Consider the motion of a particle through a medium after being initially
prepared in a certain state of motion (e.g., after being pushed by an external
force). As a result of friction with the medium, the particle will be slowed
down (i.e., its initial kinetic energy will be dissipated by heating up
the medium). The motion of such particle is described by the *generalized
Langevin equation*, which is derived in this section as follows.

Consider the Hamiltonian that describes a particle with coordinates , linearly coupled to the bath coordinates ,

(468) |

where
are the coupling constants that linearly couple
and .
The terms
and ,
introduced by Eq. (468), describe the interactions among system coordinates
and among bath coordinates, respectively.

The total force acting on the particle is

(469) |

where the fluctuating force f(t) can be readily identified from
Eq. (468),

(470) |

Note that the motion of
depends on
since, according to Eq. (468), the force acting on
is .
Therefore, f(t) is also a function of x(t). Assuming that
is a linear in x(t),

(471) |

where, according to Eq. (449),

(472) |

with

(473) |

Therefore, the equation of motion for the particle is

(474) |

Changing the integration variable
to ,
in Eq. (474), we obtain

(475) |

where the lower integration limit includes only values of x(t-t'')
with .
Integrating Eq. (475) by parts, we obtain

(476) |

Changing the integration variable
to ,
in Eq. (476), we obtain

(477) |

Eq. (477) is the *Generalized Langevin Equation*, which can
be written in terms of the *potential of mean force*

(478) |

and the fluctuating force

(479) |

as follows,

(480) |

where the third term on the r.h.s. of Eq. (480) is the generalized
frictional force, a force that is *linear* in the velocity. The connection
between the frictional force and the regression of thermal fluctuations
of ,
introduced by Eq. (480), is known as the *second fluctuation-dissipation
theorem*.

**Markovian Approximation**

Changing the interation variable ,
in Eq. (480), to
and *considering a time
much larger than the relaxation time scale for the correlation function *
(so that
and ),
we obtain

(481) |

Note that Eq. (481) becomes the traditional *Langevin Equation*,

(482) |

when .
The friction coefficient
is, therefore, determined by the regression of spontaneous thermal fluctuations
as follows

(483) |

The approximation implemented to obtain Eq. (481) involves considering
that the relaxation time for fluctuating forces in the bath is much shorter
than the time over which one observes the particle. Such approximation
removes the ``memory effects'' from the equation of motion (note that Eq.
(481) does not involve the nonlocality in time introduced by the time integral
in Eq. (480)). This approximation is thus called *Markovian approximation*
since it makes the instantaneous force independent of the state of the
particle at any previous time. Note that, according to Eq. (482),

(484) |

where ,
since .
The solution to Eq. (484) is,

exp | (485) |

Eq. (485) indicates that the average initial momentum of the particle
is dissipated into the bath at an exponential rate (i.e., the average velocity
vanishes at an exponential rate). However, it is important to note that
the condition
at
(e.g., at equilibrium) does not imply that the particle is at rest! At
equilibrium, the fluctuating force
keeps buffeting the particle and the distribution of velocities is given
by the Boltzmann distribution (see Eq. (383)). The average squared velocity
for the particle is

(486) |

and the velocity autocorrelation function is

exp exp | (487) |

since Eq. (484) is valid not only for the dynamical variable
but also for .

The motion of the particle is called *Brownian Motion*,
in honor to the botanist Robert Brown who *observed* it for the first
time in his studies of pollen. In 1828 he wrote ``the pollen become dispersed
in water in a great number of small particles which were perceived to have
an irregular swarming motion''. The theory of such motion, however, was
derived by A. Einstein in 1905 when he wrote: ``In this paper it will be
shown that ... bodies of microscopically visible size suspended in a liquid
perform movements of such magnitude that they can be easily observed in
a microscope on account of the molecular motions of heat ...''

In order to compute the average mean squared displacement of the particle we substitute the variable x(t) in Eq. (482) by , we multiply both sides of Eq.(482) by such variable and we average over the ensemble distribution as follows,

(488) |

since .
Eq. (488) is equivalent to

(489) |

which, according to Eq. (486), gives

(490) |

The solution to Eq. (490) is

exp | (491) |

Therefore, the mean squared displacement is

(492) |

At *short times* (i.e., when exp),

(493) |

i.e., *the mean squared displacement at short times is quadratic
in time*. This is the so-called *ballistic regime*, since it corresponds
to ballistic motion (motion without collisions) for a particle with velocity
equal to .

At *long times* (i.e., when exp),

(494) |

where the constant

(495) |

is the *diffusion coefficient*. Therefore, *at long times
the mean squared displacement is linear in time*. This long time limit
is the so-called *diffusional regime*. The remaining of this section
shows that the diffusion coefficient can be computed in terms of the velocity
autocorrelation function
as follows:

(496) |

Note that Eq. (495) can be readily obtained by substituting Eq.
(487) into Eq. (496).

In order to prove Eq. (496), consider the particle displacement at time t,

(497) |

and compute the time derivative of the squared displacement as follows

(498) |

which according to Eq. (448) gives,

(499) |

Changing integration variables from
to
we obtain

(500) |

since
is equal to .

Finally, since we obtain

(501) |

Eq. (496) is obtained by substituting Eq. (494) into the l.h.s.
of Eq. (501).