Abstract
We study how an object that partitions in two its thermally agitated environment relaxes to its equilibrium fluctuations. Event-driven molecular dynamics indicate that a conventional diffusive-like bound-Brownian motion is reached after a long-lived sub-diffusive regime. The two-stage behavior is not eliminated increasing the number of agitated particles, suggesting that the origin of the effect is a topologically frustrated exploration of phase-space.
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Acknowledgments
We thank Angelo Vulpiani for useful discussions. Stefano Di Sabatino is now with LPT-IRSAMC, Université Paul Sabatier - Toulouse, France. The research leading to these results was supported by funding from the Italian Ministry of Research (MIUR) through the “Futuro in Ricerca” FIRB Grant PHOCOS-RBFR08E7VA. Partial funding was received through the SMARTCONFOCAL project of the Regione Lazio and through the PRIN Project No. 2012BFNWZ2.
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Appendix
Appendix
We note that our study of fluctuations in partitioning object position \(x\) gives us direct information on its velocity correlation function. In fact, starting from \(x(0)=0\), we have
that is,
from which we obtain
i.e., after setting \(\theta =t'-t\),
Since we are interesting in the behavior for large values of \(t\) and, as we will see, the velocity correlation time tends to vanish as \(t\) increases, Eq. (5) can be approximated as
If we reasonably assume a Gaussian decay \(\langle v(t)v(t+\theta )\rangle \sim \exp {(-C(t)\theta ^2)}\) (for \(\theta <0\)) and recall that, after a short initial transient, \(\langle v^2(t) \rangle \) reaches its equilibrium value \(kT/M\), Eq. (6) gives
We now consider the two situations characterized by exponential and power law behavior of \(\langle x^2(t) \rangle \), i.e., in particular, diathermal and adiabatic partitioning objects,
In the first case, from Eq. (8), we have
while in the second case, Eq. (9) leads to
Comparing Eqs. (10,11) with Eq. (7), we obtain, respectively,
For a Gaussian behavior \(\exp {(-C(t)\theta ^2)}\) the typical decay time is given by \(C(t)^{-1/2}\), so that the velocity self-correlation times \(\tau _{c,exp}\) and \(\tau _{c,pl}\) are
Interestingly, the characteristic velocity memory times tend to vanish as \(t\) increases either following an exponential or a power-law depending on whether \(\langle x^2 \rangle \) approaches its equilibrium value \(\langle x^2\rangle _{\text {EQ}}\) with an exponential or a power-law behavior.
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DelRe, E., Di Sabatino, S., Di Porto, P. et al. Anomalous Fluctuations in the Motion of Partitioning Objects. J Stat Phys 156, 291–300 (2014). https://doi.org/10.1007/s10955-014-1001-3
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DOI: https://doi.org/10.1007/s10955-014-1001-3