Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 23. Brownian motion and continuation 247
In a stationary situation without periodic force, the probability is 1/2. Now use the expression
for r and look for a contribution, proportional to the periodic force. Again, ax 0 /kT is
considered small. We may put W 1/2 W 1 sin(t w), and get:
a
vW1 cos( vtw) r0 sin( vt) r0W1sin
( vt−w)
2kT
Identification of coefficients of cos and sin terms then yields the result:
v
tanw
;
r
0
ax b
P1
kT
B
r 0v
v2
r
0 2
(23.25)
w is a phase shift of the stochastic oscillations, W 1 is the amplitude of the oscillating contribution
of the probability. This is usually compared to the spectral function of the noise,
in this case, the Brownian motion in the potential at the frequency . This spectral function
is in this case primarily given by the Fourier transform of the slowly decaying probability
with the exponential decay function exp(r 0 t) which is 1/( 2 R 2 0). Thus, that quotient,
the signal–noise ratio, becomes proportional to v r 0 . As a function of the friction (which
also determines the strength of the irregular term, and what is called noise), it will have the
same maximum as found in the Kramers’ problem.