nonequilibrium thermodynamics theory of sequence-structure evolution. The sequence complexity follows the minimal entropy production resulting from a steady nonequilibrium state... 

Molecular fragments are the mutually open subsystems, which exhibit fluctuations in their electron densities and overall numbers of electrons. In chemistry one is interested in both the equilibrium distributions of electrons and non-equilibrium processes characterized by rates. Recently, it has been demonstrated [23] that the information theory provides all necessary tools for the local dynamical description of the density fluctuations and electron flows between molecular subsystems, which closely follows the thermodynamic theory of irreversible processes [146],... 

Glansdorff, P., I. Prigogine Thermodynamic Theory of Structure Stability and Fluctuations Wiley New York, 1971... 

The theory of fluctuation deals with the problem that a system far from equilibrium could move away from the direction of equilibrium for a short time, which is in contrast to the second law [12, 13]. The fluctuation theorem is also connected to the phenomenological formalism of reaction kinetics, in particular, to unimolecular reactions [14]. The fluctuation theorem predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales. 

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