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In the context of molecular dynamics simulations of soft or hard spheres in the fluid phase (e.g., with Lennard-Jones interactions), it is known that the velocity autocorrelation function (VACF) decays as $ t^{-d/2} $ in dimension $ d $, which is typically interpreted as viscous dissipation of momentum. Additionally, it is observed that the spatially resolved VACF exhibits peaks propagating at the acoustic velocity, with these peaks reflecting when they reach the nearest neighbor. This behavior is often interpreted as a backscattering peak.

Question:

How do these observations relate to each other precisely? Specifically, if the VACF is the spatial integral of the spatially resolved VACF, how can we understand that the integral of several peaks (where for each position the peak time is slightly shifted due to sound propagation) results in an algebraic decay of $ t^{-d/2} $?

I understand that while individual peaks correspond to sound waves and backscattering, their integrated effect over space results in a smoothed-out decay. However, I interpreted the smoothed-out decay in terms of viscous dissipation, considering it as a random walk of momentum in space, which leads to diffusive behavior. On the other hand, acoustic waves represent convective transport of momentum rather than diffusive transport.

What am I missing?

Why does summing over acoustic peaks at different positions result in a "diffusive" and "dissipative" algebraic smoothed-out decay $ t^{-d/2} $? How do these acoustic waves and their reflections translate into a decay that is typically associated with viscous dissipation?

While acoustic waves represent a coherent and directional transport of momentum, they still contribute to the overall momentum distribution in the system. Over time, the collective effect of many such waves, each propagating and reflecting off particles, leads to a more randomized distribution of momentum. This integration sums contributions from all spatial locations, each with its own time-shifted peak. As time progresses, these peaks spread out and overlap, leading to a smoother and more diffusive-like behavior in the aggregate. At short times, the VACF is influenced by the initial ballistic motion and the coherent propagation of sound waves. However, at long times, the multiple reflections, scatterings, and interactions between particles lead to a diffusive spread of momentum. The long-time decay of the VACF is governed by hydrodynamic modes, which inherently involve dissipative processes. These processes dominate the behavior at large timescales, resulting in the observed algebraic decay.

Additional Context:

The VACF $ C(t) $ is given by: $ C(t) = \langle \mathbf{v}(0) \cdot \mathbf{v}(t) \rangle $

The spatially resolved VACF $ C(\mathbf{r}, t) $ is the correlation function considering the velocity at a specific spatial position $ \mathbf{r} $: $ C(\mathbf{r}, t) = \langle \mathbf{v}(0) \cdot \mathbf{v}(\mathbf{r}, t) \rangle $

The relation between the two is: $ C(t) = \int C(\mathbf{r}, t) \, d\mathbf{r} $

Given this relationship, how does the integration of spatially and temporally shifted acoustic peaks lead to the observed viscous-like decay in the VACF?

Hints to literature discussing the relation between spatially resolved VACF and VACF would be appreciated.

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  • $\begingroup$ I don't know the slightest thing about these kind of things. So it might be pure gibberish. But to me, what you call backscatering is a linear effect. You see it with linearized hydrodynamics, you see it in the landau-placzek dynamic structure factor (hence also in the velocity velocity dynamic correlation I assume). And they are linear effect in the sense that its just a density wave at some give wavenumber. Now the long time tails, in a hydrodynamic description, you only see them through a mode coupling theory. That is, coupling between two different wavenumber. $\endgroup$
    – Syrocco
    Commented Jul 7 at 17:43
  • $\begingroup$ You do not see it in the linear regime where the velocity autocorrelation decays exponentially (eventually modulated by the sound waves). So, to me these things are really two different phenomena. I agree that they both are related to momentum transfer but I don't know if there is much to know about how the long time tails arise from the spatio temporal dependence, integrated. Since, to me, you already see sound without taking into account the effects that would lead to a long time tails. But again, I NEVER looked really at these things. You might be interested by the mode coupling review.. $\endgroup$
    – Syrocco
    Commented Jul 7 at 17:45
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    $\begingroup$ By Pommeau and Resibois: TIME DEPENDENT CORRELATION FUNCTIONS AND MODE—MODE COUPLING THEORIES $\endgroup$
    – Syrocco
    Commented Jul 7 at 17:48

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