Einstein’s fluctuation theory

Scattering losses in polymers arise from microscopic variations in the material density. By using Einstein s fluctuation theory, the intensity of the isotropic light scattering Vy° from thermally induced density fluctuations in a stmctureless liquid is expressed by ... 

The depolarization of light by dense systems of spherical atoms or molecules has been known as an experimental fact for a long time. It is, however, discordant with Smoluchowski s and Einstein s celebrated theories of light scattering which were formulated in the early years of this century. These theories consider the effects of fluctuation of density and other thermodynamic variables [371, 144]. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

The foundations for the macroscopic approach to non-equilibrium thermod5mamics are found in Einstein s theory of Brownian movement [1] of 1905 and in the equation of Langevin [2] of 1908. Uhlenbeck and Omstein [3] generalized these ideas in 1930 and Onsager [4, 5] presented his theory of irreversible processes in 1931. Onsager s theory [4] was initially deterministic with little mention of fluctuations. His second paper... 

We have previously considered the effects of net charge and of ionic strength on the slopes of the Mc /x curves for proteins when plotted as a function of ca. However, the discussion was limited to cases where the fluctuation theory, as developed by Einstein, might be expected to apply that is to solutions of ionic strength 0.003 or greater, when the net proton charge (Z2) on the protein molecule is between +25 and —25 (for serum albumin). Under such conditions Einstein s assumption that the fluctuations in neighboring volume elements are independent... 

Thermodynamic fluctuation theory characterises equilibrium fluctuation by the so-called Einstein relation connecting the probability density function g with the (appropriately defined) entropy function S ... 

Fluctuations of macroscopic variables in thermodynamic systems at equilibrium or in steady-state conditions have long been understood (1). Fluctuations are the macroscopic manifestation of the discrete nature of matter and can be exploited to gain quantitative information about the elementary components of large systems. A classical example is the measurement by Perrin of Boltzmann s constant (and Avogadro s number) from the application of Einstein s theory of the Brownian motion of colloidal particles (2). Another equally classical example is the evaluation of the charge of the electron from the measurement of shot noise in vacuum tubes (3). 

The photomicrographic measurements refer directly to polymer motion under the influence of an external force. However, measurements of migration velocity v as a function of applied electrical field E show that some of these electrophoretic measurements were made in a low-field linear regime, in which the electrophoretic mobility jx is independent of E. Linear response theory and the fluctuation-dissipation theorem are then applicable they provide that the modes of motion used by a polymer undergoing electrophoresis in the linear regime, and the modes of motion used by the same polymer as it diffuses, must be the same. This requirement on the equality of drag coefficients for driven and diffusive motion was first seen in Einstein s derivation of the Stokes-Einstein equation(16), namely thermal equilibrium requires that the drag coefficients / that determine the sedimentation rate v = mg/f and the diffusion coefficient D = kBT/f must be the same. 

To proceed further, Onsager did not take the Gibbsian path. Instead, he adopted Boltzmann s definition of entropy and Einstein s theory of fluctuations. Nonetheless, Onsager was led to the following expression for the phenomenological coefficient Ljk. ... 

The result given by Eq. (411) for Spit) is the expression assumed for the errtropy entropy production and thermodynamic driving forces. On the basis of the statistical mechanical treatment of nonequihbrium systems, we find that this approximate approach is not required. 

Provided third-order and higher variations in the entropy are neglected, the results given by Eqs. (422a)-(424) are equivalent to the results given by Einstein s theory of fluctuations. Nonetheless, Callen s formulation is more general than the theory given by Einstein. 

In Einstein s theory of fluctuations, the hnear term in the expression for the instantaneous entropy Sj, given by Eq. (421), is absent due to the reqirirement that the fluctuations be executed about an equilibrium state. In the Callen treatment of thermodynamic fluctuations, the entropy S is the entropy of the most probable state, which does not necessarily represent an eqitilibriitm state. Hence, the linear term in Eq. (421) is, in general, nonvanishing. 

For systems both at equilibrium and removed from equilibrium, Einstein s theory of fluctuations indicates that the curvature of entropy (8 s) is a quantity which should be considered when seeking to describe the stability of a system. First-order terms, according to the stability theory of differential equations, are considered as general equilibrium conditions which reflect the irreversibility of the processes involved. Second-order terms, however, measure the ability to reestablish given boundary conditions when perturbations or fluctuations occur, and it is the sign of the curvature of entropy (8 s) which dictates stability for thermodynamic equilibrium. In the range for which the local equilibrium assumption remains valid. 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

Kinetics is concerned with many-particle systems which require movements in space and time of individual particles. The first observations on the kinetic effect of individual molecular movements were reported by R. Brown in 1828. He observed the outward manifestation of molecular motion, now referred to as Brownian motion. The corresponding theory was first proposed in a satisfactory form in 1905 by A. Einstein. At the same time, the Polish physicist and physical chemist M. v. Smolu-chowski worked on problems of diffusion, Brownian motion (and coagulation of colloid particles) [M. v. Smoluchowski (1916)]. He is praised by later leaders in this field [S. Chandrasekhar (1943)] as a scientist whose theory of density fluctuations represents one of the most outstanding achievements in molecular physical chemistry. Further important contributions are due to Fokker, Planck, Burger, Furth, Ornstein, Uhlenbeck, Chandrasekhar, Kramers, among others. An extensive list of references can be found in [G.E. Uhlenbeck, L.S. Ornstein (1930) M.C. Wang, G.E. Uhlenbeck (1945)]. A survey of the field is found in [N. Wax, ed. (1954)]. 

Investigation of the Brownian motion of dispersed particles made it possible to experimentally verify the theory of fluctuations, also formulated by Einstein and Smoluchowski. Svedberg s observations of Brownian motion indicated that the number of particles confined within a small... 

In photon correlation spectroscopy (PCS), light from a low-power helium-neon laser is focused on a temperature-controlled sample cell and light scattered at a known angle to the cell is detected by a photomultiplier. The random motion of particles in the laser beam causes fluctuations in the intensity of the scattered radiation that can be analyzed with a digital correlator. The smaller the particle the more rapid the fluctuations due to more rapid motion. The time dependence of the fluctuations is used to generate a correlation function, which is the sum of fluctuations caused by all particles. Autocorrelation theory can then be used to determine the diffusion coefficient, D, for the particles and hence the particle s hydrodynamic diameter, S, from the Stokes-Einstein equation ... 

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