In classic dynamics and chaos theory we use a concept of phase space. Time evolution of an individual system corresponds to a motion of the representative point along the phase trajectory (orbit). Dynamics of an ensemble (a collective of independent identical systems distinct only with the initial conditions) may be thought as evolution of a cloud of representative points in the phase space and the density function is the distribution function governed by the Liouville equation.

Due to the uncertainty principle,
in the quantum theory the ordinary distribution function cannot be defined
*f*(*p*,*x*),
because we cannot construct the ensemble of systems possessing
simultaneously by a certain momentum *p* and a position *x*.

One possibility to introduce a quantum analog of the distribution function is to use the so-called Husimi distribution, and another is based on the Wigner function.

Husimi distribution

For a quantum system the phase plane
may be thought as a collection of cells of size
D*p*D*x*=*h*/2p,
and we speak on a probability of presence at each cell.
To define it one can estimate an overlap integral of the wave function
|y> with a coherent state
|y* _{px}*>, centered at the given point
(

**Remark: What is the coherent state?**
Following Glauber, the coherent state |y_{0}>
centered at the origin is defined
as an eigenvector of the operator
**a**=(**x**+*i***p**)/2^{1/2}
with zero eigenvalue. The equation in position representation
has a form
**x**y+(*h*/2p)*d*y/*dx*=0,
and the solution is yµexp(-*x*^{2}/2p*h*).
Then, we construct states shifted in momentum and coordinate
by arbitrary values *p* and *q*:
|y* _{pq}*=exp((

The figure below shows evolution of the
Husimi function in the quantum Arnold cat map at
*N*=131. The initial state is the Gaussian packet of
maximal localization. Gray tones code levels of the function
*f _{H}*(

Wigner function

For one-dimensional motion of the quantum particle
the Wigner function is expressed via the wave function as
y(*x*):

The value *W*(*p*,*x*) is always real.

Wigner function is used widely in statistical physics in the quantum consideration instead of the ordinary distribution function in classic statistics. The reason is that integration of the Wigner function over the firs argument yields the probability distribution for coordinate, and integration over the second argument to the distribution for momentum:

Under the periodicity conditions on the torus
the Wigner function *W* looks like a set
of delta-peaks at nodes of square lattice with step 1/2*N*
in the unit square. It is defined by a table of 2*N*x2*N*
real numbers associated with the amplitudes of the delta-peaks:

where *k* and *m* accept integer and semi-integer values
(0, 1/2, 1, 3/2,...*N*-1/2).

Summation of *w*(*k*,*m*) over *k* yields the
probability distribution for coordinate, and
summation over *m*
- the probability distribution for momentum:

It may be shown that evolution of the Wigner function for the Arnold cat map is governed by a simple expression

In other words, the values of the Wigner function associated with the nodes of
the lattice
2*N*x2*N* simply transfer along the classic orbits
on the lattice:

**Remark.** The additional term 1/2 in the equation for *k'*,
is in a sense irrelevant, it can be excluded by a shift of origin
*m*->*m*-1/2, *k*->*k*+1/2.
This redefinition is not so convenient because forces us to deal with the semi-integer
indices, although recover exact correspondence with the classic map.

In the figure below we show evolution of the Wigner function in the
Arnold cat map at *N*=13.
The initial state is the Gaussian packet.
The evolution is periodic: return to the original state occurs
after 14 steps.
A peculiar feature of a hyperbolic map is that the
essentially non-zero values of the function at intermediate steps are
dispersed more or less uniformly over the unit square.

of theoretical nonlinear

dynamics

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