An Introduction to Stochastic Processes in Physics by Don S. Lemons

Author:Don S. Lemons [Lemons, Don S.]
Language: eng
Format: epub
Published: 2011-02-19T05:00:00+00:00

P R O B L E M S

4 9

we find that p( x, t) solves the classical diffusion equation

∂ p( x, t)

δ2 ∂2

=

p( x, t) . (

∂

6.5.5)

t

2

∂ x 2

Equation (6.5.5) is mathematically equivalent to the stochastic dynamical equation (6.3.1). The latter equation governs the random variable X ( t), while the former governs its probability density p( x, t).

Deducing the diffusion equation (6.5.5) from its solution (6.5.1) reverses the usual order in modeling and problem solving. A more physically motivated derivation of (6.5.5) often starts with the observation, called Fick’s law, that a gradient in the probability density ∂ p/∂ x drives a probability density flux J

so that

∂ p

J = − D

.

(

∂

6.5.6)

x

where the proportionality constant D is called the diffusion constant. Fick’s law, like F = ma and V = IR, both defines a quantity (diffusion constant, mass, or resistance) and states a relation between variables. The diffusion constant is positive definite, that is, D ≥ 0, because a gradient always drives an oppositely directed flux in an effort to diminish the gradient. Combining Fick’s law and the one-dimensional conservation or continuity equation

∂ p

∂

+ J =

∂

0

(6.5.7)

t

∂ x

yields the diffusion equation (6.5.5) with D replacing δ2/2.

In his famous 1905 paper on Brownian motion, Albert Einstein (1879–1955) constructed the diffusion equation in yet another way—directly from the continuity and Markov properties of Brownian motion. Our approach, in section 6.3, to the mathematically equivalent result X ( t) − X (0) = N t (0, 2 Dt) has been 0

via the algebra of random variables. We use the phrase Einstein’s Brownian motion to denote both these configuration-space descriptions (involving only position x or X ) of Brownian motion. In chapters 7 and 8, we will explore their relationship to Newton’s Second Law and possible velocity-space descriptions (involving velocity v or V as well as position).

Problems

6.1. Autocorrelated Process.

Let X ( t) and X ( t ) be the instantaneous random position of a Brownian particle at times for which t ≤ t.

a. Find cov{ X ( t), X ( t )}.

b. Find cor{ X ( t), X ( t )}.

c. Evaluate cor{ X ( t), X ( t )} in the limits t / t → 0 and t / t → 1.

(Hint: Refer to the solution [6.3.6] and to self-consistency [6.2.7]. Also compare with Problem 3.4, Autocorrelation.)

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