Brownian motion in the equilibrium state

M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part I. Brownian motion in the equilibrium state, J. Chem. Soc. Faraday Trans. II, 74, 1789 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 2. Molecular motion under flow, J. Chem. Soc. Faraday Trans.II, 74, 1802 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 3. The constitutive equation, J. Chem. Soc. Faraday Trans. II, 74,1818 (1978) M. Doi and S. F. Edwards, Dynamics of concentrated polymer systems. Part 4. Rheological properties, J. Chem. Soc. Faraday Trans. II, 75,38 (1979). 

M.Doi, S.F.Edwards, Dynamics of concentrated pol3uner systems- Part 1 Brownian motion in the equilibrium state, J. Chem. Soc, Faraday Trans II B(1978), 1789-1801. 

M. Doi and S. F. Edwards, Dynamics of Concentrated Polymer Systems. Part I Brownian Motion in the Equilibrium State , J. Chem. Soa, Faraday Trans. 2 74,1789-1801 (1978). 

The period between these two dates saw the publication of two papers by Einstein (1905) 07 and of two by Smoluchowski (1906)208 on the Brownian motion. In the present context the reasoning of Einstein s second paper is of special interest. Let us assume that the N molecules discussed in connection with the expression given Eq. (78) compose a microscopically small particle suspended in a liquid which is in thermal equilibrium. In order to determine the instantaneous state of motion of... 

An important quantity characterizing the Brownian motion is the time correlation function, which is operationally defined in the following way. Suppose we measure a physic quantity A of a system of Brownian particles for many samples in the equilibrium state. Let A t) be the measured values of A at time t. Usually A(f) looks like a noise pattern as shown in Fig. 3.3a. The time correlation function Cyi (l) is defined as the average of the product A(r)A(0) over many measurements ... 

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 transient interval of time between the application of the field and saturation (Fig. 11a) lasts for less than 1.0 ps, and in this period the rise transient oscillates deeply (Fig. 11b). The oscillation of the racemic mixture is significantly deeper than that in the / enantiomer. The experimental study of transients such as these, then, migllt be a conv ent method of measuring the dynamical effect of chiral discrimination in the liquid state. Deep transient oscillations such as these have been foreseen theoretically by Coffey and coworkers using the theory of Brownian motion. The equivalent fall transients (Fig. 11b) are much loiter lived than the rise transients and are not oscillatory. They decay more quickly than the equilibrium acfs. The effect of chiral discrimination in Fig. lib is evident. Note that the system... 

The first theory giving the tj -induced decrease of the rate constant is the Kramers theory presented as early as in 1940. He explicitly treated dynamical processes of fluctuations in the reactant state, not assuming a priori the themud equilibrium distribution therein. His reaction scheme can be understood in Fig. 1 which shows, along a reaction coordinate X, a double-well potential VTW composed of a reactant and a product well with a transition-state barrier between them. Reaction takes place as a result of diffusive Brownian motions of reactants surmounting... 

Polymer networks are conveniently characterized in the elastomeric state, which is exhibited at temperatures above the glass-to-rubber transition temperature T. In this state, the large ensemble of configurations accessible to flexible chain molecules by Brownian motion is very amenable to statistical mechanical analysis. Polymers with relatively high values of such as polystyrene or elastin are generally studied in the swollen state to lower their values of to below the temperature of investigation. It is also advantageous to study network behavior in the swollen state since this facilitates the approach to elastic equilibrium, which is required for application of rubber elasticity theories based on statistical thermodynamics. ... 

It is conceivable that the equilibrium orientation of the intercalated dye in its Sj excited state differs from that in its S0 ground state and that this is what is responsible for the rapid initial relaxation. If so, the rms amplitudes of internal Brownian motion estimated above would be upper limits to the actual values. 

Neutral molecules, dissolved, dispersed or suspended in a liquid medium, are in continuous random motion (Brownian motion) with a mean free path (x) and collision diameter (xe), depending on c and vex effects. At a far separation distance, is negative, increasing to 0 at xe, where repulsion counterbalances attraction and the amphiphiles are at dynamic equilibrium in a primary minimum energy state. At x High concentrations shorten x and make the collision rate nonlinear with c, (Hammett, 1952). A separation distance of x < xe is sterically forbidden without fusion. 

The application of this model in physics and chemistry has had a long history. We shall give some examples of the early works. The work of Einstein S2) on the theory of Brownian motion is based on a random walk process. Dirac S3) used the model to discuss the time behavior of a quantum mechanical ensemble under the influence of perturbations this development enables one to discuss the probability of transition of a system from one unperturbed stationary state to another. Pauli 34) [also see Tolman 35)], in his treatment of the quantum mechanical H-theorem, is concerned with the approach to equilibrium of an assembly of quantum states. His equations are identical with those of a general monomolecular... 

The fluctuating variables aie thereby projected onto pair-density fluctuations, whose time-dependence follows from that of the transient density correlators q(,)(z), defined in (12). Tliese describe the relaxation (caused by shear, interactions and Brownian motion) of density fluctuations with equilibrium amplitudes. Higher order density averages are factorized into products of these correlators, and the reduced dynamics containing the projector Q is replaced by the full dynamics. The entire procedure is written in terms of equilibrium averages, which can then be used to compute nonequilibrium steady states via the ITT procedure. The normalization in (10a) is given by the equilibrium structure factors such that the pair density correlator with reduced dynamics, which does not couple linearly to density fluctuations, becomes approximated to ... 

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