16
L. Rondoni
E[(x(t) − x 0 )
2
] =
v
2
0 −
q
2γ
(1 − e
−γt
)
2
γ 2
+
q
2γ
2
γ
t −
2(1 − e
−γt
)
γ 2
(1.36)
which explodes linearly in time. One may then introduce the diffusion coefficient as:
D := lim
t→∞
E[(x(t) − x 0 )
2
]
2t
=
1
2
lim
t→∞
q
γ 2 +
v
2
0 −
q
2γ
γ 2 t
−
q
γ 3 t
=
q
2γ 2 =
k B T
mγ
(1.37)
where the last equality holds if equipartition does. Equation (1.37) yields the celebrated Einstein relation:
D = μk B T
(1.38)
which links the diffusion coefficient to the temperature via the mobility.
The presence of fluctuations, i.e. the mean square displacement E[(x(t) − x 0 )
2
],
and of dissipation, μ, makes Eq. (1.37) the first Fluctuation-Dissipation Relation.
It is an amazing result experimentally validated first by Perrin [9], that allows the
calculation of Avogadro’s number N A . Indeed, it is relatively simple to measure the
variance of positions, which yields D; the temprature T , the friction coefficient γ
and the mass m are known, and the Boltzmann constant can be expressed as the
ratio k B = R/N A , where R, the universal constant of gases, is also known. Then one
obtains:
D =
k B T
mγ
=
RT
6πηa
1
N A
(1.39)
where the only unknown is N A , and can thus be estimated to be N A = 6.02210
23
mol
−1 . If atoms can be counted, they definitely exist!
The above is readily generlazied to a system in a d-dimensional space, for which
one has:
˙
v i = −γv i + i (t), i = 1, . . . , d
(1.40)
In case there are no correlations among the different coordinates, one may assume:
E[ i (t)] = 0; E[ i (t)) j (t
)] = qδ i j δ(t − t
)
(1.41)
so that each variable can be treated separately. Moreover, when equipartition applies,
we can write:
E[E] =
d
i=1
1
2
mE[v
2
i ] =
d
2
k B T, and
E[
− →
x(t)
2
]
t
→
2dk B T
mγ
, for t → ∞
(1.42)
where
− →
x(t) is (x(t) − x(0)) and has dimension d.
Relation (1.39) contains in itself an incredible amount of information. In the
first place, it links the microscopic realm with the macroscopic one through k B =
L. Rondoni
E[(x(t) − x 0 )
2
] =
v
2
0 −
q
2γ
(1 − e
−γt
)
2
γ 2
+
q
2γ
2
γ
t −
2(1 − e
−γt
)
γ 2
(1.36)
which explodes linearly in time. One may then introduce the diffusion coefficient as:
D := lim
t→∞
E[(x(t) − x 0 )
2
]
2t
=
1
2
lim
t→∞
q
γ 2 +
v
2
0 −
q
2γ
γ 2 t
−
q
γ 3 t
=
q
2γ 2 =
k B T
mγ
(1.37)
where the last equality holds if equipartition does. Equation (1.37) yields the celebrated Einstein relation:
D = μk B T
(1.38)
which links the diffusion coefficient to the temperature via the mobility.
The presence of fluctuations, i.e. the mean square displacement E[(x(t) − x 0 )
2
],
and of dissipation, μ, makes Eq. (1.37) the first Fluctuation-Dissipation Relation.
It is an amazing result experimentally validated first by Perrin [9], that allows the
calculation of Avogadro’s number N A . Indeed, it is relatively simple to measure the
variance of positions, which yields D; the temprature T , the friction coefficient γ
and the mass m are known, and the Boltzmann constant can be expressed as the
ratio k B = R/N A , where R, the universal constant of gases, is also known. Then one
obtains:
D =
k B T
mγ
=
RT
6πηa
1
N A
(1.39)
where the only unknown is N A , and can thus be estimated to be N A = 6.02210
23
mol
−1 . If atoms can be counted, they definitely exist!
The above is readily generlazied to a system in a d-dimensional space, for which
one has:
˙
v i = −γv i + i (t), i = 1, . . . , d
(1.40)
In case there are no correlations among the different coordinates, one may assume:
E[ i (t)] = 0; E[ i (t)) j (t
)] = qδ i j δ(t − t
)
(1.41)
so that each variable can be treated separately. Moreover, when equipartition applies,
we can write:
E[E] =
d
i=1
1
2
mE[v
2
i ] =
d
2
k B T, and
E[
− →
x(t)
2
]
t
→
2dk B T
mγ
, for t → ∞
(1.42)
where
− →
x(t) is (x(t) − x(0)) and has dimension d.
Relation (1.39) contains in itself an incredible amount of information. In the
first place, it links the microscopic realm with the macroscopic one through k B =