General Linearized Brownian Motion

General Linearized Brownian Motion. Our discussion has not so far passed beyond the Langevin approximation, which must be modified for realistic description of the motion of molecules not vastly more massive than the molecules of the solvent medium. This is particularly necessary for the discussion of pure substances. 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

Diffusion Times. Brownian motion of molecules and particles is discussed in Section 5.2. The root-mean-square displacement of a particle is inversely proportional to the square root of its diameter. Examples are given in Table 9.4. The diffusion time for heat or matter into or out of a particle of diameter d is of the order of d2/ ()D where D is the diffusion coefficient. All this means that the length scale of a structural element, and the time scale needed for events to occur with or in such a structural element, generally are correlated. Such correlations are positive, but mostly not linear. 

Correlations between the linear displacement (of the origin O) and angular displacement in a given time interval due to Brownian motion may be expressed in terms of a generalized form of Eq. (317). It is found that (B26a)... 

Let us first explain what we mean here for generalized Langevin equation (GLE). For this, we summarize rather well-known concepts of the theory of Brownian motion and thermal fluctuations cast as stochastic processes, generated by linear stochastic equations with additive noise. 

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