Liouville equation

The Liouville equation , after Joseph Liouville , is a differential equation for the temporal development of ensembles of physical systems . The equation belongs to the field of classical statistical mechanics , but there is also an analogue in quantum mechanics . The equation of quantum mechanics is the Von Neumann equation .

The Liouville equation of classical statistical mechanics is closely related to the Liouville theorem and can be derived from it.

Liouville equation of classical statistical mechanics

In statistical physics, an ensemble of instances of a physical system is described by the probability density of the system points of the system instances in phase space ( phase space density ). Here stands for time, and and are the canonical coordinates and impulses of the system. The Liouville equation ${\displaystyle \rho \left(q_{i},p_{i},t\right)}$ ${\displaystyle t}$ ${\displaystyle q_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle N}$

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\sum _{i=1}^{N}\left({\frac {\partial \rho }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)}$

provides the change in the probability density at a given point in phase space as a function of time. Since the flow of the system points in phase space is given by the vector field according to Hamilton's equations of motion , the Liouville equation is usually written in the form ${\displaystyle \left({\dot {q}}_{i},{\dot {p}}_{i}\right)=\left(\partial H/\partial p_{i},-\partial H/\partial q_{i}\right)}$

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\sum _{i=1}^{N}\left({\frac {\partial \rho }{\partial q_{i}}}{\frac {\partial H}{\partial p_{i}}}-{\frac {\partial \rho }{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}\right)=\left\{H,\rho \right\}.}$

The curly bracket is a Poisson bracket , is the Hamilton function of the system. ${\displaystyle \{.,.\}}$ ${\displaystyle H}$

With the introduction of the Liouville operator

${\displaystyle L=\sum _{i=1}^{n}\left[{\frac {\partial H}{\partial p_{i}}}{\frac {\partial }{\partial q_{i}}}-{\frac {\partial H}{\partial q_{i}}}{\frac {\partial }{\partial p_{i}}}\right]=\{\cdot ,H\}}$

there is a third notation

${\displaystyle {\frac {\partial \rho }{\partial t}}=-{L}\rho }$.

Derived from Liouville's theorem

The set of Liouville states that the volume of an arbitrary phase space cell in the course of time is constant, d. H. the flow through the phase space is volume and even orientation preserving. The Liouville equation holds if and only if the total derivative of the probability density vanishes with respect to time,

${\displaystyle {\frac {d\rho }{dt}}={\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{N}\left({\frac {\partial \rho }{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)=0,}$

d. H. if the probability density is constant along a phase space trajectory. But the number of system points which define a moving cell in phase space is constant. The total derivative therefore disappears precisely when the volume of the cell is also constant. ${\displaystyle \rho }$ ${\displaystyle d\rho /dt}$

Quantum mechanical equation

The quantum mechanical form of the Liouville equation is also called the Von Neumann equation :

${\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {i}{\hbar }}[\rho ,H]}$.

Marked here

• ${\displaystyle i}$the imaginary unit
• ${\displaystyle \hbar }$the reduced Planck quantum of action
• ${\displaystyle \rho }$the density matrix
• ${\displaystyle H}$the Hamilton operator
• the square brackets denote the commutator .

As in the case of classical mechanics, one can formally introduce a Liouville operator , defined by its effect on an operator : ${\displaystyle L}$ ${\displaystyle A}$

${\displaystyle LA={\frac {i}{\hbar }}[A,H]}$

The Von Neumann equation is thus written:

${\displaystyle {\frac {\partial \rho }{\partial t}}=L\rho }$

With the help of the Wigner picture , a direct relationship between the Hamilton operator and the classic Poisson bracket can be derived in the semiclassical borderline case :

${\displaystyle \lim \limits _{\hbar \rightarrow 0}~{\frac {i}{\hbar }}[{\hat {A}},{\hat {B}}]=\{{A}_{w},{B}_{w}\}}$

literature

• Franz Schwabl : Statistical Mechanics . Springer, Berlin a. a. 2004, ISBN 3-540-20360-5 .
• Harald JW Müller-Kirsten : Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral . 2nd Edition. World Scientific, Singapore 2012, ISBN 978-981-4397-73-5 , pp. 29-40.
• Harald JW Müller-Kirsten : Basics of Statistical Physics . 2nd Edition. World Scientific, Singapore 2013, ISBN 978-981-4449-53-3 , Chapter 3.