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time
τ
c
. Using this slowness property we can approximate the web variable under the
integral in (
3.42
) as being dependent solely on the present time and extend the upper
limit on the integral to infinity without significant change to the integrated value of the
response function,
2
∞
0
d
w(
t
)
dt
K
t
)w(
=
ξ
0
(
t
)
−
(
t
).
(3.46)
dt
The integral can therefore be approximated as the constant
2
∞
0
dt
K
t
)
≡
2
γ
=
(
χ,
(3.47)
resulting in the linear Langevin equation
d
w(
t
)
=−
γw(
t
)
+
ξ
0
(
t
),
(3.48)
dt
where
γ
is the dissipation parameter and
χ
is the stationary susceptibility of the door-
way bath variable
The susceptibility in (
3.47
) is the same as that obtained for a
constant perturbation applied suddenly at time
t
ξ.
0.
In this linear model irreversibility is a consequence of the infinite number of degrees
of freedom in the bath. However, the statistical fluctuations are explicitly introduced
into the solution by the assumption that the initial state of the bath (environment) is not
known and consequently the initial state of the bath is specified by a probability density.
The Langevin equation (
3.48
) is valid only if we assume that the initial conditions of an
ensemble of realizations of the bath are normally distributed. The width of this Gauss
distribution is imposed by our technical judgement and determines the temperature of
the bath. It is clear that in this approach thermodynamics is introduced by fiat through
the initial conditions, and that the infinite number of variables of the (linear) bath simply
preserves microscopic thermodynamics up to finite times. Note that the linearity and
the initial normality yield Gauss statistics for the variable
=
at all times: this in turn
implies that the stochastic force acting on the web variable is Gaussian as well. The
fact of the matter is that most physical phenomena modeled by (
3.48
) do not satisfy
the conditions under which the strictly linear model was constructed and yet the linear
Langevin equation often does quite well in describing their thermodynamic behavior.
Unfortunately there does not exist a systematic procedure to include all the ways the
environment can interact with an arbitrary dynamical web and so the desirable situation
outlined above has not as yet been attained. One way this program has been pursued
has been to make the evolution of the web uncertain by inserting a random force into
the equations of motion. This procedure saves the applicability of physical theories,
but it also introduces the seed of subjectivity into the evolution of the web. It requires
that we develop mathematical tricks to treat webs with an infinite number of degrees of
freedom like the environment, where the global properties of the overwhelming majority
of degrees of freedom are chosen in a subjective manner, mostly for computational
convenience. One such trick is to assume that the environment is already in the state
of thermodynamic equilibrium before it interacts with the web, and that it remains in
equilibrium throughout the interaction because of its enormous size. Therefore it is
ξ
0
(
t
)
