Page 349 - Numerical Methods for Chemical Engineering

338 7 Probability theory and stochastic simulation

the probability of observing the system in microstate q is
1 E(q) E(q)
P(q) = exp − Q = exp − (7.114)
Q k b T k b T
q
k b is Boltzmann’s constant, the ideal gas constant R divided by Avagadro’s number. T is the
absolute temperature, in kelvin. Applying this formula to the kinetic energy distribution
of a moving particle of mass m at thermal equilibrium, we obtain the Maxwell velocity
distribution:
2
m|v|
P(v) ∝ exp − (7.115)
2k b T

Brownian dynamics and stochastic differential equations (SDEs)

We next consider an important application of probability theory to physical science, the
theory of Brownian motion, and introduce the subject of stochastic calculus. Let us consider
the x-direction motion of a small spherical particle immersed in a Newtonian fluid. As
observed by the botanist Robert Brown in the early 1800s, the motion of the particle is very
irregular, and apparently random. Let V x (t) be the x-direction velocity as a function of time.
For a particle of mass m and radius R in a fluid of viscosity µ, the equation of motion is
dV x
m =−ζ V x + F R (t) (7.116)
dt
where ζ = 6πµR is the drag constant (predicted for very small particles by Stokes’ law)
and F R (t) is a random, fluctuating force due to collisions between the particle and the fluid
molecules. Even though V x (t) fluctuates randomly, we can characterize deterministically
(t). Let V x (t 1 ) be the velocity at time t 1 and V x (t 2 )
the velocity autocorrelation function C V x
be the velocity at time t 2 . If we measure the product of these two values and take the
average, we obtain a function that should depend only upon the time elapsed between the
two measurements,

(t 2 − t 1 ) = V x (t 1 − t 2 )V x (0) (7.117)
V x (t 1 )V x (t 2 ) = C V x
(t), and at t = 0 should agree with
This function should have the property C V x (−t) = C V x
2
the average V predicted by the Maxwell velocity distribution,
x
k b T
; 2 <
(0) = V = (7.118)
C V x x
m
Also, since the velocities measured at very different times should be uncorrelated, we expect
(t) = 0. In general, we expect this correlation function to take the approximate
lim t→∞ C V x
form of an exponential decay,
(0)e −t/τ V x (7.119)
C V x (t ≥ 0) ≈ C V x
is a velocity correlation time.
where τ V x
related to the properties of the particle and fluid? Let us say that the particle
How is τ V x
is moving through the fluid at some velocity, and then at time t = 0, we turn off the random