A system is in ``equilibrium'' if its properties do not change
spontaneously over time. (In case there are external fields - such as
gravity - the material properties may vary in space, otherwise they will
be independent also of position: the system is not only in equilibrium but
In particular, in equilibrium the local energy density, i.e. the energy contained
in a volume element divided by that , as well as the
momentum density and the particle or mass density remain
constant. These densities, which refer to the three conserved quantities of
mechanics, play a special role in what follows.
Again considering the dilute gas, and assuming there are no external fields,
the densities are
By looking more closely we would see that these local densities are in
in fact not quite constant; rather, they will fluctuate about their mean.
In other words, there will be a spontaneous waxing and waning of
local gradients of those densities. Also, the experimentor
may intervene to create a gradient. For instance, one might induce
in a horizontal layer of a gas or liquid a certain -velocity
, and thereby a momentum . The difference between
the momenta in adjacent layers then defines a gradient of the momentum
density. If we left the system to itself this gradient would decrease and
eventually vanish. The property governing the speed of this decrease
is called viscosity.
If a physical quantity - such as mass, or momentum - is conserved,
any local change can only be achieved by a flow into or out of the
space region under consideration; we are then speaking of
transport processes whose speeds are governed by the
respective transport coefficients - such as the viscosity ,
the heat conductivity (for the energy transport) and the
diffusion constant (for mass transport).
In real experiments aimed at determining these transport coefficients
the respective gradient is artificially maintained. Thus there will be
a continuing flow of momentum, energy or matter in the (reverse) direction
of the gradient - which is obviously a non-equilibrium situation.
However, by a careful setup we can keep at least these flows and the
local densities and gradients constant in time. This is called a
stationary non-equilibrium situation.
DEFINITION OF TRANSPORT COEFFICIENTS
In order to define the transport coefficients , and
we consider the basic experimental setup.
Viscosity: To measure the coefficient of
momentum transport we generate a laminary flow by
placing a gas or fluid layer between two horizontal plates
and moving the upper plate with constant velocity to the
right. In this manner we superimpose a systematic -velocity
onto the random thermal motion of the molecules.
The magnitude of the thermal speed is of the order ; by
adding a ``shear velocity'' of some centimeters per second
the local equilibrium is not considerably disturbed. Thus we may assume
that we have still a Maxwell-Boltzmann distribution of velocities
at any point in the fluid, with the same value of
(bzw. ) everywhere.
Yet by imposing a velocity gradient we have slightly perturbed the
equilibrium; a certain amount of -momentum will flow against the
gradient - in our case downwards - so as to reestablish equilibrium.
The amount of momentum flowing down through a unit of area per unit
of time is called flow density . In a first (linear)
approximation this flow will be proportional to the imposed velocity
gradient, and the coefficient of proportionality is, by definition,
the transport coefficient :
This defining equation for is often called Newton's law
of viscous flow. The parameter is known as viscosity.
To generate a gradient of energy density we may place the substance between
two heat reservoirs with different temperatures. if we use the same simple
geometry as for the viscosity then the temperature (or energy) gradient
will have only the one non-vanishing component which accordingly
produces a flow of energy in the counter direction. The coefficient
of thermal conduction is defined by (Fourier's law),
Even in a homogeneous gas (or a fluid, or a solid, for that matter) the
individual molecules will alter their positions in the course of time.
This process is known as self diffusion. It is possible to
``tag'' some of the particles - for instance, by giving them a radioactive
nucleus. The density of the marked species - call it - may then once
more have a gradient that will be balanced by a particle flow
MEAN FREE PATH
We will presently attempt to calculate the above defined transport
coefficients from the microscopic dynamics of the molecules. In a
simple kinetic treatment the elementary process determining the various
flows is the free flight of a gas particle between two collisions with
other molecules. Thus the first step should be an estimation of the
average path length covered by a particle between two such collisions.
Let be the - as yet unknown - probability that a particle meets
another one before it has passed a distance . Then is
the probability for the molecule to fly a distance without undergoing
a collision. But the probability of an encounter within an infinitesimal
path element is given by
is the diameter of the particles. ( equals the fraction
of the ``target area'' covered by other particles.)
Thus the differential probability for a collision between and
which upon integration yields
The probability density
for a collision
within that interval is
The mean free path is then the first moment of this density:
The diameter of a molecule is
( Ångstrøm). According to Loschmidt (and the later but
more often quoted Avogadro) a volume of
contains, at standard conditions,
particles; the particle density is thus
, and the mean free path is
ESTIMATING THE TRANSPORT COEFFICIENTS
In the case of a dilute gas we can actually find expressions for the
transport coefficients. Assuming once more the basic geometric setting
used in most experiments, we suppose that the gradient of the respective
conserved quantity has only a component, leading to a flow that is
also directed along the axis.
In the course of the random (thermal) motion of the particles the
small systematic term is carried along. On the average,
an equal number of particles will cross the interface between two
horizontal layers from below and above. However, since the upper
layer has a larger (assuming , it will gradually
slow down, while the lower layer will take up speed.
The flow of momentum due to this mechanism can be estimated as follows:
The mean number of particles crossing the unit of area from below and from
above, respectively, is
. Each of the molecules
carries that systematic which pertains to the layer of its
previous collision, i.e.
. The momentum flow density is
Comparing this to the defining equation for the viscosity we find
It is a somewhat surprising consequence of this formula that, since
, the viscosity is apparently independent of
density! Maxwell, who was the first to derive this result, had to
convince himself of its validity by accurate experiments on dilute gases.
A similar consideration as in the case of momentum transport yields,
is the specific heat (.. molar
heat capacity; ..molar mass).
Again we assume a gradient of the relevant density - in this case,
the particle density of some species - along the direction.
and using the same reasoning as
before we find