1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
27
In the simplest case, I (ω) = I = constant, in which all Fourier components contribute the same power, the spectrum of is called white, and the spectrum of v, I v ,
is Lorentzian:
I v (ω) =
I
ω 2 + γ 2
(1.92)
In a stationary state, taking t 1 > t 2 , one gets:
(t 1 − t 2 ) = E[(t 1 ))(t 2 )] =
+∞
−∞
I e
iω(t 1 −t 2 ) dω
(1.93)
Considering a stationary state, taking t 0 = t 2 , and using the following representation
of the Dirac delta function:
δ(x) =
1
2π
+∞
−∞
e
ikx dk, which means
+∞
−∞
dx
1
2π
∞
−∞
dk f (x)e
ikx
= f (0) (1.94)
one gets:
(t 1 − t 2 ) = 2π I δ(t 1 − t 2 )
(1.95)
which shows that q = 2π I in the Langevin treatment of Sect. 1.3.1. Analogously,
the autocorrelation of v reads:
v (t) =
+∞
−∞
I
ω 2 + γ 2 e
iωt dω
(1.96)
Then, for t = t 1 − t 2 , one may write:
v (t 1 − t 2 ) = E[v(t 1 )v(t 2 )] = I
+∞
−∞
e
iω(t 1 −t 2 )
ω 2 + γ 2 dω = I
π
γ
e
−γ(t 1 −t 2 ) for t 1 > t 2
(1.97)
or
E[v(t 1 )v(t 2 )] =
π I
γ
e
−γ|t 1 −t 2 |
, ∀t 1 , t 2 ∈ R
(1.98)
So velocity correlations decay exponentially in time. Then, taking t 1 = t 2 = t, and t
large enough that the steady state has been reached, and averages do not depend on
time anymore, one obtains:
E[v
2
] =
π I
γ
⇒ I =
γ
π
E[v
2
]
(1.99)