Microscopic fluctuations

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

The small statistical sample leaves strong fluctuations on the timescale of the nuclear vibrations, which is a behavior typical of any detailed microscopic dynamics used as data for a statistical treatment to obtain macroscopic quantities. 

The relaxation time r of the mean length, = 2A Loo, gives a measure of the microscopic breaking rate k. In Fig. 16 the relaxation of the average length (L) with time after a quench from initial temperature Lq = 1.0 to a series of lower temperatures (those shown on the plot are = 0.35,0.37, and 0.40) is compared to the analytical result, Eq. (24). Despite some statistical fluctuations at late times after the quench it is evident from Fig. 16 that predictions (Eq. (24)) and measurements practically coincide. In the inset is also shown the reverse L-jump from Tq = 0.35 to = 1.00. Clearly, the relaxation in this case is much ( 20 times) faster and is also well reproduced by the non-exponential law, Eq. (24). In the absence of laboratory investigations so far, this appears the only unambiguous confirmation for the nonlinear relaxation of GM after a T-quench. 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

Further progress in understanding membrane instability and nonlocality requires development of microscopic theory and modeling. Analysis of membrane thickness fluctuations derived from molecular dynamics simulations can serve such a purpose. A possible difficulty with such analysis must be mentioned. In a natural environment isolated membranes assume a stressless state. However, MD modeling requires imposition of special boundary conditions corresponding to a stressed state of the membrane (see Refs. 84,87,112). This stress can interfere with the fluctuations of membrane shape and thickness, an effect that must be accounted for in analyzing data extracted from computer experiments. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during mechanical work. This is in essence the fluctuation theorem that was first derived by Bochkov and Kuzovlev [58-60] and later by Evans et al. [56, 57]. A derivation has also been given by Crooks [61, 62], and the theorem has been verified experimentally [63]. The present derivation is based on the author s microscopic transition probability [4]. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

These measurements for the first time allowed experimental access to the microscopic extent of cross-link fluctuations. The observed range of fluctuation is smaller than predicted by the phantom network model, for which... 

A laser beam highly focused by a microscope into a solution of fluorescent molecules defines the open illuminated sample volume in a typical FCS experiment. The microscope collects the fluorescence emitted by the molecules in the small illuminated region and transmits it to a sensitive detector such as a photomultiplier or an avalanche photodiode. The detected intensity fluctuates as molecules diffuse into or out of the illuminated volume or as the molecules within the volume undergo chemical reactions that enhance or diminish their fluorescence (Fig. 1). The measured fluorescence at time t,F(t), is proportional to the number of molecules in the illuminated volume weighted by the... 

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