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S1p. Dissipative dynamics and Lagrangians
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Any system of ordinary differential equations can be brought
into an artificial Lagrangian form, by first rewriting it in first
order form
F(q,q')=0
doubling the degrees of freedom by introducing conjugate variables p,
and then considering the Lagrangian
L(p,q)= p^T F(q,q').
In particular, this provides a Lagrangian formulation of dissipative
systems, such as the damped harmonic oscillator
m q'' + c q' + k q = 0 (m,c,k >0)
Unfortunately, the Hamiltonian in such a formulation has
nothing to do with the physical energy
E = (m q'^2 + k q^2)/2
The same holds for various other representations for the damped
harmonic oscillator found in the literature.
Lagrangians for the damped harmonic oscillator go back to
H. Bateman, Phys. Rev. 38, 815-819 (1931); the treatise
P.M. Morse and H. Feshbach,
Methods of Theoretical Physics
MacGraw-Hill, Boston 1953
discusses the procedure in Chapter 3 in terms of 'mirror images'
= additional dynamical variables needed to absorb the missing energy,
and remarks on p 313:
''The introduction of the mirror image ... is probably too artificial
a procedure to expect to obtain much of physical significance from
it.''
And indeed, the book doesn't make use of it anywhere.
Having a formal Lagrangian or Hamiltonian is no virtue in itself.
In particular, for a _quantum_ system, the Hamiltonian _must_ be the
energy. Playing around with alternative Lagrangians and Hamiltonians
may be amusing, but does not produce relevant physics.
Since dissipative equations (like the diffusion equation or the damped
harmonic oscillator) describe open systems (where energy is lost to an
unspecified environment), they cannot be described by a Schroedinger
equation.
Classically, dissipative systems are described by stochastic processes
describing dynamical semigroups. These can be classified into
stochastic differential equations (and their equivalent deterministic
Fokker-Planck equations), master equations, or a combination of both.
See the differential Chapman-Kolmogorov equation in Gardiner's Handbook
of stochastic methods - (3.4.22) in the second edition.
One of the simplest of such models - the diffusion equation - is the
particular case of a Fokker-Planck equation for Brownian motion.
Quantum mechanically, dissipative systems are described by stochastic
Schroedinger equations or, corresponding to the Fokker-Planck level,
by quantum dynamical semigroups, expressed using quantum Liouville
equations with Lindblad terms. This gives correct physics in a
dissipative environment. Many quantum optical systems
are directly modeled on the Lindblad level, where the terms have an
understandable and experimentally verifiable meaning independent of
any underlying more microscopic model.
An important recent example is that of photons on demand,
M. Keller, B Lange, K Hayasaka, W Lange and H Walther,
A calcium ion in a cavity as a controlled single-photon source,
New Journal of Physics 6 (2004), 95.
There is no trace of a Lagrangian in the modeling, and indeed, a
useful Lagrangian formulation does not exist - unless one extends the
dynamics and explicitly includes the environment.
Of course, in theory, a dissipative system is thought to be a
contracted version of a bigger conservative system which includes
the envoironment, and in simple situations, this theoretical view can
indeed be substantiated.
If one models the dissipative environment explicitly, on gets a
bigger conservative system, not a dissipative system. Of course,
this conservative system has a Hamiltonian or Lagrangian description,
but it does not describe the dissipative system alone. When one
contracts it to the degrees of freedoms of the original system,
one gets an integro-differential equation with memory, which is no
longer described by a physically meaningful Hamiltonian or Lagrangian
framework.
The reduced dynamics takes the exact form
m x''(t) + k x(t) = int_0^t G(s) x(t-s) ds + F(t).
with functions F(t) (the noise caused by the environment) and G(s)
(the memory kernel) that depend on the state of the environment.
If the interaction is of the usual, dissipative nature then both F(t)
and G(s) are extremely oscillating, even for intervals short compared
to the inverse frequency T of the oscillator. But the short time
averages of the memory Kernel have an exponentially decaying bound on
their size and become negligible after some relaxation time tau << T.
Thus it suffices in a good approximation to take the integral
from s=0 to s=tau only. This allows us to expand x(t-s) in a second
order Taylor expansion (valid since s<=tau<0, recovering the
traditional equation for the damped harmonic oscillator, including a
stochastic force term. (Its size can be related to the damping
coefficient and the temperature of the environment, a relation known
as the fluctuation-dissipation theorem.)
A thorough discussion of the reduction of microscopic conservative
large systems to dissipative subsystems of interest is given in
H Grabert,
Projection Operator Techniques in Nonequilibrium
Statistical Mechanics,
Springer Tracts in Modern Physics, 1982
at a much more general level that also applies for
many other dissipative systems.
There are cases where one needs to model the memory to capture the
essence of the reduced dynamics. But in many cases, a simpler,
memory-free description is possible and adequate. One can remove the
memory by employing a Markov approximation, and gets again a
differential equation, which defines the Lindblad (or, classicallally,
the Focker-Planck) dynamics. Again, this is no longer described by a
Hamiltonian or Lagrangian framework.
In the extended formulation with explicit environment or with memory,
already a simple damped harmonic oscillator becomes a huge and
unwieldy dynamical system which is no longer equivalent to the damped
harmonic oscillator, but includes unwanted environment terms or memory
terms. In cases where one really needs to model the memory, the system
therefore is no longer a damped harmonic oscillator. The latter is
described by a simple linear constant coefficient second order
differential equation for a single function, and has no memory.
Its analysis is very simple, and compared to that any more detailed
description is unwieldy.
In practice, the dissipative formulation therefore stands by itself
(apart from lip service paid to a hypothesized more fundamental
conservative description).
The situation is similar to that in fluid dynamics. In theory, the
Navier-Stokes equations (which are dissipative) should be derivable from
a Lagrangian. Indeed, such derivations have been given, but only for
very simple model problems such as an ideal gas. However, there is no
microscopic derivation of the Navier-Stokes equations in the practically
interesting case of water at room temperature...