Biology Reference
In-Depth Information
extensive quantities come in pairs: concentration-mass, temperature-
heat, charge density-charge, etc. Field quantities as considered in con-
tinuous models are always intensive, and quantities in discrete models are
usually extensive. Corresponding extensive and intensive quantities are
interrelated through an averaging operation. The concentration of mol-
ecules can e.g. be determined by measuring the total mass of all mole-
cules within a given volume and dividing this mass by the volume. We
imagine such an averaging volume around each point in space in order to
recover a spatially resolved concentration field. If the averaging volume
chosen is too small, entry and exit of individual molecules will lead to sig-
nificant jumps in the average. With a growing averaging volume, the con-
centration may converge to a stable value. If the volume is further
enlarged, the concentration may again start to vary due to macroscopic
spatial gradients. This behavior is illustrated in Fig. 3. Above the contin-
uum limit
, the average is converged and microscopic single-particle
effects are no longer significant. The value of the continuum limit is gov-
erned by the abundance of particles compared to the size of the averag-
ing volume. If the microscopic particles are molecules such as proteins,
λ
λ
is related to their mean free path. On length scales larger than the scale
of field variations
L
, macroscopic gradients of the averaged field become
apparent if the field is not homogeneous, i.e. if its value varies in space.
The dimensionless ratio Kn
=λ
/
L
is called the Knudsen number.
Fig. 3.
The value
u
of a volume-averaged intensive field quantity depends on the
size of the averaging volume
V
. For volumes smaller than the continuum limit
λ
,
individual particles cause the average to fluctuate. In the continuum region above
,
the volume average can be stationary or vary smoothly due to macroscopic field
gradients above the length scale of field variations
L
.
λ
