Equipartition of ideal gas energy in Fourier space

Question

Consider an ideal gas of $N$ particles at temperature $T$ enclosed in a box of volume $V$. Equipartition of energy between the velocities $v_{i}$ of the particles implies that on average the magnitude of velocity fluctuations of each particle $i$ is given by $<v^{2}_{i}> = \frac{k_{B}T}{m}$. This follows from the Maxwell-Boltzmann velocity distribution.

I wish to derive the equipartition of energy between the normal modes of the velocity field $<u^{2}_{k}> = \frac{k_{B}T}{m}$. For this I define the empirical velocity field as $u(r, t) = \frac{1}{N}\sum^{N}_{i=1} v_{i}\delta(r-r_{i})$. Due to the finite volume $V$, the field $u(r,t)$ can be decomposed into a discrete Fourier sum as $u(r, t) = \frac{1}{N}\sum_{k} u_{k} \ e^{ikr}$ over wave vectors $k=\frac{2\pi n}{L}$ with $n \in Z^{3}$. To derive $<u^{2}_{k}>$ I transform the Hamiltonian from the particle to the field representation as $H = \sum_{i} \frac{mv^{2}_{i}}{2} = \int d^{3}r \ u^{2}(r) = \sum_{k} u^{2}_{k} $ with the last step coming from the orthogonality of the normal modes. Thus I recognize the "equivalence" between $<u^{2}_{k}> $ and $<v^{2}_{i}>$.

Now coming back to $<u^{2}_{k}> = \frac{k_{B}T}{m} $, what does this imply for the spatial profile of $u(r)$ ? I had this idea that the slowest modes would be dominating the spatial profile and the higher-order contributions are less important. Is this not the case ? Should there not be some decay of $u_k$ as function of $k$ ? If such expectations can not be concluded from the equipartition theorem, how could we derive these expectations for this simple model ?

Edit

After spending more time in literature, I found several authors referring to the volumes of Landau Lifschitz on hydrodynamics and statistical mechanics, more precisely a chapter in volume 9 on hydrodynamic fluctuations. As far I understand, in this chapter the hydrodynamic equations of the velocity field were used to derive the spatial and temporal correlations of the velocity field. I guess I would be looking at this in more detail.

But I am still puzzled by the meaning of equipartition in momentum space for the simple system presented above, i.e a fluid at thermal equilibrium inside a container. What is the consequence of equipartition for this case ?

Answer 1

First: these two answers by Ron Maimon are extremely good 1 and 2 and might interest you. They don't discuss about the spatial characterization of equipartition but they do discuss about equipartion in k space. (Any answer by Ron Maimon is worth pondering!)

Equilibrium is "maximum randomness" (that is why it is so simple compared to non-eq phy stat, where you have more structure to your randomness, more conservation law, more constraints and thus less universality).

In the case of an ideal fluid, maximally random could correspond to the equipartition that indeed stating that EVERY mode has the same kinetic energy. This does not mean that every scale has the same energy because the smaller the scale the larger is the number of modes associated to this scale (this number grows as the d-disk in d dimension). In any case, since the energy is evenly distributed there is nothing special happening on the spatial part. Moreover, the interactions are invariant by translation (you do not have a external field for example) so you cant have a dependence of the velocity on $r$ after taking your average. You might be talking about correlation over some distance for the velocity field. This an other question unrelated to the equipartition. I come back on that later.

Due to the finite nature of the particle you know that there will be a finite small lenght scale below which your field approximation will break. This will give you a natural ultraviolet cut-off (a limit). Ultraviolet here is a term taken from the high energy physics corresponding: to high energy states or equivalently small wavelenght states or equivalently large $k$ states. This is nice because, in your case you have a finite amount of energy to distribute (by equipartition) in the modes $k_L<k<k_{cut-off}$ which are of course in finite number ($k_L$ is the min $k$ associated to the finite size of the box). When the ulteaviolet cut-off goes to the infinity however, you have an finite energy to distribute in an infinite number of mode: this is the ultraviolet catastrophe. Everything goes wrong! This is why, any field theory without a cut-off will demonstrate the ultraviolet catastrophe.

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Now my second comment which does not really answer your question but might give you nice insights to what is equilibriumby placing ourselves in a non equilibrium situation.

This was the equilibrium picture but if you take a fluid for example and use a stick to move it, at high reynold you will see eddies forming at the surface beakinginto smaller eddies so on and so forth until a state over which no movement can be seen. In this case, what you will see is the energy is located at small $k$ at the beginning corresponding to the initial disturbance, then the energy will move toward the smaller lenght scale until the dissipation kicks at very small lenght scale and dissipate the energy. If you move constantly your stick, you will constantly inject energy at large lenght which will travel to smaller lenght to get dissipated. A stationary state, very different from the equipartition will be reached where you have a peak at large wavelenght of energy a power law decrease wirh respect to k following the so called kolmogorov law until the lenght of viscous dissipation is reached. This picture is called the Richardson cascade. In 2D you have an inverse cascade explained by krainchan. If you wish to look at it.

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You mentioned fluctuating hydrodynamics. The picture above was simple navier stokes. If on top of the NS equations you add a fluctuating random noise which respects the fluctuation dissipation theorem you obtain the fluctuation Navier stokes equations (first derived by Landau and Lifschitz). They, basically take into account the microscopic part of the fluid and thus can take into account the equipartition (at the molecular level). This allows you to recover the equipartition because you always have a non zero velocity field (even with dissipation since you have a noise injecting energy at very small scale to mimic the collisions between water molecules instead of simply draining their collective energy with viscuous dissipation). In this case, you would also have the cascade but you would also see the equipartition (a $k^2$ increase of the k-energy ) at small lenght scale if you moved constantly your stick. If you chose to stop moving your stick, you reach the equipartition at every scale and the system finally reach equilibrium.

You are right that, with this model of fluctuation you can obtain information such as the velocity correlation or structure factor: $\langle u_ku_q\rangle $ which tells you how the velocity or density decorrelate over lenght. This is easily computed in the linear regime and in the steady state. I like this source for a granular gas personaly (they also have a global noise, on top of the "molecular" noise).