18
L. Rondoni
processes. We call Gaussian a process Z (t) indexed by a parameter t, e.g. time, if
the joint probability density of its values z 1 , z 2 , . . . , z n at times t 1 , t 2 , . . . , t n is given
by
W n (z 1 , t 1 ; . . . ; z n , t n ) = ce
−
1
2
n
i, j=1
a i j (z i −m i )(z j −m j )
(1.43)
where m k = E[Z (t k )] and A = (a i j ) is a positive definite n × n matrix, whose inverse
has entries defined by (A
−1
) i j = E[(Z (t i ) − m i )(Z (t j ) − m j )], for i, j = 1, . . . , n.
The first three cumulants of the probability distribution of a random variable Z , E[·] c ,
are defined by:
E[Z ] c = E[Z ]
E[Z
2
] c = E[Z
2
] − E[Z ]
2
E[Z
3
] c = E[Z
3
] − 3E[Z
2
]E[Z ] + 2E[Z ]
3
(1.44)
Therefore, a normal distribution N (m, σ) implies:
E[Z ] c = m, E[Z
2
] c = σ
2
, E[Z
n
] c = 0 for n ≥ 3
(1.45)
and if all cumulants of order n ≥ 3 vanish, then the distribution is normal. defining
a linear transformation of a stochastic process Z (t) as
Y (t) =
b
a
c(t, t
)Z (t
)dt
(1.46)
the cumulant of Y (t), for any order n, is given by:
E[Y
n
] c =
b
a
dt
1
b
a
dt
2 . . .
b
a
dt
n c(t 1 , t
1 )c(t 2 , t
2 ) . . . c(t n , t
n )E[Z
n
] c
(1.47)
Hence, if Z (t) is Gaussian, Y (t) is, because all cumulants of order n ≥ 3 vanish.
Then, the velocity V of the Brownian motion is a Gaussian process if the stochastic
term (t) is, because it results from an integral like (1.46). Considering that in
the t → ∞ limit, E[V ] c = 0, and σ
2
= E[V
2
] c = q/2γ, the asymptotic probability
density of V is given by:
f V (v) = ce
−
v 2
(2q/2γ) = ce
−
γv 2
q
=
m
2πk B T
e
−
mv 2
2k B T
= f M B (v)
(1.48)
if we accept the equipartition principle,
q
γ
=
2k B T
m
.
This is the celebrated Maxwell-Boltzmann distribution in 1 dimension. Because
its variance takes the form σ
2
= k B T /m, fixing the temprature of the fluid T , and
placing a large mass m in it, yields small uncertainty on the velocity: the velocity
L. Rondoni
processes. We call Gaussian a process Z (t) indexed by a parameter t, e.g. time, if
the joint probability density of its values z 1 , z 2 , . . . , z n at times t 1 , t 2 , . . . , t n is given
by
W n (z 1 , t 1 ; . . . ; z n , t n ) = ce
−
1
2
n
i, j=1
a i j (z i −m i )(z j −m j )
(1.43)
where m k = E[Z (t k )] and A = (a i j ) is a positive definite n × n matrix, whose inverse
has entries defined by (A
−1
) i j = E[(Z (t i ) − m i )(Z (t j ) − m j )], for i, j = 1, . . . , n.
The first three cumulants of the probability distribution of a random variable Z , E[·] c ,
are defined by:
E[Z ] c = E[Z ]
E[Z
2
] c = E[Z
2
] − E[Z ]
2
E[Z
3
] c = E[Z
3
] − 3E[Z
2
]E[Z ] + 2E[Z ]
3
(1.44)
Therefore, a normal distribution N (m, σ) implies:
E[Z ] c = m, E[Z
2
] c = σ
2
, E[Z
n
] c = 0 for n ≥ 3
(1.45)
and if all cumulants of order n ≥ 3 vanish, then the distribution is normal. defining
a linear transformation of a stochastic process Z (t) as
Y (t) =
b
a
c(t, t
)Z (t
)dt
(1.46)
the cumulant of Y (t), for any order n, is given by:
E[Y
n
] c =
b
a
dt
1
b
a
dt
2 . . .
b
a
dt
n c(t 1 , t
1 )c(t 2 , t
2 ) . . . c(t n , t
n )E[Z
n
] c
(1.47)
Hence, if Z (t) is Gaussian, Y (t) is, because all cumulants of order n ≥ 3 vanish.
Then, the velocity V of the Brownian motion is a Gaussian process if the stochastic
term (t) is, because it results from an integral like (1.46). Considering that in
the t → ∞ limit, E[V ] c = 0, and σ
2
= E[V
2
] c = q/2γ, the asymptotic probability
density of V is given by:
f V (v) = ce
−
v 2
(2q/2γ) = ce
−
γv 2
q
=
m
2πk B T
e
−
mv 2
2k B T
= f M B (v)
(1.48)
if we accept the equipartition principle,
q
γ
=
2k B T
m
.
This is the celebrated Maxwell-Boltzmann distribution in 1 dimension. Because
its variance takes the form σ
2
= k B T /m, fixing the temprature of the fluid T , and
placing a large mass m in it, yields small uncertainty on the velocity: the velocity