You ask a potentially interesting question, I tried to formulate an answer but need more focus. Could you tell us more about the system you have in mind? And why do you think temperature is bath noise variance?

To set the tone: in statistical mechanics T is the derivative of energy with respect to entropy (from dU = T dS + stuff related to work). So: calculate the entropy of your system (basically the logarithm of the number of microstates corresponding to a given energy), take the derivative with respect to that energy, and you have the temperature. Then show that it is related to variance in some way. See, however, below.

If you do that for the classical ideal mono atomic gas you would find that T is proportional to the variance in velocity: T = (m/3kB), but why take that as the fundamental variable? More natural would be to take the energy fluctuations, and T is not proportional to the variance in that. The variance in energy is (in general!) proportional to kBT2CV. So my first question is: how do you want to define the temperature exactly?

Leonard Susskind, in The Black Hole War, states (p 167 ff) that he only understood what temperature was after he found the following definition: "Temperature is the increase in energy of a system when you add one bit of entropy" (apart from Boltzmann's constant that is). Having taught courses in thermodynamics where I tried this on students I have to disagree vehemently (and had to stick with "temperature is what you measure on a thermometer", or a temperature scale derived from the Carnot cycle), but this is similar to the definition I gave in the previous paragraph, and it could be helpful in the system you have in mind. I still don't understand how you would relate it to bath noise, but again, maybe if you explain a bit more about the why and how of your system, we can make some progress.

Perhaps is just a modelling problem. For example, in this paper [http://arxiv.org/pdf/1407.6191v1.pdf] the author calculate the work done by a noisy correlated external force. My assumption is that because this force is not purely white they related it to a 'oriented' injection of energy in the system. But for me it is not clear why this external force shouldn't contribute to his heat definition Eq. (4) as it introduces randomness. My conclusion is that his definition is changed in the limit that his external force have vanishing correlation.

Thank you for your answer. 

There is a lot of literature on Brownian motion in a potential, and on Brownian motion with non-white noise. Maybe a good intro to the latter is Hanggi and Jung's "colored noise in dynamical systems", Adv. Chem. Phys. 89 (1995), 239-326.

 http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/160.pdf

In particular, see their equation (1.3), which combines both your questions and the further discussion after that. Section II would also be of interest to you. There are serious problems in this field which as far as I know have not been resolved to everyone's satisfaction. I think van Kampen also wrote several papers on the topic of non linear systems, (confining potentials other than the harmonic).

I now understand what you mean by bath variance and temperature, generally these things are known as fluctuation-dissipation theorems. Basically it relates the strength of the random force to viscosity and temperature. 

I'll have a look at your reference before commenting on other aspects you mention. At first sight I do not quite understand what they mean by their equations (4) to (7), so I need to delve a little into the literature they cite. I did some work on Brownian motion a very long time ago, but kept track of some of the literature. I am not sure if I understand why there would be a heat accumulation problem, the temperature of the Brownian particle should be the same as that of the bath. I did a paper on Brownian motion close to the liquid-gas critical point where the correlation length of the solvent fluctuations can be of the same order of magnitude as the size of the particle, but to be honest, we neglected temperature fluctuations in the force, and the noise was still white (although the fluctuation-dissipation theorem needed to be modified). At that time I also tried working out Brownian motion in a system with a temperature gradient (in fact close to the Rayleigh-Benard instability), but was never able to solve some fundamental problems in that case, primarily since it was a dissipative system to begin with. 

The notion of temperature is problematic if the system is not in thermodynamic equilibrium. At thermodynamic equilibrium different sources of noise are normally uncorrelated. Therefore your question may not have any proper answer. At least you need have to specify your system in more detail if there should be a meaningful answer. For instance your proposal might be valid if the correlation is very weak so that the system is close to equilibrium.

Dear Eduardo,

I have some serious problems with the ArXiv paper you gave as a reference, I think there is a fundamental inconsistency in their model. The model is reminiscent of models I worked on myself, often called the Brownian oscillator models, where we coupled reactive modes in a chemical reaction to polarization fluctuations in a dielectric medium, which behave exactly as the Ornstein-Uhlenbeck process in the current paper. In our work the coupling was different, however, and does not lead to the problem with the fluctuation-dissipation theorem I worked out in the attached file.

I have to think a little more about the things you mention in the previous contribution, but maybe this also helps. I did a lot of work on what you call "exterior influences". If you tell me more about what your goal is, I can maybe give more directed literature references. I have also a lot of unfinished work in this field, if the right problem comes up, this may be of interest (to me) as well. 

Dear Gert,

your calculation seems to state well my concern. This is very nice, thank you for taking your time. But, right now I'm taking a look at the experiment of Ciliberto. Note that the noise bath in the supplementary material have a colored spectrum.

This is an interesting experiment, but be aware that there are two heat baths at different temperatures in their case.  This means that there is a temperature gradient, it is not a system in equilibrium, and in such a system there is always dissipation and entropy production. I do not quite understand why they try to connect it to the Feynman ratchet, since in that case you try to get work from a single temperature reservoir. Not everybody believes that Feynman conclusively proved the impossibility of such a ratchet, and I have a number of papers (some of them also from Phys. Rev. Lett.) of people claiming to show the possibility of creating such a perpetual motion machine of the second kind, mostly using capacitors or magnetism though; or claims that biological systems, such as photosynthesis, can accomplish this. However, nobody as yet constructed one. I personally don't believe that it is possible, although I have no conclusive proof that you cannot get work from equilibrium fluctuations. There is a vast literature about this, google  "Maxwell's demon" to get started.

In the work I did (or we, as I always worked with others on this, notably P. Mazur and later J.T. Hynes), consistency was always a major issue. That is why the fluctuation dissipation theorems were always part of our investigations, something that is problematic in non-equilibrium situations, such as systems with a temperature gradient. It also puts restrictions on the coupling between bath and system of interest, which is why Pal's system of equations struck me immediately as problematic. There is likely some recent literature in the field of dissipative systems that I am not aware of.  

I'd be happy to discuss these things with you if you are interested. We can also do that outside the RG environment. I think I now understand your original question, since it will be an issue how to describe the fluctuations in population dynamics using some equivalent of a heat bath. The driving force in living systems is an entropy gradient after all. I'll have a look at your papers to see what you are working on exactly.

By all means. You can use my regular email address: [email protected]