1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
23
This implies that
E[x] = 0; E[v] = 0
E[x
2
] =
k B T
mw 2
o
; E[v
2
] =
k B T
m
1.3.3 Exercise
Consider the Langevin representation of the gravitational wave detector introduced
in Ref. [11]:
˙
q(t) = I (t)
˙
I (t) = −
R k +R d
L k
I −
1
L k C k
q +
2k B T e f f (R k + R d ))
(1.73)
which implies
γ =
R k + R d
L k
; w
2
o =
1
L k C k
; σ =
2k B T e f f (R d + R k )
As we took σ
2
=
γk B T
m
, the above averages can be written as
E[x] = 0; E[v] = 0
E[x
2
] =
σ
2
γw 2
o
; E[v
2
] =
σ
2
γ
Consequently, our specific model has Gaussian distributions with
E[q] = 0; E[I ] = 0
(1.74)
and
E[q
2
] = var(q) =
σ
2
γw 2
o
=
2k B T e f f (R k + R d )
(R k +R d )
L k
1
L k C k
= 2k B T e f f L
2
k C k
E[I
2
] = var(I ) =
σ
2
γ
=
2k B T e f f (R k + R d )
(R k +R d )
L k
= 2k B T e f f L k
To obtain the PDF’s of
q τ (t) =
1
τ
t+τ
t
q(s)ds and of I τ (t) =
1
τ
t+τ
t
I (s)ds
(1.75)