Entropy fluctuation approach

In the previous section we have seen that the stationary states in the linear regime are also states that extremize the internal entropy production. We shall now consider the stability of these states, and also show that the entropy production is minimized. In Chapter 14 we saw that the fluctuations near the equilibrium state decrease the entropy and that the irreversible processes drive the system back to the equilibrium state of maximum entropy. As the system approaches the state of equilibrium, the entropy production approaches zero. The approach to equilibrium can be described not only as a steady increase in entropy to its maximum value but also as a steady decrease in entropy production to zero. It is this latter approach that naturally extends to the linear regime, close to equilibrium. 

Due to the local nature of the fluctuation approach, we did not pay much attention to the origin of the currents. We want to be more precise in this respect. Therefore we separate the total entropy change, dS, into two distinctly different contributions, i.e. 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

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 ... 

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... 

Figure 2.5 The rate of energy dissipation (entropy production) near the stationary point in a system close to thermodynamic equilibrium dependence of P = Td S/dt on thermodynamic driving forces nearby stationary point Xj (A) time dependence of P(7, 3) and dP/dt 2, 4) on approaching the stationary state (B). The vertical dashed line stands for the moment of approaching the stationary state by the system, and wavy line for escaping the stationary state caused by an internal perturbation (fluctuation).

In understanding experimental studies where a particle in an optical trap could be considered as a Brownian particle, FRs based on the stochastic Langevin equations were developed. This allowed analytic expressions for the entropy production and its probability to be obtained, and numerical predictions to be made. A similar approach has been used to study a Brownian particle diffusing in a periodic potential under steady state conditions and useful information characterising the fluctuations have been obtained analytically and from numerical calculations. ... 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

The direct demodulation algorithm provides a general approach to handle a large variety of image restoration or reconstruction problems. Computer simulations and analysis results for COS-B and CGRO 7-ray data show that in comparison with traditional techniques, e.g. maximum entropy method, cross-correlation deconvolution or likelihood approach, the direct demodulation method has high sensitivity, high resolution ability and capability to effectively reduce the effect of statistical fluctuations and noise in data and to simultaneously restore both the extended and discrete features in the object. 

There are rival theories of the glass transition the Gibbs Dimarzio theory assumes that the configurational entropy of the chains approaches zero at Tg. Other researchers prefer a mode coupling theory (MCT), based on the dynamics of density fluctuations. However, it is difficult to extract a simple physical meaning from the complex equations that describe correlations between density fluctuations. Neither theory, at its current state of development, is particularly useful in understanding the properties of glassy polymers. 

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