### 5.4 Statistical Description – Boltzmann’s Equation

So far little heed was paid to the number of entities under consideration as the formalisms are abstract
enough to deal with a scaling of the degrees of freedom. When considering identical but
distinct entities in a three-dimensional setting, the associated phase space is simply a Cartesian
product (Deﬁnition 3) of the single phase spaces, thus forming an overall space of dimension .
While the formalism does not hinder the speciﬁcation of problems encompassing multiple entities, it
becomes increasingly diﬃcult to obtain solutions. Furthermore, when turning to systems composed of
particularly large numbers of entities, such as molecules or atoms in gases, it becomes quite unfeasible
to deal with them directly, as the sheer amount of boundary and/or initial conditions becomes
prohibitive.

Therefore, a description oﬀering a reduction of the overwhelming degrees of freedom is called for. This
can be demonstrated using a basic equation describing the evolution of particles using a density
depending on the phase space coordinates and time

and a
Hamiltonian , which depends on all of the particles.
By averaging by means of integration (Deﬁnition 94) an particle distribution function is
obtained

thus
constructing a projection (Deﬁnition 25)
such
that the higher-dimensional phase space appears as a ﬁber (Deﬁnition 40) over the reduced
phase space , since it is certain that
An
equation governing the evolution of this reduced density function can be obtained by integrating
Equation 5.33b, resulting in the expression
where
the left hand side containing the Hamiltonian depends on the considered particles, while
the right hand side describes the interactions with all the remaining particles by
coupling to the next higher distribution function. Therefore this equation is not closed, but
gives rise to a hierarchy of equations, the BBGKY, named after the individuals who have
independently derived this system, Bogoliubov [87], Born [88], Green [89], Kirkwood [90] and
Yvon [91 ].
Using this formulation, the restriction to just one particle yields an expression for a single particle
Hamiltonian, which results in phase space trajectories with deviations from these trajectories
attributable to collisions.

This also shows the price which has to be paid for the reduction of degrees of freedom. The
observed particle will no longer follow a simple curve through phase space and is disturbed due
to the scattering term which appears on the right hand side as is illustrated in Figure 5.3.

While the derivation naturally describes the collison term with other particles, it can also model the
interaction of the particle tracked by Boltzmann’s equation with the envrionment, thus allowing for an
interpretation as probability of a particle evolving to a given point.

While Boltzmann’s equation appears as a simpliﬁcation here, it is far from easy to obtain
solutions to this deceivingly simple equation. Therefore several diﬀerent techniques have
been developed to at least calculate estimates in diﬀerent contexts and with various levels of
accuracy.