Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

The macroscopic properties can be evaluated as averages of the microscopic

; t � �:

; t � � (94)

¼ 0: (95)

¼ 0 (97)

<sup>∂</sup>fsþ<sup>1</sup> <sup>r</sup>1…p<sup>s</sup>þ<sup>1</sup>; <sup>t</sup> � � ∂pi

:

(98)

; t � �, the

; t � �: (93)

� � with respect to the distribution function <sup>f</sup> <sup>N</sup> <sup>r</sup>i; <sup>p</sup><sup>i</sup>

dΓa ri; p<sup>i</sup>

In addition to these so-called mechanical properties there exist also thermal properties, such as entropy, temperature, and chemical potential. Instead of a dynamical variable, they are related to the distribution function. For example, the

dΓf <sup>N</sup> ri; p<sup>i</sup>

We derive an equation of motion for the distribution function f <sup>N</sup> ri; p<sup>i</sup>

∂f N ∂ri r\_<sup>i</sup> þ

We shortly remember the quantum case. Instead of the N-particle distribution function f <sup>N</sup>ð Þt , the statistical operator ρð Þt is used to indicate the probability of a microstate in a given macrostate. The equation of motion is the von Neumann equation (8). Both equations are closely related and denoted as Liouville-von Neumann equation.

þ ∑ N i¼1

To evaluate averages, instead of the N-particle distribution function <sup>f</sup> <sup>N</sup> <sup>r</sup>1; …; <sup>r</sup>N; <sup>p</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � often reduced <sup>s</sup>-particle distribution functions

ð d<sup>3</sup>

<sup>r</sup><sup>s</sup>þ<sup>1</sup>…d<sup>3</sup>

are sufficient. Examples are the particle density, the Maxwell distribution of the

We are interested in the equations of motion for the reduced distribution functions. For classical systems, one finds a hierarchy of equations. From the Liouville

we obtain the equation of motion for the reduced distribution function fs

¼ ∑ s i¼1 ð d<sup>3</sup>

This hierarchy of equations is called BBGKY hierarchy, standing for Bogoliubov,

<sup>r</sup><sup>s</sup>þ1d<sup>3</sup>

h3

<sup>p</sup><sup>s</sup>þ<sup>1</sup>

<sup>∂</sup>Vi,sþ<sup>1</sup> ∂ri

� ∑ N i6¼j ∂Vij ∂ri

∂f N ∂pi

pN

ð Þ <sup>N</sup> � <sup>s</sup> !h<sup>3</sup>ð Þ <sup>N</sup>�<sup>s</sup> <sup>f</sup> <sup>N</sup> <sup>r</sup>1; …; <sup>p</sup>N; <sup>t</sup> � � (96)

� �<sup>f</sup> <sup>N</sup> <sup>r</sup>i; <sup>p</sup><sup>i</sup>

; <sup>t</sup> � � ln <sup>f</sup> <sup>N</sup> <sup>r</sup>i; <sup>p</sup><sup>i</sup>

∂f N ∂pi p\_ i

� �

h i <sup>A</sup> <sup>t</sup> <sup>¼</sup>

Seq ¼ �k<sup>B</sup>

df <sup>N</sup> <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> <sup>N</sup> ∂t

3.2.2 Classical reduced distribution functions

fs r1; …; p<sup>s</sup> ; <sup>t</sup> � � <sup>¼</sup>

equation, Eq. (95) without external potential,

Born, Green, Kirkwood, and Young.

dfs <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup>fs ∂t þ ∑ s i¼1 vi ∂fs ∂ri � ∑ s i6¼j

24

df <sup>N</sup> <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup> <sup>N</sup> ∂t þ ∑ N i vi ∂f N ∂ri

through integration over the 3ð Þ N � s other variables:

∂Vij ∂ri

∂fs ∂pi

particle velocities, and the pair correlation function.

ð

ð

quantities a ri; p<sup>i</sup>

equilibrium entropy is given by

Non-Equilibrium Particle Dynamics

Liouville equation, see [5]:

The equation of motion (98) for the reduced distribution function fs is not closed because on the right-hand side the higher order distribution function fsþ<sup>1</sup> appears. In its turn, fsþ<sup>1</sup> obeys a similar equation that contains fsþ2, etc. This structure of a system of equations is denoted as hierarchy. To obtain a kinetic equation that is a closed equation for the reduced distribution function, one has to truncate the BBGKY hierarchy, expressing the higher order distribution function fsþ<sup>1</sup> by the lower order distribution functions <sup>f</sup> <sup>1</sup>; …; fs .
