Monthly Archives: July 2017

Response and Dissipation: Part 1 of the Fluctuation-Dissipation Theorem

I’ve referred to the fluctuation-dissipation theorem many times on this blog (see here and here for instance), but I feel like it has been somewhat of an injustice that I have yet to commit a post to this topic. A specialized form of the theorem was first formulated by Einstein in a paper about Brownian motion in 1905. It was then extended to electrical circuits by Nyquist and then generalized by several authors including Callen and Welten (pdf!) and R. Kubo (pdf!). The Callen and Welton paper is a particularly superlative paper not just for its content but also for its lucid scientific writing. The fluctuation-dissipation theorem relates the fluctuations of a system (an equilibrium property) to the energy dissipated by a perturbing external source (a manifestly non-equilibrium property).

In this post, which is the first part of two, I’ll deal mostly with the non-equilibrium part. In particular, I’ll show that the response function of a system is related to the energy dissipation using the harmonic oscillator as an example. I hope that this post will provide a justification as to why it is the imaginary part of a response function that quantifies energy dissipated. I will also avoid the use of Green’s functions in these posts, which for some reason often tend to get thrown in when teaching linear response theory, but are absolutely unnecessary to understand the basic concepts.

Consider first a damped driven harmonic oscillator with the following equation (for consistency, I’ll use the conventions from my previous post about the phase change after a resonance):

\underbrace{\ddot{x}}_{inertial} + \overbrace{b\dot{x}}^{damping} + \underbrace{\omega_0^2 x}_{restoring} = \overbrace{F(t)}^{driving}

One way to solve this equation is to assume that the displacement, x(t), responds linearly to the applied force, F(t) in the following way:

x(t) = \int_{-\infty}^{\infty} \chi(t-t')F(t') dt'

Just in case this equation doesn’t make sense to you, you may want to reference this post about linear response.  In the Fourier domain, this equation can be written as:

\hat{x}{}(\omega) = \hat{\chi}(\omega) \hat{F}(\omega)

and one can solve this equation (as done in a previous post) to give:

\hat{\chi}(\omega) = (-\omega^2 + i\omega b + \omega_0^2 )^{-1}

It is useful to think about the response function, \chi, as how the harmonic oscillator responds to an external source. This can best be seen by writing the following suggestive relation:

\chi(t-t') = \delta x(t)/\delta F(t')

Response functions tend to measure how systems evolve after being perturbed by a point-source (i.e. a delta-function source) and therefore quantify how a system relaxes back to equilibrium after being thrown slightly off balance.

Now, look at what happens when we examine the energy dissipated by the damped harmonic oscillator. In this system the energy dissipated can be expressed as the time integral of the force multiplied by the velocity and we can write this in the Fourier domain as so:

\Delta E \sim \int \dot{x}F(t) dt =  \int d\omega d\omega'dt (-i\omega) \hat{\chi}(\omega) \hat{F}(\omega)\hat{F}(\omega') e^{i(\omega+\omega')t}

One can write this more simply as:

\Delta E \sim \int d\omega (-i\omega) \hat{\chi}(\omega) |\hat{F}(\omega)|^2

Noticing that the energy dissipated has to be a real function, and that |\hat{F}(\omega)|^2 is also a real function, it turns out that only the imaginary part of the response function can contribute to the dissipated energy so that we can write:

\Delta E \sim  \int d \omega \omega\hat{\chi}''(\omega)|\hat{F}(\omega)|^2

Although I try to avoid heavy mathematics on this blog, I hope that this derivation was not too difficult to follow. It turns out that only the imaginary part of the response function is related to energy dissipation. 

Intuitively, one can see that the imaginary part of the response has to be related to dissipation, because it is the part of the response function that possesses a \pi/2 phase lag. The real part, on the other hand, is in phase with the driving force and does not possess a phase lag (i.e. \chi = \chi' +i \chi'' = \chi' +e^{i\pi/2}\chi''). One can see from the plot from below that damping (i.e. dissipation) is quantified by a \pi/2 phase lag.

ArgandPlaneResonance

Damping is usually associated with a 90 degree phase lag

Next up, I will show how the imaginary part of the response function is related to equilibrium fluctuations!

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Research Topics and the LAQ Method

As a scientific researcher, the toughest part of the job is to come up with good scientific questions. A large part of my time is spent looking for such questions and every once in a while, I happen upon a topic that is worth spending the time to investigate further. The most common method of generating such questions is to come up with a lot of them and then sift through the list to find some that are worth pursuing.

One of the main criteria I use for selecting such questions/problems is what I refer to as the “largest answerable question” or LAQ method. Because the lifetime of a researcher is limited by the human lifespan, it is important to try to answer the largest answerable questions that fall within the window of your ability. Hence, this selection process is actually tied in with one’s self-confidence and actually takes a fair amount of introspection. I imagine the LAQ method looking a little bit like this:

Image result for broad to specific triangle

One starts by asking some really general questions about some scientific topic which eventually proceeds to a more specific, answerable, concrete question. If the question is answerable, it usually will have ramifications that will be broadly felt by many in the community.

I imagine that most readers of this blog will have no trouble coming up with examples of success stories where scientists employed the LAQ method. Just about every famous scientist you can think of has probably, at least to some extent, used this method fruitfully. However, there are counterexamples as well, where important questions are asked by one scientist, but is answered by others.

I am almost done reading Erwin Schrodinger’s book What is Life?, which was written in 1943. In it, Schrodinger asks deep questions about genetics and attempts to put physical constraints on information-carrying molecules (DNA was not known at the time to be the site of genetic information). It is an entertaining read in two regards. Firstly, Schrodinger, at the time of writing, introduces to physicists some of the most pertinent and probing questions in genetics. The book was, after all, one that was read by both Watson and Crick before they set about discovering the structure of DNA. Secondly, and more interestingly, Schrodinger gets almost everything he tries to answer wrong! For instance, he suggests that quantum mechanics may play a role in causing a mutation of certain genes. This is not to say that his reasoning was not sound, but at the time of writing, there were just not enough experimental constraints on some of the assumptions he made.

Nonetheless, I applaud Schrodinger for writing the book and exposing his ignorance. Even though he was not able to answer many of the questions himself, he was an inspiration to many others who eventually were able to shed light on many of the questions posed in the book. Here is an example where the LAQ method fails, but still pushes science forward in a tangible way.

What are your strategies with coming up with good research questions? I have to admit that while the LAQ method is useful, I sometimes pursue problems purely because I find them stunning and no other reason is needed!