4.3 Diffusion Processes and Green’s Functions
35
proportional to G(x − x 1 , t) + G(x − x 2 , t), because the diffusion equation is linear.
If we had a smooth distribution ρ(x
) of heat injected into the system at t = 0 we
could describe the solution H (x, t) as the sum over all individual heat sources,
weighted by ρ(x
). But this is the convolution of the initial heat distribution with the
spreading function G(x, t)
H (x, t) =
∞
−∞
G(x − x
)ρ(x
)dx
,
(4.19)
where G(x, t) is given by (4.18) and is called a Green’s function for the diffusion
equation.
Let us briefly digress on the concept of Green’s functions by mentioning two
examples, one from physics and one from engineering. In general, Green’s functions are “helper functions”—integral kernels—that appear when constructing the
inverses of differential operators. One prominent example is the Green’s function of
the Poisson equation for the electro-static potential U , given by
U (x) =
1
ε 0
ρ(x)
(4.20)
with the charge density ρ(x) and the dielectric constant ε 0 . Here, the Green’s function
is the potential of a point charge, located at x 0 . Its charge distribution is ρ(x) =
eδ(x − x 0 ) with Dirac’s delta function δ(x) and the elementary charge e. It is wellknown from elementary electro-dynamics lectures that the potential G(x) of a point
charge is given by
G(x − x 0 ) =
e
4πε 0 |x − x 0 |
.
(4.21)
For a point-like charge the symmetry of the system is spherical and we can introduce
spherical coordinates. Then, to verify this statement, we set x 0 to the origin, such
that |x| = r and express the Laplace operator and the solution G(r ) = 1/4πε 0 r
in spherical coordinates. As an exercise, calculate G(r ) and verify that it is zero
everywhere, except at the origin, where it diverges.
The Laplace operator is linear in the charge density ρ and we find the potential from
two point charges at x 1 and x 2 as the sum of the potentials G 1 =
e
4πε 0 |x−x 1 |
and G 2 =
e
4πε 0 |x−x 2 |
from the individual charges. As a generalization, we find the potential of a
continuous charge distribution ρ as the sum over the, properly weighted, potentials
of point charges. But this is the just convolution of the charge distribution ρ(y) and
the point-charge potential—the Green’s function G(x − y). The can therefore write
U (x) =
V
G(x − y)ρ(y)d
3 y =
V
ρ(y)d
3 y
4πε 0 |x − y|
,
(4.22)
where the integration extends over a suitable volume V that covers all charges in ρ(y).
35
proportional to G(x − x 1 , t) + G(x − x 2 , t), because the diffusion equation is linear.
If we had a smooth distribution ρ(x
) of heat injected into the system at t = 0 we
could describe the solution H (x, t) as the sum over all individual heat sources,
weighted by ρ(x
). But this is the convolution of the initial heat distribution with the
spreading function G(x, t)
H (x, t) =
∞
−∞
G(x − x
)ρ(x
)dx
,
(4.19)
where G(x, t) is given by (4.18) and is called a Green’s function for the diffusion
equation.
Let us briefly digress on the concept of Green’s functions by mentioning two
examples, one from physics and one from engineering. In general, Green’s functions are “helper functions”—integral kernels—that appear when constructing the
inverses of differential operators. One prominent example is the Green’s function of
the Poisson equation for the electro-static potential U , given by
U (x) =
1
ε 0
ρ(x)
(4.20)
with the charge density ρ(x) and the dielectric constant ε 0 . Here, the Green’s function
is the potential of a point charge, located at x 0 . Its charge distribution is ρ(x) =
eδ(x − x 0 ) with Dirac’s delta function δ(x) and the elementary charge e. It is wellknown from elementary electro-dynamics lectures that the potential G(x) of a point
charge is given by
G(x − x 0 ) =
e
4πε 0 |x − x 0 |
.
(4.21)
For a point-like charge the symmetry of the system is spherical and we can introduce
spherical coordinates. Then, to verify this statement, we set x 0 to the origin, such
that |x| = r and express the Laplace operator and the solution G(r ) = 1/4πε 0 r
in spherical coordinates. As an exercise, calculate G(r ) and verify that it is zero
everywhere, except at the origin, where it diverges.
The Laplace operator is linear in the charge density ρ and we find the potential from
two point charges at x 1 and x 2 as the sum of the potentials G 1 =
e
4πε 0 |x−x 1 |
and G 2 =
e
4πε 0 |x−x 2 |
from the individual charges. As a generalization, we find the potential of a
continuous charge distribution ρ as the sum over the, properly weighted, potentials
of point charges. But this is the just convolution of the charge distribution ρ(y) and
the point-charge potential—the Green’s function G(x − y). The can therefore write
U (x) =
V
G(x − y)ρ(y)d
3 y =
V
ρ(y)d
3 y
4πε 0 |x − y|
,
(4.22)
where the integration extends over a suitable volume V that covers all charges in ρ(y).