ABSTRACT
It w ill be described phenomenologically by decay rates and
noises in kinetic equations. We shall illustrate these basic
features by considering the Brownian motion of a harmonic
oscillator.
Let us forget about critical phenomena for the
moment. Consider a harmonic oscillator with the fam iliar
Hamiltonian
42 6 INTRODUCTION TO DYNAMICS
2 1 2 H = f — + 4 Kx . (11. 1)
2m 2
Let us introduce the notation
We call q^ and q^ modes. We shall re fe r to the
plane as the phase space. The motion of the osc illa tor is
thus represented by the motion of a point in the phase
space. In the absence of any other force, the veloc ity
(Vj, v^) of the point in phase space is
v 2 = q l / m ’
BROWNIAN MOTION 427
which traces out an ellipse. Now suppose that the oscillator
is immersed in a viscous fluid, i. e. , in contact with a ther-
mal reservo ir . The effect of the rese rvo ir can be approxi-
mately accounted for by a damping on the oscillator and a
random force, as given by the phenomenological kinetic
equations:
9qz — - v2 (11.4b)
where Γ^/Τ is a constant and ( - Γ /T) öH/öq^ = -F^q^/mT
is simply a frictional force. It gives a velocity in phase
a random function of time, is the random force of "noise. n
What is the reason for defining the damping term with an
extra factor l / T ? The reason is to have the dimensionless
combination H/T instead of H appear in the kinetic equa-
tion. Experience has told us that H and T most often
appear together as H/T. The average value and the c o r -
relation function of the noise are assumed to be
space pointing in the direction of decreasing energy, ζ ^(t),
where the average is taken over the assumed Gaussian
probability distribution of ζ , and 2D^ is a constant.
