1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
25
which can be assumed to be the case in general, since the subtraction of a constant
from a stochastic process does not alter its significance.
The average strength of the Fourier component a n is defined by:
E[|a n |
2
] = E[α
2
n ] + E[β
2
n ]
(1.80)
and the average intensity I T around frequency ω is defined by:
I T (ω; ω))ω =
ω n ∈[ω,ω+ω)
E
|a n |
2
(1.81)
where ω must be larger than the difference of two consecutive angular frequencies,
2π/T for the interval [ω, ω + ω) to contain at least one angular frequency. In
the case in which T is very large, ω can be small, and the Fourier components
indexed by the angular frequencies within [ω, ω + ω) are appriximately equal, if
the process is sufficiently regular. One may then write:
I T (ω; ω))ω ≈ E
|a ω ]
2
T
2π
ω
(1.82)
where ˆ
ω is any angular frequency in [ω, ω + ω), e.g. the smallest, and T /2π is
the number of frequencies per frequency unit, so that T ω/2π for very large T
approximately equals the number of frequencies in [ω, ω + ω).
Given the process Z and the accepted tolerance ω, one may choose T large
enough that the approximate equalities can be treated as equalities, and a n as a
practically continuous function of ω. Mathematically, this requires the large T limit
with fixed z Dω to be taken first, so that arbitrarily small ω can be taken after, for
a n to approximate better and better a given value a ω . As T increases, the power of
the signal in a the fixed interval [ω, ω + ω) should reach a finite value, therefore
the power of each single frequency must tend to 0. If this is the case, and it is the
case for standard physical applications, one may divide by ω and take the ω → 0
limit, thus obtaining the intensity spectrum of Z as:
I (ω) =
lim
ω→0
ˆ
ω∈[ω,ω+ω)
lim
T →∞
I T ( ˆ
ω, ,ω) =
lim
ω→0
ˆ
ω∈[ω,ω+ω)
lim
T →∞
T
2π
E
|a ˆ
w |
2
(1.83)
The terminology can be understood thinking about time dependent currents in electrical circuits. Letting I be the current, R the resistence and V the electric potential,
the dissipated power P under Ohm’s law is given by
P(t) = I (t)V (t) = R |I (t)|
2
(1.84)
Expressing I in terms of its Fourier components, I (t) =
n a n exp(iω n t), the average dissipated power then becomes:
25
which can be assumed to be the case in general, since the subtraction of a constant
from a stochastic process does not alter its significance.
The average strength of the Fourier component a n is defined by:
E[|a n |
2
] = E[α
2
n ] + E[β
2
n ]
(1.80)
and the average intensity I T around frequency ω is defined by:
I T (ω; ω))ω =
ω n ∈[ω,ω+ω)
E
|a n |
2
(1.81)
where ω must be larger than the difference of two consecutive angular frequencies,
2π/T for the interval [ω, ω + ω) to contain at least one angular frequency. In
the case in which T is very large, ω can be small, and the Fourier components
indexed by the angular frequencies within [ω, ω + ω) are appriximately equal, if
the process is sufficiently regular. One may then write:
I T (ω; ω))ω ≈ E
|a ω ]
2
T
2π
ω
(1.82)
where ˆ
ω is any angular frequency in [ω, ω + ω), e.g. the smallest, and T /2π is
the number of frequencies per frequency unit, so that T ω/2π for very large T
approximately equals the number of frequencies in [ω, ω + ω).
Given the process Z and the accepted tolerance ω, one may choose T large
enough that the approximate equalities can be treated as equalities, and a n as a
practically continuous function of ω. Mathematically, this requires the large T limit
with fixed z Dω to be taken first, so that arbitrarily small ω can be taken after, for
a n to approximate better and better a given value a ω . As T increases, the power of
the signal in a the fixed interval [ω, ω + ω) should reach a finite value, therefore
the power of each single frequency must tend to 0. If this is the case, and it is the
case for standard physical applications, one may divide by ω and take the ω → 0
limit, thus obtaining the intensity spectrum of Z as:
I (ω) =
lim
ω→0
ˆ
ω∈[ω,ω+ω)
lim
T →∞
I T ( ˆ
ω, ,ω) =
lim
ω→0
ˆ
ω∈[ω,ω+ω)
lim
T →∞
T
2π
E
|a ˆ
w |
2
(1.83)
The terminology can be understood thinking about time dependent currents in electrical circuits. Letting I be the current, R the resistence and V the electric potential,
the dissipated power P under Ohm’s law is given by
P(t) = I (t)V (t) = R |I (t)|
2
(1.84)
Expressing I in terms of its Fourier components, I (t) =
n a n exp(iω n t), the average dissipated power then becomes: