Dynamical Theories Of Brownian Motion by Edward Nelson

By Edward Nelson

These notes are in keeping with a process lectures given by means of Professor Nelson at Princeton through the spring time period of 1966. the topic of Brownian movement has lengthy been of curiosity in mathematical chance. In those lectures, Professor Nelson strains the historical past of past paintings in Brownian movement, either the mathematical conception, and the traditional phenomenon with its actual interpretations. He maintains via fresh dynamical theories of Brownian movement, and concludes with a dialogue of the relevance of those theories to quantum box conception and quantum statistical mechanics.

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We let x(0) = x0 , v(0) = v0 . Then the mean of the process is etA x0 . v0 The covariance matrix of the Wiener process 2c = 0 B is 0 0 . 2, but the formulas are complicated and not very illuminating. The covariance for equal times are listed by Chandrasekhar [21, original page 30]. The Smoluchowski approximation is dx(t) = − ω2 x(t)dt + dw(t), β where w is a Wiener process with diffusion coefficient D. This has the same form as the Ornstein-Uhlenbeck velocity process for a free particle. , the highly overdamped case).

If we include the random fluctuations due to Brownian motion, this suggests the equation dx(t) = K dt + dw(t) β where w is the Wiener process with diffusion coefficient D = kT /mβ. If there were no diffusion we would have, approximately for t β −1 , dx(t) = (K/β)dt, and if there were no force we would have dx(t) = dw(t) . If now K depends on x and t, but varies so slowly that it is approximately constant along trajectories for times of the order β −1 , we write dx(t) = K x(t), t dt + dw(t). β This is the basic equation of the Smoluchowski theory; cf.

The random variables w(t ˜ i ) obtained when this transformation is applied to the w(ti ) are orthonormal since Ew(ti )w(tj ) = 50 CHAPTER 9 2D min(ti , tj ). Thus g˜, the probability density function of the w(t ˜ i ), is the unit Gaussian function on ❘n . 6) to the x(ti ). 6). We use the notation Cov for the covariance of two random variables, Cov xy = Exy − ExEy. 2 and the remark following it, Cov x(ti )x(tj ) = Cov w(ti )w(tj ) + εij , where |εij | ≤ 3Dβ −1 . 3) holds. 4) holds. The mean of x˜(ti ) for i > 1 is, in absolute value, smaller than 1 (e−βti−1 − e−βti )|v0 |/β[2D(ti − ti−1 )] 2 .

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