Re: Hamiltonian vs. Energy

On Mar 18, 4:33 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:
Chris H. Fleming schrieb:

On Mar 16, 1:24 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:
Chris H. Fleming schrieb:

On Mar 11, 1:17 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:
The correct quantization of dissipative systems is in terms of
Lindblad master equations, or, in more mathematical terms, completely
positive semigroups.
The Lagrangian and Hamiltonian approach is most certainly still
applicable.
Only one must model the dissipative environment and then trace out
it's degrees of freedom from the master equation. When that is done,
Lindblad terms usually only arise in some kind of bandwidth limit
where the system energies are much less than the environment energies
(a high temperature reservoir for instance).
If you model the dissipaticve environment explicitly, you have a
bigger conservative system, not a dissipative system. Of course, the
conservative system is Hamiltonian, but it does not describe the
dissipative system alone.

Maybe you misunderstood what I meant by "dissipative environment".

Of course the entire system + environment is conservative. But that
doesn't mean there is no dissipation in the system. Open systems are
not necessarily conservative.

In the limit of a large environment, you get true dissipation in the
system.
The Poincare recurrence time becomes infinite and energy that is lost
to the environment never comes back. It is a reservoir.

By "dissipative environment" I merely mean an environment which causes
dissipation in the system.

And by a dissipative system, I mean a system in whose description only
the variables of the system, and not that of the environment appear.


So though you start with a conservative system + environment,
eventually you trace out the environment and do end up with a truly
dissipative system when the environment is a large reservoir.


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
memory terms.

That would be the HPZ master equation and it is equivalent to a
parametrically damped oscillator with diffusive noise. I wouldn't
consider it unwieldy considering that fairly general (finite
temperature, finite coupling) solutions have been found.

And I don't know what you mean by *unwanted* memory terms. I suppose
you mean unwanted in that they make the maths difficult. Physically
they are certainly wanted. They are absolutely necessary for low
temperature physics.

There are cases where one wants to model the memory. But then it is no
longer a damped harmonic oscillator. The latter is universally agreed
to be described by a 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.


I can't claim to completely understand what you are saying, but it
doesn't seem correct however I interpret it.

For a single oscillator coupled to a thermal reservoir of oscillators,
one gets most generally a nonlocally damped oscillator with colored
noise. It is also equivalent to parametric damped oscillator with
colored noise. (The Langevin equation will look nonlocal with noise
while the master equation will look parametric with diffusion. The
homogeneous (no noise/diffusion) solution matrices are related
algebraically)

There is a limit in which the damping is local (parametric goes to
simple) and there is a limit in which the noise is white. Both of
these things I would classify as "memory" and they are two different
limits. One can (perturbatively) have a simple damped oscillator, but
still with colored noise. Lindblad terms cannot describe this.

In either case, I don't think it is fair to say that it is unwieldy
given that it has been done. Certainly the calculation is challenging
to do non-perturbatively when one has both nonlocal dissipation and
colored noise. Such non-perturbative calculations are not yet in the
literature but will be soon... followed by more if there is any
demand.


Wen you contract it to the degrees of freedoms of the original system,
you get an integro-differential equation with memory, which is no longer
described by a Hamiltonian or Lagrangian framework.

If you then remove the memory by employing the Markov approximation
you get 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.

Correct, but it was *derived* from a Lagrangian. That is critical to
getting correct noise and dissipation. The subtlety is lost in the
Markov limit where you get local dissipation and white noise, which is
no different from the old classical models.
The correct quantization of dissipative systems starts with a
Lagrangian and ends with a master equation (or Langevin equation, or
Stochastic Schroedinger equation, ...).

Perhaps in theory, but not in practice. 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.

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...


It is completely dependent upon the regime of interest. For high
temperature environments, Lindblad terms are perfectly acceptable. And
quantum mechanically speaking high temperature isn't necessarily that
high in the conventional sense. But at low temperature such terms are
either completely unphysical or a mixed limit of sorts... and still
probably unphysical.

As far as modeling, one isn't "required" to invoke a Lagrangian to
insert Lindblad terms into the master equation because they are system
independent. You only have to make sure you are in the correct regime.
Then they can automatically be said to come from a high temperature
(or some other relatively short memory) reservoir which has an
associated Lagrangian. It would not be difficult to construct a model
environment to reproduce the master equation in your reference and
then link the phenomenological coefficients to microscopic causes.
That sort of thing is well known and there is no point in doing it.

.

Relevant Pages

  • Re: Hamiltonian vs. Energy
    ... The Lagrangian and Hamiltonian approach is most certainly still ... Only one must model the dissipative environment and then trace out ... In the limit of a large environment, you get true dissipation in the ...
    (sci.physics.research)
  • Re: Hamiltonian vs. Energy
    ... Lindblad master equations, or, in more mathematical terms, completely ... Only one must model the dissipative environment and then trace out ... it's degrees of freedom from the master equation. ... In the limit of a large environment, you get true dissipation in the ...
    (sci.physics.research)
  • Re: Non-lagrangian from lagrangian
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