Fluctuations from Equilibrium Values

The kinetic theory relied on converting the momentum transfer from individual collisions (which are very abrupt) into an average pressure. This will only be valid if the pressure we observe is the average of many events on an everyday timescale—in which case the fluctuations are small. This is a reasonable approximation, as we can illustrate by an example which might reflect an attempt to measure these fluctuations. 

For example, suppose we measure the pressure with a simple U-tube manometer filled with mercury. Suppose the manometer is set up with a 1 cm diameter tube exposed to 1 atm nitrogen at room temperature (298K) on one end, and exposed to vacuum on the other end (which of course will be approximately 760 mm higher). The observed pressure can only change when the column of mercury has time to flow the device (and any other measuring device) will have a nonzero response time. A reasonable estimate for the response time of a manometer might be 0.1 seconds, so the amount the pressure will appear to fluctuate will depend on the number of collisions with the top of the column in that time. 

The average collision generates a momentum transfer of 2 m ( vy ) (Equation 7.7) which for N2 is roughly 

Finally, the rule of thumb given in Chapter 4 is that fluctuations in random processes scale roughly as the square root of the number of events. So the number of collisions will fluctuate by about V2.8 x 1022 1011, and the pressure will fluctuate by about 1 part in 1011—in other words, it will stay the same in the first ten or eleven digits of its value  

Realistically, fluctuations are larger than this because the atmosphere is not at equilibrium both air currents and temperature variations will generate larger effects. But these examples show that the macroscopic average effect (the pressure) can be quite uniform, even though each molecule provides its contribution to the pressure only in the instant it collides with the walls, and thus the individual contributions are not at all uniform. Statistical averaging has dramatically simplified the apparent behavior. 

---

Toupin and Lax consider the problem of permanent and induced dipoles on cubic lattice sites (or continuum) with the latter represented by harmonic oscillators as in Van Vleck s early work described in 1 3 The device of introducing fluctuations from equilibrium displacements works for harmonic oscillators because the integrations over the formula to evaluate averages are for all values from oO to and unchanged by the shifts in origin A similar device is not possible for proper averages over possible permanent dipole moments as the ranges are restricted by the N constraints becomes so if these are replaced by the... 

Next we consider how to evaluate the factor 6p. We recognize that there is a local variation in the Gibbs free energy associated with a fluctuation in density, and examine how this value of G can be related to the value at equilibrium, Gq. We shall use the subscript 0 to indicate the equilibrium value of free energy and other thermodynamic quantities. For small deviations from the equilibrium value, G can be expanded about Gq in terms of a Taylor series ... 

The functional form of U R) differs from one diatomic molecule to another. Accordingly, we wish to find a general form which can be used for all molecules. Under the assumption that the intemuclear distance R does not fluctuate very much from its equilibrium value so that U R) does not deviate greatly from its minimum value, we may expand the potential U R) in a Taylor s series about the equilibrium distance R ... 

For simplicity, it is assumed that the equilibrium value of the macrostate is zero, x = 0. This means that henceforth x measures the departure of the macrostate from its equilibrium value. In the linear regime, (small fluctuations), the first entropy may be expanded about its equilibrium value, and to quadratic order it is... 

Now consider a pair of reservoirs in equilibrium with respect to the extensive parameters Xj and Xk, with instantaneous values of Xj and Xk. Let 6Xj denote a fluctuation from the instantaneous value. The average value of 6Xj is zero, but the average of its square (SXj)2 = ((6Xj)2) 0. Likewise, the average correlation moment (5Xj5Xk) / 0. 

To remove momentum fluctuations from the problem, BCAH assume that in the creeping flow limit, in which a system of small mass interacts strongly with a thermally equilibrated solvent, the distribution of values for the momenta for fixed coordinate values stays very near a state of local equilibrium, in which... 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

At thermal equilibrium, the helical fraction and all other quantities characterizing the conformation of a helix-forming polypeptide are fluctuating from time to time about certain mean values which are uniquely determined by three basic parameters s, a, and N. The rates of these fluctuations depend on how fast helix units are created or disappear at various positions in the molecular chain. Recently, there has been great interest in estimating the mean relaxation times of these local helix-coil interconversion processes, and several methods have been proposed and tested. In what follows, we outline the theory underlying the dielectric method due to Schwarz (122, 123) as reformulated by Teramoto and Fujita (124). 

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... 

Thus we have found the distribution of the fluctuations around the macroscopic value. They have been computed to order Q 1/2 relative to the macroscopic value n, which will be called the linear noise approximation. In this order of Q the noise is Gaussian even in time-dependent states far from equilibrium. Higher corrections are computed in X.6 and they modify the Gaussian character. However, they are of order 2 1 relative to n and therefore of the order of a single molecule. 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

AT1 usually varied from 0.5°K during the first 100 steps to 2.5 or 5°K for the last 100 steps. The maximum value of AT depended on the number of molecules being followed and the expected temperature or kinetic energy fluctuations at equilibrium for this number of molecules. The effect on temperature fluctuations of applying the above method in the first 100 steps of the equilibration phase is again illustrated in Figure 1. Note that... 

Figures 4.29 and 4.30 display the co-evolution of the formamide dipole moment and of the oxygen(formamide)-oxygen(water) radial distribution function during the polarization process [12]. As the dipole moment of the formamide increases, the position of the first peak of the RDF is shifted inward and its height increases. Once the dipole moment has reached its equilibrium value, it begins to fluctuate. Fluctuations are related to the statistical error associated with the finite length of the simulations. From Figures 4.29 and 4.30 it is clear that (1) ASEP/MD permits one to simultaneously equilibrate the...

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. 

Very large affinity values may cause instability, and lead to new states that are no larger homogeneous in space. This causes a discontinuous decrease of entropy, and has important consequences in oscillating chemical reactions. Such reactions are far from equilibrium, and present undamped fluctuation on a macroscopic scale. Oscillations around a stationary state are possible as long as the total entropy production is positive. 

It is important to realize that a perturbation is not necessarily related to any external action on the system. Molecular fluctuations lead inevitably to small variations of the macroscopic quantities from their equilibrium values. There is in fact a relation between the probability of a fluctuation and the production of entropy which accompanies it but we shall defer full discussion of this problem until we examine irreversible processes in greater detail. 

---