Equilibrium fluctuation

Two major features must be introduced into the standard thermodynamic framework in order to apply thermodynamics at meso-scales fluctuations and local (coordinate-dependent) properties. Thermodynamics of equilibrium fluctuations is a well developed science and we will briefly address this topic in this section. Incorporating the local inhomogeneities is another task of mesoscopic thermodynamics. In this section we introduce a phenomenological approach, which is restricted to fluids with smoothly varying properties, known as local or quasi thermodynamics, and which dates back to van der Waals.  

Fluctuations are spontaneous and random deviations of thermodynamic properties from their average equilibrium values. These deviations are caused by thermal molecular motion. Macroscopic thermodynamics ignores fluctuations because they do not affect thermodynamic properties in the thermodynamic limit and they are usually insignificant in finite macroscopic systems. However, the situation changes when the system becomes very small or when it is near the limit of thermodynamic stability. In these two cases, fluctuations may become very large and may play a significant role in determining thermodynamic properties. 

A general approach for introducing fluctuations into thermodynamics is given by statistical mechanics. Let us consider an arbitrary, small portion of an isolated fluid. This small portion, referred to as the system , has a fixed volume V and is in equilibrium with the surrounding fluid at temperature T and chemical potential p. The thermal molecular motion of the fluid particles causes fluctuations of the thermodynamic properties of the system. These fluctuations exist in violation of the Second Law since they decrease the total entropy St of the fluid. Hence, the probability density of a fluctuation is 

Law can be used to express the heat flow Q in terms of p and the energy density M as g = V(Su-p5p). The probably density of a fluctuation then becomes 

Assuming that the fluctuations are small, du can be expanded to second order in the quantities 6s and 6p such that 

---

Figure 38. Classification of nonequilibrium fluctuations. (Reprinted from M. Asanuma and R. Aogaki, Non-equilibrium fluctuation theory on pitting dissolution. I. Derivation of dissolution current equations." J. Chem. Phys. 106,9938,1997. Copyright 1997, American Institute of Physics.)...

According to these equations, in kinetically controlled reactions the mean-square amplitude is about 10 V, while in reactions occurring under diffusion control it is almost an order of magnitude smaller. Thus, the size of electrochemical (thermal) equilibrium fluctuations is extremely small. 

The dielectric response of a solvated protein to a perturbing charge, such as a redox electron or a titrating proton, is related to the equilibrium fluctuations of the unperturbed system through linear response theory [49, 50]. In the spirit of free energy... 

In the above sense, the system may be considered as a thermodynamically closed system that will attain equilibrium if a non-equilibrium fluctuation were produced by some external means. 

Thus, the pressure difference on the two sides of the step is proportional to the difference of the inverse cubes of the terrace widths (neglecting possible intereactions with more distant steps). Again in the overdamped limit, the step velocity f)x/<5t is proportional to the pressure from the terrace behind the step minus the pressure from the terrace ahead of the step. Since the motion is again step diffusion, the prefactor ought to contain the same transport coefficient as that for equilibrium fluctuations, fa for EC or Ds Cs , for TD, in either case divided by keT. Alternatively, this can be described as a current produced by the gradient of achemicalpotential associated witheachstep(Rettori and Villain, 1988). 

A fiequently-used approximation in modeling SD and other dynamical processes in hquids is that of linear response When apphed to SD it corresponds to assuming that nonequilibrium response of the system to the perturbation AE turned on at r = 0 can be approximated in terms of equilibrium fluctuations of AE in the absence of the perturbation, i.e., for the system con-taiiting the solvent and the ground-state (subscript 0) chromophore ... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Thus the way in which the macroscopic average adjusts itself to the new field B is determined by the transition probability of the equilibrium fluctuations in the new field B. 

For definiteness consider a closed, isolated physical system. If at t = 0 the quantity Y has the precise value y0 the probability density P(y, t) is initially 5(y — y0). It will tend to Pe(y) as t increases. If y0 is macroscopically different from the equilibrium value of Y it means that y0 is far outside the width of Pe(y), because macroscopically observed values are large compared to the equilibrium fluctuations. We also know from experience that the fluctuations remain small during the whole process. That means that P y, t), for each t, is a sharply peaked function of y. The location of this peak is a fairly well-defined number, having an uncertainty of the order of the width of the peak, and is to be identified with the macroscopic value y(t). For definiteness one customarily adopts the more precise definition... 

Exercise. Compute from (4.14) the jump moments, taking for F the Maxwell distribution. Show that (4.1) and (4.2) hold when V is small compared to the average speed of the gas molecules, and can therefore be used to describe equilibrium fluctuations if Mpm. 

In this section we examine the higher orders beyond the linear noise approximation. They add terms to the fluctuations that are of order relative to the macroscopic quantities, i.e., of the order of a single particle. They also modify the macroscopic equation by terms of that same order, as has been anticipated in (V.8.12) and (4.8). These effects are obviously unimportant for most practical noise problems, but cannot be ignored in two cases. First, they tell us how equilibrium fluctuations are affected by the presence of nonlinear terms in the macroscopic equation, in particular how... 

Exercise. Find the equilibrium fluctuations at the critical point of the Schlogl reaction (X.3.6). 

Next consider the correlation of the equilibrium fluctuations at two different points and different times... 

As one wishes to deduce from such experiments the microscopic equilibrium properties of the material under study, it is essential to first establish a link to the equilibrium fluctuations AM(l) of the macroscopic dipole moment M(f) = —that is, the sum of the permanent molecular moments... 

When out-of-equilibrium dynamic variables are concerned, as will be the case in the following sections of this chapter, the equilibrium fluctuation-dissipation theorem is not applicable. In order to discuss properties such as the aging effects which manifest themselves by the loss of time translational invariance in... 

D. Herisson and M. Ocio, Off-equilibrium fluctuation—dissipation relation in a spin glass. Eur. Phys. J. B 40, 283 (2004). 

Producing an excitation of thdr angular velocity, that is, far-from-equilibrium fluctuation. 

---